Strong electrolytes. Activity. Ionic strength. Nutrition rules: daily calorie requirements, energy balance Ion activity is determined by the formula
Electrochemistry
Ion activity. Ionic strength of the solution. Dependence of the ion activity coefficient on the ionic strength of the solution. Debye-Hückel theory.
Activity (ions) - effective concentration, taking into account the electrostatic interaction between ions in solution. Activity differs from concentration by some amount. The ratio of activity (a) to the concentration of a substance in solution (c, in g-ion / l) is called the activity coefficient: γ \u003d a / c.
Ionic strength of solution - measure of intensity electric field created by ions in solution. Half the sum of the products of the concentration of all ions in a solution and the square of their charge. The formula was first derived by Lewis:
where cB are the molar concentrations of individual ions (mol/l), zB are the ion charges
The summation is carried out over all types of ions present in the solution. If two or more electrolytes are present in the solution, then the total total ionic strength of the solution is calculated. For electrolytes in which multiply charged ions are present, the ionic strength usually exceeds the molarity of the solution.
The ionic strength of the solution has great importance in the Debye-Hückel theory of strong electrolytes. The basic equation of this theory (Debye-Hückel limiting law) shows the relationship between the ion activity coefficient ze and the ionic strength of the solution I in the form: solvent constant and temperature.
The ratio of activity (a) to the total concentration of a substance in solution (c, in mol / l), that is, the activity of ions at a concentration of 1 mol / l, is called activity factor :
In infinitely dilute aqueous solutions of non-electrolytes, the activity coefficient equal to one. Experience shows that as the concentration of the electrolyte increases, the values of f decrease, pass through a minimum, and then increase again and become significantly greater than unity in strong solutions. Such behavior of the dependence of f on concentration is determined by two physical phenomena.
The first is especially pronounced at low concentrations and is due to the electrostatic attraction between oppositely charged ions. Attractive forces between ions prevail over repulsive forces, i.e. in solution, a short-range order is established, in which each ion is surrounded by ions of the opposite sign. The consequence of this is an increase in the bond with the solution, which is reflected in a decrease in the activity coefficient. Naturally, the interaction between ions increases with increasing their charges.
With increasing concentration, the activity of electrolytes is increasingly influenced by the second phenomenon, which is due to the interaction between ions and water molecules (hydration). In this case, in relatively concentrated solutions, the amount of water becomes insufficient for all ions and gradual dehydration begins, i.e. the connection of ions with the solution decreases, therefore, the activity coefficients increase.
Some regularities concerning activity coefficients are known. So, for dilute solutions (up to approximately m = 0.05), the relation 1 - f = k√m is observed. In somewhat more dilute solutions (m ≈ 0.01), the values of f do not depend on the nature of the ions. This is due to the fact that the ions are located at such distances from each other, at which the interaction is determined only by their charges.
At higher concentrations, along with the charge, the activity value begins to be affected by the radius of the ions.
To assess the dependence of activity coefficients on concentration in solutions where several electrolytes are present, G. Lewis and M. Randall introduced the concept of ionic strength I, which characterizes the intensity of the electric field acting on ions in a solution. The ionic strength is defined as half the sum of the terms obtained by multiplying the molalities of each ion, mi, by the square of its valence, Zi:
I = 1/2∑miZi. (IX.18)
DEBYE-HUKKEL THEORY , statistical theory of dilute solutions of strong electrolytes, which allows you to calculate the coefficient. ion activity. It is based on the assumption of complete dissociation of the electrolyte into ions, which are distributed in the solvent, considered as a continuous medium. Each ion by the action of its electric charge polarizes the environment and forms around itself a certain predominance of ions of the opposite sign - the so-called. ionic atmosphere. In the absence of external electric field ionic atmosphere has a spherical. symmetry and its charge is equal in magnitude and opposite in sign to the charge of the center that creates it. and she. Potential j total electric. fields created by the center. ion and its ionic atmosphere at a point located at a distance r from the center. ion, m.b. calculated if the ionic atmosphere is described continuous distribution charge density r near the center. and she. For the calculation, the Poisson equation is used (in the SI system):
n2j = -r/ee0,
where n2 is the Laplace operator, e is the dielectric. solvent permeability, e0 - electric. constant (vacuum permittivity). For each i-th kind of ions, r is described by the function of the Boltzmann distribution; then, in the approximation that considers ions as point charges (the first approximation of D.-H.T.), the solution of the Poisson equation takes the form: 
where z is the charge number center. ion, rd - so-called. Debye screening radius (radius of the ionic atmosphere). At distances r > rd, the potential j becomes negligible, i.e., the ionic atmosphere shields the electric. center field. and she.
