Despite the fact that thermodynamics does not take into account the processes occurring in real solutions, for example, the attraction and repulsion of ions, the thermodynamic laws derived for ideal solutions can be applied to real solutions if we replace concentrations with activities.

Activity ( a) - is the concentration of a substance in a solution at which the properties of this solution can be described by the same equations as the properties of an ideal solution.

The activity can be either less or greater than the nominal concentration of the substance in solution. The activity of a pure solvent, as well as a solvent in not too concentrated solutions, is taken to be equal to 1. The activity of a solid in the precipitate or a liquid that is immiscible with a given solution is also taken as 1. In an infinitely dilute solution, the activity of the solute coincides with its concentration.

The ratio of the activity of a substance in a given solution to its concentration is called activity coefficient.

The activity coefficient is a kind of correction factor that shows how much reality differs from the ideal.

Deviations from ideality in solutions of strong electrolytes

A particularly noticeable deviation from ideality occurs in solutions of strong electrolytes. This is reflected, for example, in their boiling and melting points, vapor pressure above the solution and, which is especially important for analytical chemistry, in the values ​​of the constants of various equilibria occurring in such solutions.

To characterize the activity of electrolytes, use:

For electrolyte A m B n:

A value that takes into account the influence of concentration (C) and charge ( z ) of all ions present in a solution on the activity of the solute is called ionic strength ( I ).

Example 3.1. 1.00 l of aqueous solution contains 10.3 g of NaBr, 14.2 g of Na 2 SO 4 and 1.7 g of NH 3. What is the ionic strength of such a solution?

0.100 mol/l

0.100 mol/l

C(Na +) = 0.300 mol/l, C(Br -) = 0.100 mol/l, C(SO 4 2-) = 0.100 mol/l

I = 0.5× = 0.400 mol/l

Rice. 3.1. Effect of ionic strength on the average ionic activity coefficient of HCl

In Fig. Figure 3.1 shows an example of the effect of ionic strength on the activity of an electrolyte (HCl). A similar dependence of the activity coefficient on ionic strength is also observed in HClO 4, LiCl, AlCl 3 and many other compounds. For some electrolytes (NH 4 NO 3, AgNO 3), the dependence of the activity coefficient on ionic strength is monotonically decreasing.

There is no universal equation that can be used to calculate the activity coefficient of any electrolyte at any ionic strength. To describe the dependence of the activity coefficient on ionic strength in very dilute solutions (up to I< 0,01) можно использовать Debye-Hückel limit law

where A is a coefficient depending on the temperature and dielectric constant of the medium; for an aqueous solution (298K) A » 0.511.

This equation was obtained by the Dutch physicist P. Debye and his student E. Hückel based on the following assumptions. Each ion was represented as a point charge (i.e., the size of the ion was not taken into account) surrounded in solution ionic atmosphere- a region of space of a spherical shape and a certain size, in which the content of ions of the opposite sign in relation to a given ion is greater than outside it. The charge of the ionic atmosphere is equal in magnitude and opposite in sign to the charge of the central ion that created it. There is an electrostatic attraction between the central ion and the surrounding ionic atmosphere, which tends to stabilize the ion. Stabilization leads to a decrease in the free energy of the ion and a decrease in its activity coefficient. In the limiting Debye-Hückel equation, the nature of the ions is not taken into account. It is believed that at low values ​​of ionic strength the activity coefficient of an ion does not depend on its nature.

As the ionic strength increases to 0.01 or more, the limiting law begins to produce larger and larger errors. This happens because real ions have a certain size, so they cannot be packed as tightly as point charges. As the ion concentration increases, the size of the ionic atmosphere decreases. Since the ionic atmosphere stabilizes the ion and reduces its activity, a decrease in its size leads to a less significant decrease in the activity coefficient.

To calculate activity coefficients at ionic strengths of the order of 0.01 - 0.1, you can use extended Debye-Hückel equation:

where B » 0.328 (T = 298K, a expressed in ), a- empirical constant characterizing the size of the ionic atmosphere.

At higher ionic strength values ​​(up to ~1), the activity coefficient can be quantified using Davis equation.

