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Example of calculating the molarity of ions in an aqueous solution

Dissolving electrolytes in water separates them into oppositely charged ions, which allow the resulting solution to conduct electricity. Some examples of common electrolytes are different types of salts, such as sodium chloride and potassium nitrate, acids such as sulfuric and nitric acids, and some bases such as sodium hydroxide, among others.

In the following sections it is explained in detail by means of examples, how to calculate the molar concentration of ions in solution for different types of electrolytes, including both strong and weak electrolytes.

Why is it important to be able to calculate the molar concentration of ions in solution?

For various reasons, it is necessary to determine or calculate the molar concentration of these ions when preparing a solution. On the one hand, the total concentration of ions allows us to have an idea of ​​their ability to conduct electricity. On the other hand, the total concentration of ions also influences the ionic strength of a solution, which affects the chemical equilibria of different real systems such as weak acids and weak bases.

Finally, the concentration of different ions is very important in the field of biology and biochemistry. This is because the concentrations of ions such as sodium and potassium, as well as chloride and other anions, are important factors in determining membrane potential, the tendency for an ion to pass spontaneously across one side of the membrane. to the other, and a multitude of other transport phenomena of great importance for the proper functioning of the cell.

Calculation of ion concentration in strong electrolyte solutions

A strong electrolyte is an ionic substance that, when dissolved in water, becomes completely ionized. This means that the dissociation reaction is irreversible, and all the formula units of the compound separate to give rise to the maximum possible number of ions in solution.

For this reason, in the cases of strong electrolytes, the calculation of the ion concentration consists of a simple stoichiometric calculation, depending on the balanced or balanced chemical reaction. Take the following case as an example.

Example of calculating the concentration of ions for a strong electrolyte.


Calculate the molar concentration of phosphate ions and the molar concentration of potassium ions in a solution prepared by dissolving 10.00 grams of potassium phosphate in 500.0 mL of solution.


These types of problems can be solved by following a series of ordered steps. Some steps will be unnecessary depending on the data provided by the statement, but generally speaking, you can always use:

Step #1: Extract the data and unknowns, determine the relevant molecular weights, and perform the necessary unit transformations.

This is almost always the first step in solving any type of problem. In this case, the statement indicates that the solution is prepared by dissolving 10.00 g of potassium phosphate (K 3 PO 4 ) , which corresponds to the mass of the solute.

Since the molarity of the ions is requested, we will need at some point the molar mass of the salt which is:

Example of the calculation of the molar concentration of ions

The statement also indicates that 500.00 mL of solution will be prepared, which corresponds to the volume of the solution. Since they ask for molarity, this volume must be transformed to liters.

Example of the calculation of the molar concentration of ions

Step #2: Calculate the molar concentration of the electrolyte. This is also often referred to as the analytical concentration.

In general, it is easier to calculate the concentration of ions in a salt from the molar concentration of the salt itself. We do this using the molarity formula and the data presented above.

Example of the calculation of the molar concentration of ions

Where C K3PO4 refers to the molar concentration of the salt.

AUTHOR’S NOTE: In general, it is customary to use C to represent any analytical concentration in any concentration unit. By analytical concentration we mean concentrations calculated from the measured amounts of solutes, solvents, and solutions. This is to distinguish them from the concentrations of the different species after a chemical reaction or when establishing chemical equilibria.

Step #3: Write the balanced dissociation equation

In this case, it is a strong electrolyte, so the reaction is irreversible (an equilibrium is not established):

Example of the calculation of the molar concentration of ions

Step #4: Use the stoichiometric relationships obtained from the balanced equation to determine the concentration of the ions of interest.

Once the equation is written, all that is needed is to use stoichiometry to determine the concentrations of the ions. We can do the stoichiometric calculations directly using the molar concentration instead of the moles, since all the calculations we are carrying out refer to a single solution in which the volume is not changing, so the concentration is directly proportional to the moles of each species.

This means that the concentrations of the two ions are determined by:

Example of the calculation of the molar concentration of ions Example of the calculation of the molar concentration of ions

Calculation of ion concentration in weak electrolyte solutions

In the case of weak electrolytes, the fundamental difference is that the dissociation reaction is reversible, and only a small fraction of the solute molecules dissociate to form free ions. For this reason, to calculate the ion concentration in these cases, the chemical equilibrium must be solved.

Example of calculating the ion concentration for a weak electrolyte.


Calculate the molar concentration of acetate ions and hydronium ions in a solution prepared by dissolving 10.00 grams of acetic acid in 500.0 mL of solution, knowing that the acid has an acidity constant of 1.75 .10 -5 .


Since this case deals with a solution of acetic acid, which is a weak electrolyte, we must proceed to solve the ionic equilibrium that is established by dissolving this solute in water. The first steps are the same as above, but from step 4 onwards the procedure changes. Here’s how:

Step #1: Extract the data and unknowns, determine the relevant molecular weights, and perform the necessary unit transformations.

The mass of the solute is again 10.00g and the volume of the solution is also 500.0 mL, which is equivalent to 0.5000 L as we saw earlier. The molecular weight of acetic acid (CH 3 COOH) is 60.052 g/mol.

Step #2: Calculate the molar concentration of the electrolyte.

Using the data presented above, the initial or analytical molar concentration of acetic acid is:

Example of the calculation of the molar concentration of ions

Step #3: Write the balanced dissociation equation

Unlike the previous case, because it is a weak electrolyte, the reaction is reversible, so an equilibrium is established:

Example of the calculation of the molar concentration of ions Example of the calculation of the molar concentration of ions

Step #4: Solve the chemical equilibrium to determine the concentrations of all species.

This part of the process is completely different from the previous ones, since the final concentrations of the ions cannot be determined directly from the initial concentration of the acid by stoichiometry, since these concentrations must also satisfy the equilibrium condition given by the law of mass action.

In this particular case, the equilibrium condition is determined by the expression of the equilibrium constant:

Example of the calculation of the molar concentration of ions

The following ICE table relates the initial concentrations to the final ones. In this case, since we do not know in advance how much acid actually dissociates, then the change in its concentration must be expressed as an unknown (X). Then, by stoichiometry, it is established that X must also be formed from acetate ions and from protons:

Concentrations CH3COOH _ _ H + CH 3 COO – initials _ 0.3330M 0 0 change _ –X +X +X and balance 0.3330–X X X

To find the unknown, X, it is enough to use the equation of the acidity constant:

Example of the calculation of the molar concentration of ions

This equation can be rewritten as:

Example of the calculation of the molar concentration of ions

which is a second degree equation whose solution, after substituting the value of the acidity constant, is:

Example of the calculation of the molar concentration of ions

As we can see in the ICE table, the concentration of both ions is, in this case, equal to X, so we can write

Example of the calculation of the molar concentration of ions

The concentration of both ions is equal to 2.41.10 -3 molar.


Bolívar, G. (2020, July 9). Weak electrolytes: concept, characteristics, examples. Recovered from https://www.lifeder.com/electrolitos-debiles/

Brown, T. (2021). Chemistry: The Central Science (11th ed.). London, England: Pearson Education.

Chang, R., Manzo, Á. R., Lopez, PS, & Herranz, ZR (2020). Chemistry (10th ed.). New York City, NY: MCGRAW-HILL.