In the absence of an external electric field, the ionic atmosphere has spherical symmetry, and its charge is equal in magnitude and opposite in sign to the charge of the central ion that creates it. In this theory, almost no attention is paid to the formation of pairs of oppositely charged ions by direct interaction between them.
Any physical or mental activity requires energy, so the calculation of the daily calorie intake per day for a woman or man should take into account not only gender, weight, but also lifestyle.
We spend energy daily on metabolism (metabolism at rest) and on movement (exercise). Schematically it looks like this:
Energy \u003d E basal metabolism + E physical activity
Basal metabolic energy, or basal metabolic rate (BRM)- Basal Metabolic Rate (BMR) - this is the energy needed for the life (metabolism) of the body without physical activity. A basic level of metabolic rate, which depends on the weight, height and age of the person. The taller a person, and the greater his weight, the more energy is needed for metabolism, the higher the basic metabolic rate. Conversely, lower, 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 with a weight of 70 kg, a height of 170 cm, 28 years old, requires for basic metabolism (basal metabolism)
= 447,593 + (9.247 * 70) + (3,098 *170) - (4.330 *28)
\u003d 447.593 + 647.29 + 526.66–121.24 \u003d 1500.303 kcal
You can also check the table: Daily energy consumption of the adult population without physical activity according to the norms of the physiological needs of the population in 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 expended by the body per day
Recall that total energy is the sum of basal metabolic energy (or basal metabolic rate) and energy that goes into movement (physical activity).
To calculate the total energy expenditure, taking into account physical activity, there is Physical activity coefficient.
What is Physical Activity Factor (CFA)
Physical activity coefficient (CFA) \u003d 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 base metabolic rate.
The more intense the physical activity, the higher the coefficient of physical activity will be.
- People who move very little have CFA = 1.2. For them, the total energy expended by the body will be calculated: E \u003d BRM * 1.2
- People who do light exercise 1-3 days a week have a CFA of 1.375. So the formula: E \u003d BRM * 1.375
- People who perform moderate exercise, namely 3-5 days a week, have a CFA of 1.55. Formula for calculation: E \u003d BRM * 1.55
- People who do heavy exercise 6-7 days a week have a CFA of 1.725. Formula for calculation: E \u003d BRM * 1.725
- People who do very heavy exercise twice a day, or workers with large physical activity, have CFA=1.9. Accordingly, the formula for calculating: E \u003d BRM * 1.9
So, in order to calculate the total amount of energy spent per day, it is necessary to multiply the basal metabolic rate according to age and weight (basal metabolic rate) by the coefficient of physical activity 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 that enters the body and the energy that the body spends.
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 the consumed food is greater than the energy needed for the life of the body. In a state of positive energy balance, a person gains extra pounds.
Negative energy balance is when less energy is received than the body has expended. To lose weight, you need to create a negative energy balance.
Solutions of strong electrolytes do not obey the law of mass action, as well as the laws of Raoult and van't Hoff, because 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 to introduce 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 the 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 \u003d c and is expressed in grams of 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 bonding of electrolyte particles. Another concept is connected with the concept of "activity" - "activity coefficient" ( f), which characterizes the degree of deviation of the properties of real solutions from the properties of ideal solutions; it is a value that reflects all the phenomena occurring in the 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 solution, and the analytical concentration is the total molar concentration of ions, determined, for example, by titration.
The activity coefficient of ions depends not only on the concentration of ions 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 the solution (m,) is the magnitude of the electric field in the solution, which is a measure of the electrostatic interaction between all ions in the solution. It is calculated according to 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. Non-dissociated 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 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 KC1.
0.01; Z K= ZCl - = 1
Consequently,
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 a solution of barium nitrate Ba (NO 3) 2.