State Educational Institution of Higher Professional Education "Ural State Technical University - UPI named after the first President of Russia"

Department of Electrochemical Production Technology

Calculation of activity coefficients

Guidelines for implementation in the discipline “Introduction to the theory of electrolyte solutions”

for students studying

direction 240100 – chemical technology and biotechnology (profile technology of electrochemical production)

Ekaterinburg

Compiled by:

Professor, Doctor of Chemistry sciences

Professor, Doctor of Chemistry sciences,

Scientific editor professor dr. chem. Sciences Irina Borisovna Murashova

Calculation of activity coefficients: Guidelines for performing calculation work in the discipline “Introduction to the theory of electrolyte solutions”/, . Ekaterinburg: USTU-UPI 2009.12p.

The guidelines set out the basics for calculating activity coefficients. The possibility of calculating this value based on various theoretical models is shown.

Bibliography: 5 titles. 1 Table

Prepared by the Department of Technology of Electrochemical Production.

Options for coursework assignments

Bibliography

INTRODUCTION

Theoretical ideas about the structure of solutions were first formulated in the Arrhenius theory of electrolytic dissociation:

1. Electrolytes are substances that, when dissolved in appropriate solvents (for example, water), disintegrate (dissociate) into ions. The process is called electrolytic dissociation. Ions in solution are charged particles that behave like ideal gas molecules, that is, they do not interact with each other.

2. Not all molecules disintegrate into ions, but only a certain fraction, which is called the degree of dissociation

Where n is the number of decayed molecules, N is the total number of molecules of the dissolved substance. 0<б<1

3. The law of mass action is applied to the process of electrolytic dissociation.

The theory does not take into account the interaction of ions with water dipoles, that is, ion-dipole interaction. However, it is precisely this type of interaction that determines the physical basis for the formation of ions and explains the causes of dissociation and the stability of ionic systems. The theory does not take into account ion-ion interaction. Ions are charged particles and therefore influence each other. Neglecting this interaction leads to a violation of the quantitative relationships of the Arrhenius theory.

Because of this, the theory of solvation and the theory of interionic interaction subsequently emerged.

Modern ideas about the mechanism of formation of electrolyte solutions. Equilibrium electrodes

The process of ion formation and the stability of electrolyte solutions (ionic systems) cannot be explained without taking into account the interaction forces between ions and solvent molecules (ion-dipole interaction) and ion-ion interaction. The entire set of interactions can be formally described using ion activities (ai) instead of concentrations (Ci)

where fi is the activity coefficient of the i-th type of ions.

Depending on the form of expression of concentrations, there are 3 scales of activity networks and activity coefficients: molar c-scale (mol/l or mol/m3); m – molal scale (mol/kg); N – rational scale (the ratio of the number of moles of a dissolved substance to the total number of moles in the volume of solution). Accordingly: f, fm, fN, a, am, aN.

When describing the properties of electrolyte solutions, the concepts of salt activity are used

(2)

and average ionic activity

where , a and are the stoichiometric coefficients of the cation and anion, respectively;

C is the molar concentration of the dissolved substance;

- average activity coefficient.

Basic provisions of the theory of solutions of strong electrolytes by Debye and Hückel:

1. Only electrostatic forces act between ions.

2. When calculating the Coulomb interaction, it is assumed that the dielectric constant of the solution and the pure solvent are equal.

3. The distribution of ions in a potential field obeys Boltzmann statistics.

In the theory of strong electrolytes by Debye and Hückel, 2 approximations are considered when determining the activity coefficients.

As a first approximation, when deriving the expression for the average activity coefficient, it is assumed that the ions are material points (ion size) and electrostatic interaction forces act between them:

, (4)

Activity coefficient on a rational scale (N – concentration expressed in mole fractions);

T - temperature;

e – dielectric constant of the medium (solvent);

- ionic strength of the solution, mol/l, k – number of types of ions in the solution;

.

To calculate the activity coefficient on a molal scale, use the relationship

Molal concentration of solute, mol/kg;

Molar mass of solvent, kg/mol.

Calculation of the average activity coefficient as a first approximation is valid for dilute solutions of strong electrolytes.