Garcia, J. (2002). Concentrations in clinical solutions: theory and interconversions. Rev. costarric. science. med. , 23 , 81–88. Retrieved from https://www.scielo.sa.cr/scielo.php?script=sci_arttext&pid=S0253-29482002000100008

Sarica, S. (nd). Ion concentration with examples. Retrieved from https://www.chemistrytutorials.org/ct/es/44-Concentraci%C3%B3n_de_iones_con_ejemplos

What you need to know about consecutive numbers

Consecutive numbers are numbers that, when counting, follow each other and are in order. For example: 1, 2, 3, 4…, or 59, 58, 57, 56… We can also divide them between consecutive even numbers and consecutive odd numbers.

What are consecutive numbers

As mentioned before, consecutive numbers are numbers that follow each other in order without jumping. In addition to consecutive numbers that vary by one unit, consecutive numbers can also be even or odd.

How to get a consecutive number

To obtain a consecutive number, one unit must be added to the previous number. That is, using this equation:

number: n

Consecutive number = n + 1.

“n” can be any integer. For example: To find out what the consecutive number of 185 is, we add 1 to it and we get 186.

Consecutive even numbers

To obtain a consecutive even number, two units must be added to the previous even number. This can be expressed with the following equation:

Even number: 2 . no

Consecutive even number = 2 · n + 2

Here too “n” can be any integer. For example, some consecutive even numbers are: 8 and 10 (if n=4), or 46 and 48 (if n=23).

Consecutive odd numbers

A consecutive odd number can be obtained by adding two units to the previous odd number. You can use the equation:

Odd number: 2 n – 1

Consecutive odd number = (2 · n − 1) + 2

In this case “n” is also any integer. Some examples of consecutive odd numbers are 1 and 3 (for n=1), or 77 and 79 (for n=39).

consecutive multiples

Math problems are often based on properties of consecutive odd or even numbers. Or also in consecutive numbers that are increasing in multiples of three, such as 3, 6, 9, 12. In this example, the numbers 3, 6, 9 are not consecutive numbers, but consecutive multiples of 3. In other cases, the problems are about consecutive even numbers (2, 4, 6, 8) or consecutive odd numbers (7, 9, 11). Here you take an even number and then the next even number, or else an odd number and the next odd number.

If “x” is one of the numbers, the algebraic representation of the consecutive numbers would be: x + 1, x + 2, x + 3…

If the problem to solve is about consecutive even numbers, it is important that the first number you choose is even. To do this, the first number must be 2.x instead of x. But keep in mind that the next consecutive even number is not 2x + 1 (because this would give an odd number), but 2x + 2, 2x + 4, 2x + 6, and so on.

Similarly, consecutive odd numbers would be expressed: 2x + 1, 2x + 3, 2x + 5…

Math problems with consecutive numbers

Here are two math problems to practice consecutive numbers:

Example 1:

Suppose the sum of two consecutive numbers is 15. What would those numbers be? 

To solve this problem we have to consider that given any number, let’s call it “x”, its consecutive number will be x+1. Therefore, the sum between x and x+1 must be equal to 23. We put this in an equation and solve:

Equation :

x + (x + 1) = 23

2x + 1 = 23

2x = 22


So, your numbers are 11 (value of x) and 12 (value of x+1).

Example 2:

Now imagine that in the previous example we had chosen the consecutive numbers differently: for example, that the first number was x -3 and the second number was x -4 (note that these numbers are still consecutive numbers: one comes directly after the first). other). Do you get the same consecutive numbers?

To solve this problem we follow the same reasoning as in the previous case: the sum of the two consecutive numbers must be equal to 23.

Equation :

(x – 3) + (x – 4) = 23

2x – 7 = 23

2x = 30

x = 15

Here you can see that x is equal to 15, while in the previous problem, x was equal to 11. However, the value of x is only used to calculate the consecutive numbers, it is not necessarily one of the consecutive numbers. To determine the consecutive numbers we substitute the value of x in the expression we use to define each number: x – 3 and x – 4.

  • 15 – 3 = 12
  • 15 – 4 = 11

As you can see, it has the same answer as in the previous problem.

It may be easier if you choose different variables for your consecutive numbers. For example, if you have to solve a problem involving the product of five consecutive numbers, you can calculate it using either of the following two methods:

x (x + 1) (x + 2) (x + 3) (x + 4)
(x – 2) (x – 1) (x) (x + 1) (x + 2)

As you can see, the second equation is easier to calculate since it can take advantage of the properties of the difference of squares.

Exercises to practice consecutive numbers

Here are more consecutive number exercises. Try to solve them with the methods taught above.

  • What are the five consecutive numbers whose total sum is zero?
    • Solution= -2, -1, 0, 1, 2
  • What are the two consecutive odd numbers that have a product of 143.
    • Solution= 11, 13
  • There are four consecutive even numbers that add up to 148. What are those numbers?
    • Solution= 34, 36, 38, 40
  • What are the three consecutive multiples of six that add up to 126?
    • Solution= 36, 42, 48
  • If the sum of four consecutive integers is 54, what are those numbers?
    • Solution= 12, 13, 14, 15
  • The sum of five consecutive even integers is 110. What are those numbers?
    • Solution= 18, 20, 22, 24, 26
  • What are the two consecutive numbers whose product is 600. What are those numbers?
    • Solution= 24, 25
  • If you do a subtraction between the product of two consecutive numbers and the sum of the same two numbers, the result is 19. What are those numbers?
    • Solution= -4 and -3 or 5 and 6


  • López Mateos, M. Basic Mathematics. (2017). Spain. CreateSpace.
  • dk. The math book. (2020). Spain. dk.

Was Sie über fortlaufende Nummern wissen müssen

Fortlaufende Zahlen sind Zahlen , die beim Zählen aufeinander folgen und in einer Reihenfolge sind. Zum Beispiel: 1, 2, 3, 4 …, oder 59, 58, 57, 56 … Wir können sie auch in aufeinanderfolgende gerade Zahlen und aufeinanderfolgende ungerade Zahlen aufteilen.

Was sind fortlaufende nummern

Wie bereits erwähnt, sind fortlaufende Zahlen Zahlen, die ohne Sprung aufeinander folgen. Neben fortlaufenden Zahlen, die um eine Einheit variieren, können fortlaufende Zahlen auch gerade oder ungerade sein.

So erhalten Sie eine fortlaufende Nummer

Um eine fortlaufende Nummer zu erhalten, muss eine Einheit zur vorherigen Nummer hinzugefügt werden. Das heißt, mit dieser Gleichung:

Zahl: n

Fortlaufende Nummer = n + 1.

“n” kann eine beliebige Ganzzahl sein. Beispiel: Um herauszufinden, was die fortlaufende Zahl 185 ist, addieren wir 1 dazu und erhalten 186.

Aufeinanderfolgende gerade Zahlen

Um eine fortlaufende gerade Zahl zu erhalten, müssen zwei Einheiten zur vorherigen geraden Zahl addiert werden. Dies kann mit der folgenden Gleichung ausgedrückt werden:

Gerade Zahl: 2 . nein

Fortlaufende gerade Zahl = 2 · n + 2

Auch hier kann “n” eine beliebige ganze Zahl sein. Einige aufeinanderfolgende gerade Zahlen sind zum Beispiel: 8 und 10 (wenn n=4) oder 46 und 48 (wenn n=23).

