Approximate formulas for calculating the activity coefficients f for different ionic strengths τ of the solution. Nutrition rules: daily calorie requirements, 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 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

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 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)

Hence,

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 by 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. Hence, 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.

Activity components of the solution is the concentration of the components, calculated taking into account their interaction in the solution. The term "activity" was proposed in 1907 by the American scientist Lewis as a quantity, the use of which will help to describe the properties of real solutions in a relatively simple way.

Instruction

There are various experimental methods for determining the activity of solution components. For example, by increasing the boiling point of the test solution. If this temperature (denoted as T) is higher than the boiling point of the pure solvent (To), then the natural logarithm of the activity of the solvent is calculated by the following formula: lnA = (-? H / RT0T) x? T. Where, ?H is the heat of evaporation of the solvent in the temperature range between To and T.

You can determine the activity of the components of the solution by lowering the freezing point of the test solution. In this case, the natural logarithm of the solvent activity is calculated using the following formula: lnA = (-?H/RT0T) x?T, where, ?H is the freezing heat of the solution in the range between the freezing point of the solution (T) and the freezing point of the pure solvent (To ).

Calculate the activity using the method of studying the equilibrium of a chemical reaction with a gas phase. Suppose you are undergoing a chemical reaction between a melt of some metal oxide (denoted by the general formula MeO) and a gas. For example: MeO + H2 = Me + H2O - that is, the metal oxide is reduced to pure metal, with the formation of water in the form of water vapor.

In this case, the equilibrium constant of the reaction is calculated as follows: Кр = (pH2O x Ameo) / (рН2 x Ameo), where p is the partial pressure of hydrogen and water vapors, respectively, A are the activities of the pure metal and its oxide, respectively.

Calculate the activity by calculating the electromotive force of a galvanic cell formed by an electrolyte solution or melt. This method is considered one of the most accurate and reliable for determining activity.

The turnover of capital is the speed at which funds pass through the various stages of production and circulation. The greater the velocity of capital circulation, the more profit the organization will receive, which indicates the growth of its business activity.

Instruction

Asset turnover in turnover is calculated by dividing the amount of revenue by the average annual value of assets.

where A is the average annual value of assets (total capital) -
B - revenue for the analyzed period (year).

The found indicator will indicate how many turnovers are made by the funds invested in the property of the organization for the analyzed period. With the growth of the value of this indicator, the business activity of the company increases.

Divide the duration of the analyzed period by the turnover of assets, thereby you will find the duration of one turnover. When analyzing, it should be taken into account that the lower the value of this indicator, the better for the organization.

Use tables for clarity.

Calculate the coefficient of fixing current assets, which is equal to the average sum of current assets for the analyzed period, divided by the organization's revenue.

This ratio indicates how much working capital is spent on 1 ruble of sold products.

Now calculate the duration of the operating cycle, which is equal to the duration of the turnover of raw materials, plus the duration of the turnover of finished products, plus the duration of the turnover of work in progress, as well as the duration of the turnover of receivables.

This indicator should be calculated for several periods. If a trend towards its growth is noticed, this indicates a deterioration in the state of the company's business activity, because. at the same time, the turnover of capital slows down. Therefore, the company's need for cash increases, and it begins to experience financial difficulties.

Remember that the duration of the financial cycle is the duration of the operating cycle minus the duration of the accounts payable turnover.

The lower the value of this indicator, the higher the business activity.

The coefficient of stability of economic growth also affects the turnover of capital. This indicator is calculated according to the formula:

(Chpr-D)/ Sk

where Npr - net profit of the company;
D - dividends;
Sk - own capital.

This indicator characterizes the average growth rate of the organization. The higher its value, the better, as it indicates the development of the enterprise, the expansion and growth of opportunities to increase its business activity in subsequent periods.

Helpful advice

The concept of "activity" is closely related to the concept of "concentration". Their relationship is described by the formula: B \u003d A / C, where A is activity, C is concentration, B is “activity coefficient”.

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.

How many calories should you eat per day

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. The basic metabolic rate is a value that 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
\u003d 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) = 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 hard exercise twice a day, or hard workers, have a CFA of 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 food consumed 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.

A comprehensive analysis of quite numerous methods for calculating activity is one of the main sections of the modern thermodynamic theory of solutions. The necessary information can be found in the dedicated manuals. Only some of the simplest methods for determining activity are briefly considered below:

Calculation of the activity of solvents from the pressure of their saturated vapors. If the volatility of the pure phase of the solvent and its decrease caused by the presence of dissolved substances are sufficiently studied, then the activity of the solvent is calculated directly from the ratio (10.44). The saturation vapor pressure of a solvent often differs significantly from volatility. But experience, and theoretical considerations, show that the deviation of vapor pressure from volatility (if we talk about the ratio remains approximately the same for solutions of not too high concentration. Therefore, approximately

where is the saturation vapor pressure over the pure solvent, whereas the saturation vapor pressure of the solvent over the solution. Since the decrease in saturated vapor pressure over solutions has been well studied for many solvents, the ratio turned out to be practically one of the most convenient for calculating the activity of solvents.