In the second approximation, Debye and Hückel took into account that the ions have a finite size equal to a. Ion size refers to the minimum distance that ions can approach each other. The size values ​​of some ions are presented in the table.

Table 1. Values ​​of parameter a, characterizing the size of ions

		F-, Cl-, Br-, I-, CN-, NO2-, NO3-, OH-, CNS-
		IO3-, HCO3-, H2PO4-, HSO3-, SO42-
		PO43-, Fe(CN)63-
Rb+, Cs+, NH4+, Tl+, Ag+
Ca2+, Cu2+, Zn2+, Sn2+, Mn2+, Fe2+, Ni2+, Co2+
Pb2+, Sr2+, Ba2+, Ra2+, Cd2+, Hg2+,
Fe3+, Al3+, Cr3+, Sc3+, Y3+, La3+, In3+, Ce3+,

As a result of thermal movement, ions in the electrolyte solution are located around an ion, arbitrarily chosen as the central one, in the form of a sphere. All ions of the solution are equivalent: each is surrounded by an ionic atmosphere and, at the same time, each central ion is part of the ionic atmosphere of another ion. The hypothetical ionic atmosphere has an equal and opposite charge relative to the charge of the central ion. The radius of the ionic atmosphere is denoted as .

If the sizes of the cation and anion are close, then using the second approximation of Debye and Hückel, the average activity coefficient can be determined:

, (6)

Where , . (7)

Expressions for the activity coefficients of the cation and anion have the form:

And

From the known activity coefficients of individual ions, the average ion activity coefficient can be calculated: .

The theory of Debye and Hückel is applicable to dilute solutions. The main disadvantage of this theory is that only the Coulomb interaction forces between ions are taken into account.

Calculation of activity coefficients according to Robinson-Stokes and Ikeda.

In deriving the equation for the average activity coefficient, Robinson and Stokes learned from the fact that ions in solution are in a solvated state:

where - the activity of the solvent depends on the osmotic coefficient (μ), ;

The number of solvent molecules associated with one solute molecule; bi is the hydration number of the i-th ion.

Ikeda proposed a simpler formula for calculating the molal average ion activity coefficient

The Robinson-Stokes equation allows you to calculate the activity coefficients of 1-1 valence electrolytes up to a concentration of 4 kmol/m3 with an accuracy of 1%.

Determination of the average ionic activity coefficient of an electrolyte in a mixture of electrolytes.

For the case when there are two electrolytes B and P in a solution, Harned’s rule is often satisfied:

, (10)

where is the average ionic activity coefficient of electrolyte B in the presence of electrolyte P

Average ion activity coefficient B in the absence of P,

- total molality of the electrolyte, which is calculated as the sum of the molal concentrations of electrolytes B and P,

Here hB and hP are the number of solvent molecules associated with one molecule of electrolyte B and P, respectively, and are the osmotic coefficients of electrolytes B and P.

Subjects of coursework in the discipline

for part-time students

Option No.	Electrolyte	Concentration, mol/m3	Temperature, 0C

For more accurate calculations based on the law of mass action, activities are used instead of equilibrium concentrations.

This value was introduced to take into account the mutual attraction of ions, the interaction of a solute with a solvent, and other phenomena that change the mobility of ions and are not taken into account by the theory of electrolytic dissociation.

Activity for infinitely dilute solutions is equal to the concentration:

For real solutions, due to the strong manifestation of interionic forces, the activity is less than the concentration.

Activity can be considered as a value characterizing the degree of connectivity of electrolyte particles. Thus, activity is an effective (active) concentration, manifesting itself in chemical processes as an actual active mass, in contrast to the total concentration of a substance in solution.

Activity coefficient. Numerically, activity is equal to concentration multiplied by a factor called the activity coefficient.

The activity coefficient is a value that reflects all the phenomena present in a given system that cause changes in the mobility of ions, and is the ratio of activity to concentration: . With infinite dilution, the concentration and activity become equal, and the value of the activity coefficient is equal to unity.

For real systems, the activity coefficient is usually less than one. Activities and activity coefficients related to infinitely dilute solutions are marked with an index and designated accordingly.