Aufeinanderfolgende ungerade Zahlen

Eine fortlaufende ungerade Zahl erhält man, indem man zwei Einheiten zur vorherigen ungeraden Zahl addiert. Sie können die Gleichung verwenden:

Ungerade Zahl: 2 n – 1

Fortlaufende ungerade Zahl = (2 · n − 1) + 2

In diesem Fall ist “n” auch eine beliebige ganze Zahl. Einige Beispiele für aufeinanderfolgende ungerade Zahlen sind 1 und 3 (für n=1) oder 77 und 79 (für n=39).

aufeinanderfolgende Vielfache

Mathematische Probleme basieren oft auf Eigenschaften aufeinander folgender ungerader oder gerader Zahlen. Oder auch in aufeinanderfolgenden Zahlen, die in Vielfachen von drei ansteigen, wie 3, 6, 9, 12. In diesem Beispiel sind die Zahlen 3, 6, 9 keine aufeinanderfolgenden Zahlen, sondern aufeinanderfolgende Vielfache von 3. In anderen Fällen ist die Bei den Aufgaben geht es um aufeinanderfolgende gerade Zahlen (2, 4, 6, 8) oder aufeinanderfolgende ungerade Zahlen (7, 9, 11). Hier nehmen Sie eine gerade Zahl und dann die nächste gerade Zahl oder eine ungerade Zahl und die nächste ungerade Zahl.

Wenn “x” eine der Zahlen ist, wäre die algebraische Darstellung der fortlaufenden Zahlen: x + 1, x + 2, x + 3…

Wenn es bei der zu lösenden Aufgabe um aufeinanderfolgende gerade Zahlen geht, ist es wichtig, dass die erste gewählte Zahl gerade ist. Dazu muss die erste Zahl 2.x statt x sein. Aber bedenke, dass die nächste aufeinanderfolgende gerade Zahl nicht 2x + 1 ist (weil dies eine ungerade Zahl ergeben würde), sondern 2x + 2, 2x + 4, 2x + 6 und so weiter.

Ebenso würden aufeinanderfolgende ungerade Zahlen ausgedrückt werden: 2x + 1, 2x + 3, 2x + 5 …

Mathematische Aufgaben mit fortlaufenden Zahlen

Hier sind zwei mathematische Aufgaben, um aufeinanderfolgende Zahlen zu üben:

Beispiel 1:

Angenommen, die Summe zweier aufeinanderfolgender Zahlen ist 15. Was wären diese Zahlen? 

Um dieses Problem zu lösen, müssen wir bedenken, dass für jede Zahl, nennen wir sie „x“, die fortlaufende Nummer x+1 ist. Daher muss die Summe zwischen x und x+1 gleich 23 sein. Wir setzen dies in eine Gleichung und lösen:

Gleichung :

x + (x + 1) = 23

2x + 1 = 23

2x = 22


Ihre Zahlen sind also 11 (Wert von x) und 12 (Wert von x+1).

Beispiel 2:

Stellen Sie sich nun vor, dass wir im vorigen Beispiel die fortlaufenden Zahlen anders gewählt hätten: Zum Beispiel, dass die erste Zahl x -3 und die zweite Zahl x -4 war (beachten Sie, dass diese Zahlen immer noch fortlaufende Zahlen sind: Eine kommt direkt nach der ersten ) andere). Bekommst du die gleichen fortlaufenden Nummern?

Um dieses Problem zu lösen, folgen wir der gleichen Argumentation wie im vorherigen Fall: Die Summe der beiden aufeinanderfolgenden Zahlen muss gleich 23 sein.

Gleichung :

(x – 3) + (x – 4) = 23

2x – 7 = 23

2x = 30

x = 15

Hier sehen Sie, dass x gleich 15 ist, während in der vorherigen Aufgabe x gleich 11 war. Der Wert von x wird jedoch nur zur Berechnung der fortlaufenden Nummern verwendet, es ist nicht unbedingt eine der fortlaufenden Nummern. Um die fortlaufenden Zahlen zu bestimmen, ersetzen wir den Wert von x in dem Ausdruck, den wir verwenden, um jede Zahl zu definieren: x – 3 und x – 4.

  • 15 – 3 = 12
  • 15 – 4 = 11

Wie Sie sehen können, hat es die gleiche Antwort wie in der vorherigen Aufgabe.

Es kann einfacher sein, wenn Sie andere Variablen für Ihre fortlaufenden Nummern wählen. Wenn Sie beispielsweise ein Problem lösen müssen, bei dem es um das Produkt aus fünf aufeinanderfolgenden Zahlen geht, können Sie es mit einer der beiden folgenden Methoden berechnen:

x (x + 1) (x + 2) (x + 3) (x + 4)
(x – 2) (x – 1) (x) (x + 1) (x + 2)

Wie Sie sehen können, ist die zweite Gleichung einfacher zu berechnen, da sie die Eigenschaften der Quadratdifferenz ausnutzen kann.

Übungen zum Üben fortlaufender Zahlen

Hier sind weitere fortlaufende Nummernübungen. Versuchen Sie, sie mit den oben gelehrten Methoden zu lösen.

  • Wie lauten die fünf aufeinanderfolgenden Zahlen, deren Gesamtsumme Null ist?
    • Lösung= -2, -1, 0, 1, 2
  • Welche zwei aufeinanderfolgenden ungeraden Zahlen haben das Produkt 143?
    • Lösung = 11, 13
  • Es gibt vier aufeinanderfolgende gerade Zahlen, die zusammen 148 ergeben. Was sind das für Zahlen?
    • Lösung = 34, 36, 38, 40
  • Wie lauten die drei aufeinanderfolgenden Vielfachen von sechs, die zusammen 126 ergeben?
    • Lösung = 36, 42, 48
  • Wenn die Summe von vier aufeinanderfolgenden ganzen Zahlen 54 ist, was sind das für Zahlen?
    • Lösung = 12, 13, 14, 15
  • Die Summe von fünf aufeinanderfolgenden geraden ganzen Zahlen ist 110. Was sind das für Zahlen?
    • Lösung = 18, 20, 22, 24, 26
  • Wie lauten die beiden aufeinanderfolgenden Zahlen, deren Produkt 600 ist? Was sind das für Zahlen?
    • Lösung = 24, 25
  • Wenn Sie eine Subtraktion zwischen dem Produkt zweier aufeinanderfolgender Zahlen und der Summe derselben beiden Zahlen durchführen, ist das Ergebnis 19. Was sind das für Zahlen?
    • Lösung= -4 und -3 oder 5 und 6


  • López Mateos, M. Grundlegende Mathematik. (2017). Spanien. CreateSpace.
  • dk. Das Mathebuch. (2020). Spanien. dk.

What is the mass number? Definition and examples

The mass number, also called the mass number, is the sum of the protons and neutrons contained in an atomic nucleus . It is usually represented by the letter “A” and in the periodic table it is indicated by a number that is generally above the name of the element.

What is the mass number

The mass number or mass number indicates the total number of particles that are in the atomic nucleus. Therefore, it can be defined as the sum of the protons and neutrons of an atom. It is indicated as a subscript to the left of the element. For example, the mass number of sodium is represented like this: 23 Na.

Electrons are not counted in the mass number because they have a much smaller mass than protons and neutrons, and therefore have almost no influence on it.

Differences between mass number and atomic number

Although these two concepts can sometimes be confused, there are some differences between the atomic number and the mass number:

  • The atomic number is the total number of protons in an atom , while the mass number also includes neutrons.
  • To identify both numbers, the letters that come from German are used: “Z” for the atomic number, from the German Zahl , and means “number”. Instead, for the mass number the letter “A” is used, which comes from the German term Atomgewicht and means “atomic weight”. However, it is important to note that the mass or mass number is not the same as the atomic mass. Atomic mass is measured in amu , that is, atomic mass units.
  • The mass number is usually greater than the atomic number, usually twice.
  • The mass number varies in each isotope. Therefore, the mass number of the most stable isotope is used as a reference.