Dissociation scheme: Ba (NO 3) 2 ↔ Ba 2+ + 2NO 3 -
[Ba 2+] \u003d 0.005, \u003d 2 0.005 \u003d 0.01 (g-ion/l)
Consequently,
The ionic strength of a dilute electrolyte solution of the type KtAn 2 and Kt 2 An is: 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 type Kt 2+ An 2- is: m = 4 With.
In general, for an electrolyte of the type Kt n + a An m - b the ionic strength of the solution can be calculated by the formula: m = (a· · p 2 + b· · t 2),
where a, b- indices at ions, and n+ and t - - ion charges, and - ion concentrations.
If two or more electrolytes are present in the solution, then the total ionic strength of the solution is calculated.
Note. Reference books on chemistry give differentiated activity coefficients for individual ions or for groups of ions. (See: Lurie Yu. Yu. Handbook of analytical chemistry. M., 1971.)
With an increase in the concentration of the solution with complete dissociation of the 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 the 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 that have the same ionic strength. However, this law only applies to very dilute aqueous solutions, with ionic strength up to 0.02 g-ion/l. With a further increase in concentration, and consequently, 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 for different ionic strengths
At present, a table of approximate values of activity coefficients is used for analytical calculations.
The dependence of the activity coefficients of ions 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 BUT- multiplier, the value of which depends on temperature (at 15°C, BUT = 0,5).
At values of the ionic strength of the solution 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\u003d - 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 to calculate ion activities.
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 ionic strength closest to it, listed in Table 2.2, is 0.01. Therefore, without a large error, we can take for potassium ions f K += 0.90; for aluminum ions f Al 3+ = 0.44, and for sulfate ions f SO 2-4 = 0.67.
3. 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\u003d 0.001 0.44 \u003d 0.00044 \u003d 4.4 10 -4 (g-ion/l)
a SO2-4= 2cf\u003d 2 0.001 0.67 \u003d 0.00134 \u003d 1.34 10 -3 (g-ion/l)
In those 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.
Example 4 solution using the interpolation method.
1. Find the activity coefficient of potassium ions f K +.
With the ionic strength of the solution equal to 0.005, f K + is 0.925, and with the ionic strength of the solution equal to 0.01, f K +, is equal to 0.900. Therefore, the difference in the ionic strength of the solution m, equal to 0.005, corresponds to the difference f K +, 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. Consequently, f K + = 0,925 - 0,020 = 0,905
2. Find the activity coefficient of aluminum ions f Al3+. With an ionic strength of 0.005, f Al 3+ is 0.51, and with an ionic strength of 0.01, f Al 3+ is equal to 0.44. Therefore, the difference in ionic strength m, equal to 0.005, corresponds to the 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+ \u003d 0.510 - 0.056 \u003d 0.454
We also find the activity coefficient of sulfate ions.
Problem 529.
Calculate the approximate value of the ion activity K+ and SO 4 2- in 0.01 M K solution 2 SO 4 .
Solution:
Dissociation equation K 2 SO 4 has the form:
K 2 SO 4 ⇔ 2K + + SO 4 2-.
The activity of an ion (mol/l) is related to its molecular concentration in solution by the relation: = fCM.
Here f is the ion activity coefficient (dimensionless value), C M is the ion concentration. The activity coefficient depends on the charge of the ion and the ionic strength of the solution, which is equal to half the sum of the products of the concentration of each ion and the square of the charge of the ion:
The ionic strength of the solution is:
I = 0.5 = 0.5(0.02 . 1 2) + (0,01 . 2 2) = 0,03.
The activity coefficient of K + and SO 4 2- ions is found by the formula, we get:
Now we calculate the activity of K + and SO 4 2- ions from the relation = fCM we get:
(K+)=0.02 . 0.82 = 0.0164 mol/l; (SO 4 2-) = 0.01 . 0.45 = 0.0045 mol/l.
Answer:(K +) = 0.0164 mol/l; (SO 4 2-) \u003d 0.0045 mol / l.
Problem 530.
Calculate the approximate value of the activity of Ba 2+ and Cl - ions in 0.002 N. BaCl 2 solution.
Solution:
M (BaCl 2) \u003d C E (BaCl 2)
C M \u003d C H \u003d 2 .