An equation applied to real solutions. If we substitute the activity value instead of the concentration value of a given substance into the equation characterizing the equilibrium of a reaction, then the activity will express the influence of this substance on the equilibrium state.

Substituting activity values ​​instead of concentration values ​​into equations resulting from the law of mass action makes these equations applicable to real solutions.

So, for the reaction we get:

or, if you substitute the values:

In the case of applying equations arising from the law of mass action to solutions of strong electrolytes and concentrated solutions of weak electrolytes or to solutions of weak electrolytes in the presence of other electrolytes, it is necessary to substitute activities instead of equilibrium concentrations. For example, the electrolytic dissociation constant of an electrolyte type is expressed by the equation:

In this case, the electrolytic dissociation constants determined using activities are called true or thermodynamic electrolytic dissociation constants.

Activity coefficient values. The dependence of the activity coefficient on various factors is complex and its determination encounters some difficulties, therefore, in a number of cases (especially in the case of solutions of weak electrolytes), where greater accuracy is not required, analytical chemistry is limited to the use of the law of mass action in its classical form.

The values ​​of the activity coefficients of some ions are given in Table. 1.

TABLE 1. Approximate values ​​of average activity coefficients f at different ionic strengths of the solution

Any physical or mental activity requires energy, so calculating the daily calorie intake per day for a woman or man should take into account not only gender, weight, but also lifestyle.

Every day we spend energy on metabolism (resting metabolism) and movement (physical activity). Schematically it looks like this:

Energy = E basal metabolic rate + E physical activity

Basal metabolic energy, or basal metabolic rate (BMR)- Basal Metabolic Rate (BMR) – this is the energy needed for the functioning (metabolism) of the body without physical activity. Basic metabolic rate is a value that depends on a person’s weight, height and age. The taller a person is and the greater his weight, the more energy is needed for metabolism, the higher the basal metabolic rate. Conversely, shorter, thinner people will have a lower basal metabolic rate.

For men
= 88.362 + (13.397 * weight, kg) + (4.799 * height, cm) - (5.677 * age, years)
For women
= 447.593 + (9.247 * weight, kg) + (3.098 * height, cm) - (4.330 * age, years)
For example, a woman weighing 70 kg, height 170 cm, 28 years old, requires for basic metabolism (basal metabolism)
= 447,593 + (9.247 * 70) + (3,098 *170) - (4.330 *28)
=447.593+647.29+526.66–121.24=1500.303 kcal

You can also check the table: Daily energy consumption of the adult population without physical activity according to the Norms of physiological needs of the population for basic nutrients and energy.

A physically inactive person spends 60–70% of daily energy on basal metabolism, and the remaining 30–40% on physical activity.

How to calculate the total amount of energy consumed by the body per day

Recall that total energy is the sum of basal metabolic energy (or basal metabolic rate) and energy used for movement (physical activity).
To calculate the total energy expenditure taking into account physical activity, there is Physical activity rate.

What is physical activity quotient (PAI)

Physical activity coefficient (PAL) = Physical Activity Level (PAL) is the ratio of total energy expenditure at a certain level of physical activity to the basal metabolic rate, or, more simply, the value of the total energy expended divided by the basal metabolic rate.

The more intense the physical activity, the higher the physical activity ratio will be.

• People who move very little have CFA = 1.2. For them, the total energy expended by the body will be calculated: E = BRM * 1.2
• People who do light exercise 1-3 days a week have a CFA of 1.375. So the formula: E=BRM*1.375
• People performing moderate exercise, namely 3-5 days a week, have a CFA of 1.55. Formula for calculation: E=BRM*1.55
• People who do heavy exercise 6-7 days a week have a CFA of 1.725. Formula for calculation: E=BRM*1.725
• People who perform very strenuous exercise twice a day, or workers with heavy physical activity, have a CFA of 1.9. Accordingly, the formula for calculation: E = BRM * 1.9

So, to calculate the total amount of energy spent per day, you need to: multiply the basal metabolic rate according to age and weight (Basic metabolic rate) by the physical activity coefficient according to the physical activity group (Physical activity level).

What is energy balance? And when will I lose weight?