How to Calculate Mass Number

To easily calculate the mass number of an element you can use the following equation:

A (mass number) = Z (atomic number) + N (number of neutrons)

In the same way, to know the number of neutrons in an atom, the following operation can be carried out: A – Z = N.

Both the mass number and the atomic number can be found on the periodic table of the elements.

Examples of mass numbers

In order to differentiate the mass number and the atomic number, in addition to recognizing the number of neutrons in an element, it is important to take into account some examples of mass numbers. For example, chlorine ( 37  17  Cl) has a mass number of 37 and an atomic number of 17. This means that its nucleus contains 17 protons and 20 neutrons.

Another common example is carbon ( 13 C). Its mass number is 13. To know, then, the number of neutrons in the carbon (C) atom, you just have to subtract the number of protons (atomic number) from the mass number. In this way, we can know that carbon-13 has 7 neutrons, since its atomic number is 6.

Other examples are:

  • Oxygen ( 16 O)
  • Uranium ( 238 U)
  • Calcium ( 40 Ca)
  • Iron ( 56 Fe)
  • Aluminum ( 27 Al)
  • Neon ( 20 Ne)
  • Hydrogen ( 1 H). This is an exception, and the atomic number is indicated, because it has no neutrons.


Additionsregeln in Wahrscheinlichkeit und Statistik

Die Additionsregeln in Wahrscheinlichkeit und Statistik beziehen sich auf die verschiedenen Arten, wie wir bekannte Wahrscheinlichkeiten von zwei oder mehr verschiedenen Ereignissen kombinieren können, um die Wahrscheinlichkeit neuer Ereignisse zu bestimmen, die durch die Vereinigung dieser Ereignisse gebildet werden .

In der Statistik und Wahrscheinlichkeit kennen wir oft die Wahrscheinlichkeit, dass bestimmte Ereignisse (z. B. Ereignis A und B) separat eintreten, aber nicht die Wahrscheinlichkeit, dass sie gleichzeitig eintreten oder dass das eine oder andere eintritt. Hier kommen die Additionsregeln ins Spiel.

Zum Beispiel: Wir können die Wahrscheinlichkeit kennen, beim Würfeln mit zwei Würfeln eine Sechs zu würfeln, nennen wir es P (würfelnde 6), und die Wahrscheinlichkeit, dass beide Würfel auf geraden Zahlen landen, nennen wir es P (gerade Zahlen).

Dies ist relativ einfach. Aber manchmal interessiert uns auch die Wahrscheinlichkeit, dass beim Würfeln zweier Würfel beide eine gerade Zahl ergeben oder dass die Summe sechs ergibt. In der statistischen Notation und in der Gruppentheorie wird dieses „oder“ mit dem Symbol U dargestellt, das die Vereinigung zweier Ereignisse angibt, und in diesem Fall würde diese Wahrscheinlichkeit wie folgt dargestellt:

unbekannt zu finden

Solche Wahrscheinlichkeiten lassen sich aus den Einzelwahrscheinlichkeiten und einigen Zusatzdaten mit Hilfe der Additionsregeln berechnen.

Es ist zu beachten, dass es sowohl von der Anzahl der betrachteten Ereignisse als auch davon abhängt, ob sich diese Ereignisse gegenseitig ausschließen oder nicht, welche Additionsregel wir jeweils verwenden sollten. Die Additionsregeln für einige einfache Fälle sind unten beschrieben.

Fall 1: Additionsregel für disjunkte oder sich gegenseitig ausschließende Ereignisse

Zwei Ereignisse heißen sich gegenseitig ausschließend, wenn das Eintreten des einen die Möglichkeit des Eintretens des anderen ausschließt. Das heißt, es handelt sich um Ereignisse, die nicht gleichzeitig auftreten können. Zum Beispiel beim Werfen eines Würfels, bei dem das Ergebnis, bei dem 4 herauskommt, ausschließt, dass eines der anderen 5 möglichen Ergebnisse herausgekommen ist.

Wenn wir zwei oder mehr Ereignisse (A, B, C …) als sich gegenseitig ausschließend betrachten, besteht die Vereinigungswahrscheinlichkeit einfach aus der Summe der Einzelwahrscheinlichkeiten jedes dieser Ereignisse. Das heißt, in diesem Fall ist die Vereinigungswahrscheinlichkeit gegeben durch:

Additionsregel für disjunkte oder sich gegenseitig ausschließende Ereignisse

Dies lässt sich am einfachsten anhand eines Venn-Diagramms nachvollziehen. Hier wird der Probenraum durch eine rechteckige Fläche dargestellt; während die Wahrscheinlichkeit jedes Ereignisses durch Sektoren innerhalb dieses größeren Bereichs dargestellt wird. In einem Venn-Diagramm werden sich gegenseitig ausschließende Ereignisse als getrennte Bereiche angesehen, die sich weder berühren noch überlappen.

Additionsregel für disjunkte oder sich gegenseitig ausschließende Ereignisse Venn-Diagramm

Bei dieser Art von Diagrammen besteht die Berechnung der Vereinigungswahrscheinlichkeit darin, die Gesamtfläche zu erhalten, die von allen Ereignissen eingenommen wird, deren Wahrscheinlichkeiten wir berücksichtigen. Im Fall des vorherigen Bildes bedeutet dies, die Gesamtfläche der Sektoren A, B und C zu erhalten, dh die blaue Fläche in der folgenden Abbildung.


Es ist leicht zu erkennen, dass die Vereinigungswahrscheinlichkeit einfach die Summe der drei Bereiche ist, wenn die Ereignisse disjunkt sind, wie im Fall der beiden obigen Bilder.

Beispiel 1: Berechnung der Wahrscheinlichkeit, beim Würfeln ein gerades Ergebnis zu erzielen

Angenommen, wir würfeln und möchten die Wahrscheinlichkeit für eine gerade Zahl ermitteln. Da die einzig möglichen geraden Zahlen auf einem 6-seitigen Würfel 2, 4 und 6 sind, wollen wir wirklich wissen, wie wahrscheinlich es ist, dass der Würfel auf 2, 4 oder 6 fällt, da dies in beiden Fällen der Fall wäre sind in eine gerade Zahl gefallen.

Die Wahrscheinlichkeit, einen der 6 Köpfe zu bekommen, ist 1/6 (solange es ein fairer Würfel ist). Außerdem sind, wie wir gerade gesehen haben, die drei Ergebnisse sich gegenseitig ausschließende Ereignisse, denn wenn 2 gewürfelt wurden, hätten 4 oder 6 nicht gewürfelt werden können, und so weiter. Unter diesen Bedingungen ist die Vereinigungswahrscheinlichkeit gegeben durch:

Beispiel für die Vereinigungswahrscheinlichkeit disjunkter Ereignisse Beispiel für die Vereinigungswahrscheinlichkeit disjunkter Ereignisse

Fall 2: Additionsregel für zwei Ereignisse, die sich nicht gegenseitig ausschließen

Wenn A und B Ereignisse sind, die gemeinsame Ergebnisse haben, d. h. sie können zur gleichen Zeit eintreten, dann schließen sich die Ereignisse nicht gegenseitig aus. In diesem Fall sieht das Venn-Diagramm so aus:

Additionsregel für zwei sich nicht gegenseitig ausschließende Ereignisse Venn-Diagramm

Wie zu sehen ist, gibt es einen Bereich des Probenraums, in dem beide Ereignisse gleichzeitig auftreten. Wenn wir die Vereinigungswahrscheinlichkeit, also P(AUB), bestimmen wollen, müssen wir die Fläche finden, die im Venn-Diagramm rechts in der vorherigen Abbildung angegeben ist.