0.002 = 0.004 mol/l.
The dissociation equation for barium chloride has the form:
BaCl 2 ⇔ Ba 2+ + 2Cl -.
The activity of an ion (mol/l) is related to its molecular concentration in solution by the relation: = fC M .
Here f is the ion activity coefficient (dimensionless value), C M is the ion concentration. The activity coefficient depends on the charge of the ion and the ionic strength of the solution, which is equal to half the sum of the products of the concentration of each ion and the square of the charge of the ion:
The ionic strength of the solution is:
I = 0.5 = 0.5(0.004 . 2 2) + (0,008 . 1 2) = 0,024.
The activity coefficient of Ba2+ and Cl- ions is found by the formula, we get:
Now we calculate the activity of Ba 2+ and Cl - ions from the relation = fC M we get:
(Ba2+) = 0.004 . 0.49 = 0.0196 mol/l; (Cl-) = 0.008 . 0.84 = 0.00672 mol/l.
Answer:(Ba 2+) = 0.0196 mol/l; (Cl -) \u003d 0.00672 mol / l.
Problem 531.
Find the approximate value of the activity coefficient of a hydrogen ion in a 0.0005 M solution of H 2 SO 4 containing, in addition, 0.0005 mol/l HCI. Think that sulphuric acid completely dissociates in both steps.
Solution:
The total concentration of hydrogen ions is the sum of the H 2 SO 4 concentration and the HCl concentration. Acids dissociate according to the scheme:
H 2 SO 4 ⇔ 2H + + SO 4 2-;
HCl ⇔ H + + Cl -
It follows from the equations that the concentration of hydrogen ions in sulfuric acid is 2 times higher than that of acids and will be: 2 . 0.0005 = 0.001 mol/l. The total concentration of hydrogen ions in the solution will be:
0.001 + 0.0005 = 0.0015 mol/l.
The ion activity coefficient is calculated by the formula:
where f is the ion activity coefficient (dimensionless value), I is the ionic strength of the solution, Z is the charge of the ion. The ionic strength of the solution is calculated by the equation:
Here the concentration of the ion in the solution, we get:
I = 0.5 = 0.002.
Let us calculate the activity coefficient of the hydrogen ion.
Despite the fact that thermodynamics does not take into account the processes that occur 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) - such a concentration of a substance in a solution, using which the properties of a given solution can be described by the same equations as the properties of an ideal solution.
The activity can be either less or more than the nominal concentration of the substance in the solution. The activity of a pure solvent, as well as a solvent in not too concentrated solutions, is taken equal to 1. The activity of a solid substance in the precipitate, or a liquid immiscible with a given solution, is also taken as 1. In an infinitely dilute solution, the activity of the solute is the same as its concentration.
The ratio of the activity of a substance in a given solution to its concentration is called activity factor.
The activity coefficient is a kind of correction factor that shows how much the 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 temperatures, vapor pressure over 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 the solution on the activity of the solute is called ionic strength ( I ).
Example 3.1. At 1.00 l aqueous solution contains 10.3 g NaBr, 14.2 g Na 2 SO 4 and 1.7 g NH 3 . What is the ionic strength of this solution?
0.100 mol/l
0.100 mol/l
C (Na +) \u003d 0.300 mol / l, C (Br -) \u003d 0.100 mol / l, C (SO 4 2-) \u003d 0.100 mol / l
I = 0.5× = 0.400 mol/l
Rice. 3.1. Effect of ionic strength on the mean ionic activity coefficient of HCl
On fig. 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 the ionic strength is monotonically decreasing.
There is no universal equation by which it would be possible to calculate the activity coefficient of any electrolyte at any value of 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 this 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 the ionic strength, the activity coefficient of the ion does not depend on its nature.
As the ionic strength increases to 0.01 or more, the limiting law begins to give more and more errors. This is because real ions have a certain size, so they cannot be packed as tightly as point charges. With an increase in the concentration of ions, 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 the activity coefficients for 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 Œ), a is an empirical constant characterizing the dimensions of the ionic atmosphere.
At higher values of ionic strength (up to ~1), the quantitative assessment of the activity coefficient can be carried out according to the Davis equation.