Energy balance is the difference between the energy entering the body and the energy expended by the body.

Equilibrium in the energy balance is when the energy supplied to the body with food is equal to the energy expended by the body. In this situation, the weight remains stable.
Accordingly, a positive energy balance is when the energy received from consumed food is greater than the energy needed for the body’s functioning. In a state of positive energy balance, a person gains extra pounds.

A negative energy balance is when less energy is received than the body expends. To lose weight you need to create a negative energy balance.

The non-subordination of solutions of strong electrolytes to the law of mass action, as well as the laws of Raoult and Van't Hoff, is explained by the fact that these laws apply to ideal gas and liquid systems. When deriving and formulating these laws, the force fields of particles were not taken into account. In 1907, Lewis proposed introducing the concept of “activity” into science.

Activity (α) takes into account the mutual attraction of ions, the interaction of a solute with a solvent, the presence of other electrolytes, and phenomena that change the mobility of ions in solution. Activity is the effective (apparent) concentration of a substance (ion), according to which ions manifest themselves in chemical processes as a real active mass. Activity for infinitely dilute solutions is equal to the molar concentration of the substance: α = c and is expressed in gram ions per liter.

For real solutions, due to the strong manifestation of interionic forces, the activity is less than the molar concentration of the ion. Therefore, activity can be considered as a quantity characterizing the degree of connectivity of electrolyte particles. The concept of “activity” is also associated with another concept - “activity coefficient” ( f), which characterizes the degree of deviation of the properties of real solutions from the properties of ideal solutions; it is a quantity that reflects all phenomena occurring in a solution that cause a decrease in the mobility of ions and reduce their chemical activity. Numerically, the activity coefficient is equal to the ratio of activity to the total molar concentration of the ion:

f =	a
c

and the activity is equal to the molar concentration multiplied by the activity coefficient: α = cf.

For strong electrolytes, the molar concentration of ions (With) calculated based on the assumption of their complete dissociation in solution. Physical chemists distinguish between active and analytical concentrations of ions in a solution. The active concentration is the concentration of free hydrated ions in a solution, and the analytical concentration is the total molar concentration of ions, determined, for example, by titration.

The ion activity coefficient depends not only on the ion concentration of a given electrolyte, but also on the concentration of all foreign ions present in the solution. The value of the activity coefficient decreases with increasing ionic strength of the solution.

The ionic strength of a solution (m,) is the magnitude of the electric field in a solution, which is a measure of the electrostatic interaction between all ions in the solution. It is calculated using the formula proposed by G. N. Lewis and M. Rendel in 1921:

m =		(c 1 Z 2 1 + c 2 Z 2 2 + ...... + c n Z 2 n)

Where c 1 , c 2 and c n - molar concentrations of individual ions present in solution, a Z 2 1, Z 2 2 and Z 2 n - their charges squared. Undissociated molecules, as having no charges, are not included in the formula for calculating the ionic strength of a solution.

Thus, the ionic strength of a solution is equal to half the sum of the products of the concentrations of ions and the squares of their charges, which can be expressed by the equation: µ = i Z i 2

Let's look at a few examples.

Example 1. Calculate ionic strength 0.01 M potassium chloride solution KS1.

0.01; Z K= Z Cl - = 1

Hence,

i.e. the ionic strength of a dilute solution of a binary electrolyte of the KtAn type is equal to the molar concentration of the electrolyte: m = With.

Example 2. Calculate ionic strength 0.005 M barium nitrate solution Ba(NO 3) 2.

Dissociation scheme: Ba(NO 3) 2 ↔ Ba 2+ + 2NO 3 -

[Ba 2+] = 0.005, = 2 0.005 = 0.01 (g-ion/l)

Hence,

The ionic strength of a dilute solution of electrolyte type KtAn 2 and Kt 2 An is equal to: m = 3 With.

Example 3. Calculate ionic strength 0.002 M zinc sulfate solution ZnSO 4.

0.002, Z Zn 2+ = Z SO 4 2- = 2

Hence, the ionic strength of an electrolyte solution of the Kt 2+ An 2- type is equal to: m = 4 With.