Es ist leicht zu erkennen, dass wir in diesem Fall, wenn wir nur die Bereiche von A und B addieren, den gemeinsamen Bereich zweimal zählen, sodass wir einen Bereich (read, Wahrscheinlichkeit) erhalten, der größer ist als wir wollen. Um diesen übermäßigen Fehler zu korrigieren, muss nur die von den Ereignissen A und B geteilte Fläche abgezogen werden, die der Schnittwahrscheinlichkeit entspricht:

Additionsregel für zwei Ereignisse, die sich nicht gegenseitig ausschließen

Dieser Ausdruck für die Vereinigungswahrscheinlichkeit gilt auch für den vorigen Fall, da die Wahrscheinlichkeit ihres gleichzeitigen Auftretens (die Schnittwahrscheinlichkeit) null ist, da sie sich gegenseitig ausschließen.

Beispiel 2: Berechnung der Wahrscheinlichkeit, beim Würfeln ein gerades Ergebnis oder eine Zahl kleiner als 4 zu erhalten

In diesem Fall haben beide Ereignisse das Ergebnis 2 gemeinsam, das sowohl gerade als auch kleiner als 4 ist, sodass die Vereinigungswahrscheinlichkeit wie folgt lautet:

Additionsregel für zwei Ereignisse, die sich nicht gegenseitig ausschließen Additionsregel für zwei Ereignisse, die sich nicht gegenseitig ausschließen

Fall 3: Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen

Ein weiterer etwas komplexerer Fall liegt vor, wenn 3 Ereignisse auftreten, die sich nicht gegenseitig ausschließen, wie z. B. das im folgenden Venn-Diagramm dargestellte:

Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen

In diesem Fall zählt die Summe der drei Bereiche doppelt die Schnittzonen zwischen A und B, zwischen B und C und zwischen C und D, und zählt dreimal die Schnittzonen der drei Ereignisse A, B und C. Wenn wir das tun wie zuvor und die Schnittflächen zwischen jedem Ereignispaar von der Summe der drei Flächen subtrahieren, subtrahieren wir das Dreifache der Fläche des Zentrums, also muss es als Schnittwahrscheinlichkeit der drei Ereignisse addiert werden. Schließlich ist die allgemeine Additionsregel für drei nicht-exklusive Ereignisse gegeben durch:

Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen

Wie zuvor gilt dieser Ausdruck allgemein für jeden Satz von drei Ereignissen, unabhängig davon, ob sie disjunkt sind oder nicht, da in diesem Fall die Schnittpunkte leer sind und das Ergebnis derselbe Ausdruck des ersten Falls ist.

Beispiel 3: Berechnung der Wahrscheinlichkeit, bei einem 20-seitigen Würfel eine gerade Zahl, eine Zahl kleiner als 10 oder eine Primzahl zu erhalten

In diesem Fall gibt es drei Ereignisse, die Ergebnisse teilen und auch Ergebnisse enthalten, die nicht geteilt werden, sodass die Vereinigungswahrscheinlichkeit durch den oben genannten Ausdruck gegeben ist.

Die Wahrscheinlichkeiten der einzelnen Ereignisse sind:

Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen

Nun sind die Schnittwahrscheinlichkeiten:

Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen

Wenden Sie nun die Gleichung für die Vereinigungswahrscheinlichkeit an:

Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen Beispiel der Additionsregel für drei Ereignisse, die sich nicht gegenseitig ausschließen


Enhance a book club with study questions and discussion

A book club is a creative strategy that promotes reading and writing. It is made up of a group of people who meet periodically to discuss common readings that have been previously agreed upon.

The objective of the book club is to help its members, through dialogue, to form a reading habit and to increase the enjoyment that this activity can bring them. Such a dialogue can be guided through questions that promote or enhance the group conversation. A helpful guide to discussion questions for a book club is presented below.

general inquiries

  • What were your expectations for this book? Did the book meet them?
  • What did you like the most and the least about this book?
  • Did reading this book remind you of another? To which?
  • Share a favorite quote from the book. Why do you highlight this quote?
  • What did you think of the length of the book? If it’s too long, what would you take away?
  • What feelings did this book provoke in you?
  • Did this book make you think of any work of art (music, painting, sculpture)? In which?
  • What do you think of the title of the book? What other title would you choose?
  • How does the title relate to the content of the book?
  • What do you think of the book cover? How well does it convey the content of the book?
  • Did the book end as you expected? Why?
  • What part of the book did you find most interesting?
  • What would be the five words you would use to summarize this book?
  • Would you recommend this book to other readers? Why?

Questions related to the author

  • Would you read another book by this author? Why?
  • If you had the opportunity to ask the author of this book one question, what would it be?
  • Why do you think the author chose to tell this story?

Questions related to characters and settings

  • Which characters in the book did you like the most and which the least?
  • If you were making a movie about this book, who would you cast?
  • Which character from the book would you like to meet? Why?
  • Which places in the book would you like to visit? Why?
  • What aspects of the story do you think would have been different if they had happened in a different time or place?

Questions related to the literary genre or type of work


  • What elements of the fiction that the author proposes could become real?
  • Do you think the author adequately constructed the world described in the book? Why?
  • Did you find the characters believable? Did they remind you of anyone?
  • If you were a fan of this book and you had to write one for other fans from the original text, what kind of story would you like to tell?


  • Did you know anything about the true story this book was based on before reading it?
  • Do you feel that the book helped to improve your knowledge and understanding of the subject?
  • After reading this true story, are you interested in investigating the narrated facts on your own, to compare them and delve into them? Why?
  • Do you think a story based on real life should stay true to the facts, or can it take some liberties in telling what happened? Why?

Compilations of short stories such as short stories and essays

  • Which story did you like the most and which the least? Why?
  • Do you find similarities between the stories presented in the book? Which are?
  • Do you think there is any element that connects the stories presented in the book? Which?  
  • Do you think any of the stories could be expanded into a full book?

Children’s and youth literature

  • If this book had a sequel, how would you like it to continue?
  • If you were the author, what would you change about the protagonist or any other character in the book?
  • If you were the author, what would you change about the story told?
  • Which is your favourite character? Why?
  • What is the most generous/ambitious/funny/honest/selfish/mysterious/cowardly/dangerous/creative thing…that your favorite character in the story does?
  • Does your favorite character act well or badly in the story? Why?
  • What was the character you liked the least? Why?
  • How are your favorite character and your least favorite character different?
  • Is there something you don’t want to forget about the book? Why?

Selection of questions with a “Tell me” approach

This approach, where all questions start with “Tell me…”, was put forward by Aidan Chambers. It is considered a simple and close way for children to start a conversation. Some questions proposed by the author are the following.

  • Is there something in the book you didn’t understand?
  • What kind of book did you think it was going to be the first time you saw it?… Now that you’ve read it, is it what you expected?
  • If the author asked you what you would improve about the book, what would you say?
  • Would you like to know how, when, where or why the author wrote this book? Why?
  • As you read the book, did you come across words or phrases that you liked? Which?
  • While reading the book, did you come across words or phrases that you didn’t like? Which?
  • Is there anything that happens in this book that happened to you?
  • While you were reading, did you imagine what the characters, places or events were like?
  • Did you read the book all at once or in parts? Why?
  • Would you like to read the book again? Why?
  • If you were to recommend the book, what would you say to make someone else want to read it?


Tobar, K., Riobuneo, M. The reading club as a creative strategy to promote creative reading in comprehensive education students . Research Journal , 42(94): 1-19, 2018.

Raquel. 10 questions to talk about books with your children . Village Books, nd

Raquel. Book Conversations (Aidan Chambers’ Dime Approach) . Village Books, nd

Anthony Giddens: biography of the British sociologist

Anthony Giddens, named Baron Giddens by the British monarchy in 2002, is a renowned 84-year-old British sociologist known, among other things, for his social theory of structuring, for the concepts of double hermeneutics, the duality of the social structure and for his social democratic theory of the third way. He is one of the most influential English sociologists of the 20th century, undoubtedly the most influential of those still alive, and a thinker of the stature of John Maynard Keynes. Giddens received the Prince of Asturias Award for Social Sciences in 2002.

Giddens’ life has been marked by academic, professional and intellectual success. Not only has he risen to the top of academia, serving as director of the prestigious London School of Economics , but through his insights and intellectual contributions, he has influenced notable political figures, including the former Social Democratic Prime Minister of England, Tony Blair.

Anthony Giddens is also known to be a very prolific academic writer. He has contributed no fewer than 40 titles that have been translated into more than 30 languages, many of which have been published in multiple editions. Many of his books are already considered classics for teaching sociology. In addition, in 1985 he was one of the founders of Polity Press publishing house .

Next, we will talk about the life and work of Anthony Giddens, as well as his most important contributions to modern sociology.

Birth and childhood of Anthony Giddens

Lord Anthony Giddens was born in Edmonton, North London, England on January 18, 1938 into a middle-class English family. His father was a worker for the London transport company, the London Passenger Transport Board , and he was the first in his family to attend university.

Giddens studied at some of the most prestigious educational institutions in England and the world. He began his studies in sociology at the small University of Hull, graduating in 1959 at the age of 21. He later earned an MA from the London School of Economics and Political Science and a PhD from King’s College, Cambridge University.

Academic and professional achievements

Giddens’ academic work began when he was a student, but intensified after he obtained a position as professor of social psychology at the University of Leicester in 1961, where he began to work on his own social theories.

Eight years later, he obtains a position as a professor at the University of Cambridge, where he did his doctorate. There he took a more active role in developing the field of sociology in England, participating in the creation of the Social and Political Science Committee. In 1987, 18 years after entering Cambridge, he was made a full professor.

Among the academic positions he held during his life, one of the most important was his tenure as director of the London School of Economics and Political Science for 6 years between 1997 and 2003.

Contributions of Anthony Giddens to modern sociology

With 34 published books and more than 50 years studying social structures, Giddens has made many contributions to contemporary sociology. In addition, she has also influenced a large number of sociologists in different parts of the world. However, the contributions for which Giddens is best known are two: the social theory of structuring and the social democratic theory of the third way.

The theory of structuring

Giddens’ name will always be associated with the theory of structuring. He developed it during the 70s of the 20th century and published it in 1984 in his work De él La constitución de la sociedad . The theory describes the creation of social systems based on the analysis of the structures of societies and the agents involved, giving equal weight or importance to both.

Additionally, with his theory, Giddens seeks to go even beyond the structure-agency dualism of society by separating social systems from their structure from the conceptual point of view, so that social systems do not, in themselves, possess structure, but rather they get it from social practices.

The “third way”

The third way (original title: The Third Way. The Renewal of Social Democracy ) is one of the most important political and social contributions that Anthony Giddens made to society in general and to English society and politics in particular. This book, published for the first time in 1998 in English, and in Spanish in 1999, seeks to reconcile the values ​​and ideas of the center-left of the social democratic movements and parties with the free market and the current global capitalist context.

In principle, the book seeks to demonstrate that neither socialism nor pure capitalism are adequate to ensure a balance in society at a global level, but that another path is needed (the third way) made up of an updated version of social democracy.

The ideas set out in the book The Third Way served as inspiration for the government of British Prime Minister Tony Blair, who appointed Anthony Giddens as adviser during his 10-year term, from 1997 to 2007. These ideas also formed an important part of the ideas policies of US President Bill Clinton, who also met with Giddens on several occasions to discuss his political views.

Giddens bibliography

  1. Capitalism and modern social theory. An analysis of Marx’s writings. Durkheim and Max Weber (Original title Capitalism and Modern Social Theory: An Analysis of the Writings of Marx, Durkheim and Max Weber , 1971)
  2. Politics and Sociology in Max Weber (Original title: Politics and Sociology in the Thought of Max Weber , 1972)
  3. The class structure in advanced societies (Original title: The Class Structure of the Advanced Societies , 1973)
  4. The New Rules of Sociological Method: A Positive Critique of Interpretive Sociologies (Original title: New Rules of Sociological Method , 1976)
  5. Political Theory (Original title: Political Theory , 1977)
  6. Emile Durkheim (1978)
  7. Central Problems in Social Theory (Original title: Central Problems in Social Theory, 1979)
  8. A Contemporary Critique of Historical Materialism (Original title: A Contemporary Critique of Historical Materialism , 1981)
  9. Sociology (Original title: Sociology, textbook of the sociological discipline, 1982)
  10. Profiles and Critiques in Social Theory (Original title: Profiles and Critiques in Social Theory, 1983)
  11. The Constitution of Society (Original title: The Constitution of Society, 1984)
  12. The Nation-State and Violence (Original title: The Nation-State and Violence , (1985)
  13. Consequences of Modernity (Original title: The Consequences of Modernity , 1990)
  14. Modernity and Self Identity: The Self and Society in Contemporary Times (Original title: Modernity and Self Identity , 1991)
  15. Beyond Left and Right: The Future of Radical Politics (Original title: Beyond Left and Right: The Future of Radical Politics , 1994)
  16. The transformation of intimacy: sexuality, love, and eroticism in modern societies (Original title: The Transformation of Intimacy: Sexuality, Love, and Eroticism in Modern Societies , 1995)
  17. In Defense of Sociology (1996)
  18. Durkheim on Politics and the State (1996)
  19. Politics, Sociology, and Social Theory : Reflections on Classical and Contemporary Social Thought (Original title: Politics, Sociology, and Social Theory: Encounters with Classical and Contemporary Social Thought , 1997)
  20. The Third Way: The Renewal of Social Democracy (Original title: The Third Way: The Renewal of Social Democracy , 1998)
  21. A World Runaway: The Effects of Globalization on Our Lives (Original title: Runaway World , 1999)
  22. The third way and its critics (Original title: The Third Way and its Critics, 2000)
  23. On the Edge: Living with Global Capitalism (Original title: On the Edge: Living with Global Capitalism, 2001)
  24. The Global Third Way Debate (Original title: The Global Third Way Debate , 2001)
  25. Where Now for New Labour? (2002)
  26. The Progressive Manifesto: New Ideas for the Center-Left (Original title: The Progressive Manifesto: New Ideas for the Center-Left, 2003)
  27. The New Egalitarianism (Original title: The New Egalitarianism , 2005)
  28. Europe in the Global Age (Original title: Europe in the Global Age , 2006)
  29. Global Europe, Social Europe (Original title: Global Europe, Social Europe , 2006)
  30. To You, Mr. Brown (Original title: Over to you, Mr. Brown , 2007)
  31. The politics of climate change (Original title: The Politics of Climate Change , 2010)
  32. Studies in Social and Political Theory (Original title: Studies in Social and Political Theory, 2014)
  33. Green Growth, Smart Growth (Original title: Green Growth, Smart Growth, 2015)
  34. Turbulent and Mighty Continent (Original title: Turbulent and Mighty Continent , 2015)
  35. The Consequences of Modernity (Original title: The consequences of Modernity, 2018)
  36. Profiles and Critiques in Social Theory (Original title: Profiles and Critiques of Social Theory , 2021)
  37. Introduction to Sociology (Original title: Introduction to Sociology , 2021)
  38. Essential Concepts in Sociology (Original title: Essential Concepts in Sociology , 2021)
  39. Sociology – Introductory Readings (Original title: Sociology – Introductory Readings, 4 th Ed , 2022)