In general, for an electrolyte of the Kt n + type a An m - b ionic strength of a solution can be calculated using the formula: m = (A· · n 2 + b· · t 2),

Where a, b - indices for ions, and n + And T - - ion charges, and - ion concentrations.

If two or more electrolytes are present in a solution, the total ionic strength of the solution is calculated.

Note. Chemistry reference books give differentiated activity coefficients for individual ions or for groups of ions. (See: Lurie Yu. Yu. Handbook of Analytical Chemistry. M., 1971.)

With increasing solution concentration and complete dissociation of electrolyte molecules, the number of ions in the solution increases significantly, which leads to an increase in the ionic strength of the solution and a significant decrease in the activity coefficients of ions. G. N. Lewis and M. Rendel found the law of ionic strength, according to which the activity coefficients of ions of the same charge are the same in all dilute solutions having the same ionic strength. However, this law applies only to very dilute aqueous solutions, with ionic strength up to 0.02 g-ion/l. With a further increase in concentration, and therefore the ionic strength of the solution, deviations from the law of ionic strength begin, caused by the nature of the electrolyte (Table 2.2).

Table 2.2 Approximate values ​​of activity coefficients at different ionic strengths

Currently, for analytical calculations, a table of approximate values ​​of activity coefficients is used.

The dependence of ion activity coefficients on the ionic strength of the solution for very dilute electrolyte solutions is calculated using the approximate Debye-Hückel formula:

lg f = - AZ 2 ,

Where A- a multiplier, the value of which depends on temperature (at 15°C, A = 0,5).

For solution ionic strengths up to 0.005, the value of 1 + is very close to unity. In this case, the Debye-Hückel formula

takes on a simpler form:

lg f= - 0.5 · Z 2.

In qualitative analysis, where one has to deal with complex mixtures of electrolytes and where great accuracy is often not required, Table 2.2 can be used when calculating the activities of ions.

Example 4. Calculate the activity of ions in a solution containing 1 l 0,001 mole potassium aluminum sulfate.

1. Calculate the ionic strength of the solution:

2. Find the approximate value of the activity coefficients of these ions. So, in the example under consideration, the ionic strength is 0.009. The closest ionic strength given in Table 2.2 is 0.01. Therefore, without a large error, we can take for potassium ions fK+= 0.90; for aluminum ions f Al 3+ = 0.44, and for sulfate ions f SO 2- 4 = 0.67.

3. Let's calculate the activity of ions:

A K+= cf= 0.001 0.90 = 0.0009 = 9.0 10 -4 (g-ion/l)

a Al 3+ = cf= 0.001 0.44 = 0.00044 = 4.4 10 -4 (g-ion/l)

a SO 2- 4 = 2cf= 2 0.001 0.67 = 0.00134 = 1.34 10 -3 (g-ion/l)

In cases where more rigorous calculations are required, the activity coefficients are found either using the Debye-Hückel formula or by interpolation according to Table 2.2.

Solution of Example 4 using the interpolation method.

1. Find the activity coefficient of potassium ions fK+.

When the ionic strength of the solution is 0.005, fK+ is equal to 0.925, and with the ionic strength of the solution equal to 0.01, fK+, is equal to 0.900. Therefore, the difference in the ionic strength of the solution m, equal to 0.005, corresponds to the difference fK+, equal to 0.025 (0.925-0.900), and the difference in ionic strength m , equal to 0.004 (0.009 - 0.005), corresponds to the difference fK+, equal X.

From here, X= 0.020. Hence, fK+ = 0,925 - 0,020 = 0,905

2. Find the activity coefficient of aluminum ions f Al 3+ . At an ionic strength of 0.005, f Al 3+ is 0.51, and at an ionic strength of 0.01, f Al 3+ is 0.44. Therefore, a difference in ionic strength m equal to 0.005 corresponds to a difference f Al 3+ equal to 0.07 (0.51 - 0.44), and the difference in ionic strength m equal to 0.004 corresponds to the difference f Al 3+ equal X.

where X= 0.07 0.004/ 0.005 = 0.056

Means, f Al 3+ = 0.510 - 0.056 = 0.454

We also find the activity coefficient of sulfate ions.