Anthony Giddens . (2019). All Biographies. https://todobiografias.net/anthony-giddens/

EcuRed. (nd). Anthony Giddens . https://www.ecured.cu/Anthony_Giddens

García Guerrero, E. (2021, May 7). Sociological Contributions of Anthony Giddens . Youtube. https://www.youtube.com/watch?v=4kCK4Jm5YBs

González, G. (2020, January 17). Anthony Giddens . lifer. https://www.lifeder.com/anthony-giddens/

Lucena Giraldo, J. (sf). Giddens, Anthony (1938-VVVV) . Biographies.com. https://www.mcnbiografias.com/app-bio/do/show?key=giddens-anthony

Zunino, H. (2000, August 1). THE “THEORY OF STRUCTURING” AND URBAN STUDIES. AN INNOVATIVE APPROACH TO STUDY THE TRANSFORMATION OF CITIES? Electronic Journal of Geography and Social Sciences. 69(74). http://www.ub.edu/geocrit/sn-69-74.htm

Hydrophobic substances: definition and examples

A substance is hydrophobic if it has the property of hydrophobicity. This means that it cannot be dissolved in or mixed with water. Oil is the most common example of hydrophobic substances.

hydrophobic substances

The word “hydrophobia” comes from the Greek and means phobia of water. There is a disease with that name, which is also called rabies. In chemistry, a substance that has the property of hydrophobicity is called hydrophobic, that is, it repels water, or does not mix or dissolve in it . They are also known as hydrophobic substances.

The hydrophobic molecules that these substances contain are usually nonpolar molecules. Nonpolar molecules are not electrically charged , so they lack attraction. Water, on the other hand, is an electrically polar substance, which has a positive and a negative pole. Not being able to interact with water, nonpolar molecules group together, and the amount of water around them increases. On the other hand, in apolar solvents such as organic solvents , hydrophobic substances dissolve easily.

There are also superhydrophobic materials, which are practically impossible to get wet. The surfaces of these elements are highly resistant to moisture and are considered self-cleaning.

Hydrophobicity and lotus effect

Hydrophobicity is the most characteristic property of hydrophobic substances: the quality that prevents them from being soluble in water . It occurs when a molecule cannot interact with water. Upon coming into contact with it, the nonpolar molecule breaks the hydrogen bonds of the water molecules, forming a network-shaped structure. This gives it more organization than free water molecules and allows them to stick together. A very simple example to observe this phenomenon is to place a few drops of oil in a cup. The oil drops will seek to clump together even if we don’t move the container.

Currently, hydrophobicity is of great scientific interest, especially in the field of nanotechnology, due to the innumerable applications that superhydrophobic elements can have in everyday life and technology.

Since 1963, for example, the “ lotus effect ”, a self-cleaning property of superhydrophobic materials, has been studied. The name derives from the lotus plant, which naturally exhibits this property. To know the hydrophobicity of a surface, its contact angle with the water is measured. The greater the contact angle, the greater the hydrophobicity.

Difference Between Hydrophobic and Lipophilic

The terms hydrophobic and lipophilic are sometimes used interchangeably, as if they meant the same thing. However, they are different concepts. As mentioned before, hydrophobic substances repel or do not mix with water. On the other hand, lipophilic substances are those that have a certain affinity with fats. In any case, most hydrophobic substances, except fluorocarbons and silicones, are at the same time lipophilic. That is, they can also easily bind to fats .

Examples of hydrophobic substances

There are various hydrophobic substances or materials in their natural state, and also artificial. Some of the most common examples are:

  • Hydrophobic substances : here we can include oils, petroleum, fats and alkanes, as well as other organic compounds.
  • Superhydrophobic materials: coatings, kitchen elements with Teflon, fabrics and paints. They are also used to collect dew or for agricultural irrigation. They are generally made with layers of silicones or fluorocarbons. In nature, these materials are found in some insects. Also, in plants such as lotus, nasturtiums, alchemilla, nopal and cane.


  • Tuñon, I. Statistical Molecular Chemistry . 2008. Spain. Synthesis.
  • Vollhardt, P. and Schore. organic chemistry . 2006 (5th edition). Spain. Omega
  • Fernández Cañete, A. (2003). Study of hydrophobicity and self-cleaning in materials with surface nanotreatments. (Final Degree Project, Autonomous University of Barcelona). Barcelona. Autonomous University of Barcelona.

What does saturated mean in chemistry?

The term saturation in chemistry can refer to different concepts, depending on the context in which it is used. There are four most commonly used definitions of saturated or saturated, as listed below.

Definition of saturated or saturated in the context of solutions

A saturated solution is one that does not admit the dissolution of a greater amount of solute. In other words, it is a solution that already has the maximum concentration of solute that it can support, and therefore the solubility equilibrium between the pure solute and the solute in solution has been established.

Definition of saturated or saturated in the context of solutions

The concept of saturated solution applies to any type of solution, both in solid and liquid state. If in a mixture it is possible to clearly distinguish two phases, one of which is a pure solute, it can be said that there is a saturated solution, since, otherwise, the solute would continue to dissolve in the solvent until it completely disappeared.

How do you get a saturated solution?

Solutions can be saturated in different ways.

  1. The most common and direct way is to add more and more solute, until the point is reached where no more can be dissolved in the solvent and some of the solute remains undissolved.
  2. Another way is to dissolve the solute in the hot solvent to increase its solubility, and then allow the system to cool. On cooling, the solubility can drop below the hot solute concentration, causing the solution to become supersaturated and the solute to crystallize out of solution. When equilibrium is established, a saturated solution will be obtained.
  3. A solution can be saturated by adding a precipitating agent that is nothing more than a salt that reacts with the solute to form another less soluble salt that precipitates. Once the precipitation stops, you will have a saturated solution.
  4. Finally, another way to obtain a saturated solution is by preparing a concentrated solution of the solute and then mixing the solution with another solvent in which the solute is less soluble. This reduces the solubility of the solute to the point where it precipitates. The resulting solution will be saturated.

Examples of saturated solutions

  • Brine is a mixture of water with salt in which not all the salt can be dissolved, so the liquid phase is a saturated solution.
  • When sugar crystals form at the bottom of the honey, it is because the solution was supersaturated and precipitated. Therefore, the remaining liquid phase is a saturated solution.

Definition of saturated in organic chemistry

In organic chemistry, the term saturated is used in relation to an organic compound. Two types of organic compounds are recognized: saturated and unsaturated. Saturated organic compounds are those whose atoms are only linked to each other by means of simple covalent bonds . For this reason, these compounds have the maximum number of hydrogen atoms attached to the chain of carbon atoms, hence the use of the term saturated.

Definition of saturated in organic chemistry (saturated hydrocarbons)

In other words, saturated compounds are those that have a “saturated” structure in hydrogen atoms, since they couldn’t have more without violating the octet rule or carbon tetravalence.

Examples of saturated organic compounds

  • Alkanes are the best example of saturated compounds. They are hydrocarbons with the general formula C n H 2n+2 , and define the number of hydrogens in a saturated compound.
  • Alcohols are also saturated compounds and their general formula only differs from alkanes by the presence of an oxygen (C n H 2n+2 O).
  • Ethers have the same general formula as alcohols (C n H 2n+2 O) and therefore are also saturated compounds.

Definition of saturated in relation to absorbent materials

Absorbent materials such as fibers, foams, or hydrogels will normally admit a limited amount of water or other solvent. Once they have absorbed that maximum amount of water, it is usually said that the material is saturated. So you can define saturated in this context as a material that has absorbed the maximum amount of water or other solvent that it can absorb.

Definition of saturated in relation to saturated hydrogel absorbent materials

Examples of saturated absorbent materials

  • A sponge completely soaked in water
  • Fully hydrated hydrogel beads such as those used as a substrate for planting.

Definition of saturated in chemical catalysis

Both homogeneous (such as enzymes) and heterogeneous (such as palladium catalysts in catalytic hydrogenation) catalysts have a limited ability to bind to substrate molecules at the same time. This is because there is a limited amount of active sites or dissolved catalyst molecules in the solution.

When the substrate concentration is high enough, all the active sites on the catalyst are occupied and the catalyst is said to be saturated. In other words, a saturated catalyst can be defined as the one that already has the maximum amount of substrate bound to the catalytically active sites. Under these conditions, increasing the concentration of the substrate does not increase the number of molecules that bind to the catalyst, so the speed of the reaction becomes independent of said concentration.

Examples of saturated catalyst systems

  • A faulty car catalytic converter (which fails to transform all the harmful combustion gases into less toxic products) is normally saturated.

Definition of saturated in chemical catalysis

  • An enzyme that is working at its maximum rate at a given temperature and pH and whose rate is not affected by an increase in substrate concentration is an example of a saturated homogeneous catalyst.

Definition of saturated in chemical catalysis


Brown, T. (2021). Chemistry: The Central Science, 11/ed. (11th ed.). London, England: Pearson Education.

Carey, F., & Giuliano, R. (2014). Organic Chemistry ( 9th ed.). Madrid, Spain: McGraw-Hill Interamericana de España SL

Chang, R., Manzo, Á. R., Lopez, PS, & Herranz, ZR (2020). Chemistry (Spanish Edition) (10. a ed.). New York City, NY: MCGRAW-HILL.

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hydrocarbons. (n.d.). Retrieved June 29, 2021, from https://espanol.libretexts.org/@go/page/1972

Merriam-Webster. (n.d.). Saturated . In Merriam-Webster.com dictionary.

What is an atom? Explanation and examples

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In chemistry, the atom is defined as the smallest possible unit that is part of a chemical element and that maintains all its properties and its characteristic composition . This concept was formalized by John Dalton in his atomic theory, established in 1808, in which he stated that all matter is made up of different types of atoms, and that all the atoms that are part of a chemical element are identical to each other, but different from all the atoms of the other elements.

This means that the element hydrogen, for example, is made up of a type of atoms (hydrogen atoms), which are different from the atoms that make up helium, carbon, and all the other elements on the periodic table.

Being the particles that make up the elements, the different types of atoms are represented by the chemical symbol of the respective element of which they are a part. Thus, H is used in chemistry to represent both the element hydrogen and the hydrogen atom; the same for the other elements of the periodic table.

The origin of the term “atom”

Despite being around for over 200 years, the concept of the atom is much older, dating back to a Greek philosopher named Democritus. Democritus reasoned that if we take a piece of matter and split it in half, then take one of the halves and split it in half and keep repeating the same process, it should come up with a particle that we can’t split any further. He called this indivisible particle atomon , which literally means “that cannot be further divided”, that is, indivisible.

When he put forward his atomic theory, Dalton actually thought he had found the fundamental indivisible unit of matter, so he used the term coined by Democritus. Dalton’s theory gave meaning to the laws of definite proportions and multiple proportions, thus he laid the foundation for the development of chemistry from that time forward.

A more modern concept of the atom

Today, we know that chemical atoms are not actually Democritus’ atoms, since they are made up of smaller particles, that is, atoms can be divided. These particles, called subatomic particles s, include protons, neutrons, and electrons, as well as a whole collection of more exotic particles like neutrinos, positrons, gluons, and others.

Even one of the premises of Dalton’s concept of atom must be modified, since it turns out that not all atoms that are part of a chemical element are equal to each other. This is because there is, for each element, a set of natural isotopes made up of atoms with very similar chemical properties but which are different from each other since they have different numbers of neutrons in their nucleus, and therefore, different masses.

Based on this, the concept of atom could be specified a little more, to avoid some ambiguities, as follows:

An atom is defined as the smallest possible electrically neutral unit that is part of a chemical element and that maintains its characteristic properties based on the number of protons in its nucleus.

examples of atoms

From the first concept of atom, any element that appears on the periodic table is made up of a particular type of atom. Thus, examples of atoms are the atoms of H, He, C, N, Fe, and the others.

Since not all hydrogen atoms are the same, it might be a bit ambiguous to talk about “the” hydrogen atom, “the” carbon atom, etc. To avoid this ambiguity, we can specify the particular isotope. So some examples of atoms could be:

  • The hydrogen-1 atom, or 1 H (also called protium).
  • The hydrogen-2 atom, or 2 H (also called deuterium).
  • The carbon-14 atom, or 14 C.
  • The iron-56 atom, or 56 Fe.

Examples of particles that are NOT atoms

It is important to learn to distinguish between different types of particles that are sometimes confused with atoms:

  • Subatomic particles like protons, neutrons, electrons, and so on, are not atoms. None of these particles satisfies either of the two concepts that define the atom.
  • Monatomic ions such as the H + ion , Na + , Cl – and so on, are not examples of atoms since they are not neutral. They are, by definition, species with a net electrical charge, which rules them out on the basis of the second concept of an atom. The same can be said of polyatomic ions such as nitrate (NO 3 – ) or peroxide ion (O 2 2- ).
  • Molecular elements such as molecular hydrogen (H 2 ), molecular nitrogen (N 2 ), white phosphorus (P 4 ), etc., are made up of identical atoms linked together, but they are not atoms in themselves; They are molecules that can be broken down into individual atoms.
  • The same can be said of any compound, since, by definition, they are made up of more than one chemical element, and therefore more than one type of atom.

Clarification about the hydrogen atom

Hydrogen can sometimes cause a bit of confusion due to the fact that its most common isotope, protium or 1H, consists of just one proton surrounded by one electron . However, it should be noted that (neutral) hydrogen does fit the concept of an atom, since it is a neutral particle with 1 proton, which identifies it as a hydrogen atom.

By losing its only electron, it is no longer considered an atom to become an ion.


Chang, R., Manzo, Á. R., Lopez, PS, & Herranz, ZR (2020). Chemistry ( 10th ed.). New York City, NY: MCGRAW-HILL.

Flowers, P., Neth, EJ, Robinson, WR, Theopold, K., & Langley, R. (2019). Chemistry: Atoms First 2e . Houston, Texas: Open Stax. Retrieved from https://openstax.org/books/chemistry-atoms-first-2e/pages/1-2-phases-and-classification-of-matter

Flowers, P., Theopold, K., Langley, R., Robinson, WR, (2019). Chemistry 2e . Retrieved from  https://openstax.org/books/chemistry-2e/pages/1-1-chemistry-in-context