Calculation of the activity of a solute from equilibrium in two solvents. Let substance B be dissolved in two solvents that do not mix with each other. And suppose that activity (as a function of B concentration) is studied; let us denote it Then it is not difficult to calculate the activity of the same substance B in another solvent A for all equilibrium concentrations. It is clear that in this case it is necessary to proceed from the equality of the chemical potentials of substance B in equilibrium phases. However, the equality of potentials does not mean that the activities are equal. Indeed, the standard states of B in solutions are not the same; they differ in different energies of interaction of particles of substance B with solvents, and these standard states, generally speaking, are not in equilibrium with each other. Therefore, the volatilities B in these standard states are not the same, but for the equilibrium concentrations we are considering and A, the volatilities B in these phases are identical. Therefore, for all equilibrium concentrations, the ratio of activities is inversely proportional to the ratio of volatilities B in standard states

This simple and convenient method of calculating the activity of a substance in one solvent from the activity of the same substance in another solvent becomes inaccurate if one of these solvents is noticeably miscible with the other.

Determination of the activity of metals by measuring the electromotive force of a galvanic cell. Following Lewis [A - 16], let us explain this by the example of solid solutions of copper and silver. Let one of the electrodes, a galvanic cell, be made of completely pure copper, and the other

electrode - from a solid solution of copper and silver concentration of copper of interest to us. Due to the unequal values of the chemical potential of copper in these electrodes, an electromotive force arises that, with the valency of the current carriers of the electrolyte solutions of oxide copper for non-oxide copper, is related to the difference in the chemical potentials of copper by the relation

where is the Faraday number; activity of the pure phase of copper Taking into account the numerical values (10.51) can be rewritten as follows:

Calculation of the activity of the solvent from the activity of the solute. For a binary solution (substance B in solvent A) according to the Gibbs-Duhem equation (7.81) with and taking into account (10.45)

Since in this case then and therefore

Adding this relation to (10.52), we get

Integrating this expression from the pure phase of the solvent when to the concentration of the solute Considering that for the standard state of the solvent we find

Thus, if the dependence of the activity of the solute B on its mole fraction is known, then by graphical integration (10.52) it is possible to calculate the activity of the solvent.

Calculation of the activity of a solute from the activity of the solvent. It is easy to see that to calculate the activity of a solute, the formula is obtained

symmetrical (10.52). However, in this case, it turns out that graphical integration is difficult to perform with satisfactory accuracy.

Lewis found a way out of this difficulty [A - 16]. He showed that the substitution of a simple function

reduces formula (10.53) to a form convenient for graphical integration:

Here is the number of moles of substance B in solvent A. If the molecular weight of the solvent, then

Calculation of the activity of the solvent from the solidification points of the solution. Above, the dependence of activity on the composition of solutions was considered, and it was assumed that the temperature and pressure are constant. It is for the analysis of isothermal changes in the composition of solutions that the concept of activity is most useful. But in some cases it is important to know how activity changes with temperature. One of the most important methods for determining activities is based on the use of the temperature change in activity - by the solidification temperatures of solutions. It is not difficult to obtain the dependence of activity on temperature in differential form. To do this, it is enough to compare the work of changing the composition of the solution at from the standard state to concentration with the work of the same process at or simply repeat the reasoning that leads to formula (10.12) for volatility.

The chemical potential of solutions is analytically determined through activity in exactly the same way as for pure phases through volatility. Therefore, for activities, the same formula (10.12) is obtained, in which the place is occupied by the difference between the partial enthalpies of the component in the considered state and in its standard state:

Here, the derivative with respect to temperature is taken at a constant composition of the solution and a constant external pressure. If the partial heat capacities are known, then by the relation it can be assumed that after substitution into (10.54) and integration leads to the formula

Lewis showed by examples [A - 16] that for metallic solutions the approximate equation (10.55) is valid with an accuracy of several percent in the temperature range of 300-600 ° K.

We apply formula (10.54) to the solvent A of the binary solution near the point of solidification of the solution, i.e., assuming that the Higher

the melting point of the pure solid phase of the solvent will be denoted by and the decrease in the solidification point of the solution will be denoted by

If we take the pure solid phase as the standard state, then the value will mean the increment in the partial enthalpy of one mole of the solvent during melting, i.e., the partial heat

melting Thus, according to (10.54)

If we accept that

where is the molar heat of fusion of a pure solvent at the heat capacity of substance A in liquid and solid states, and if, when integrating (10.56), we use the expansion of the integrand in a series, we get

For water as a solvent, the coefficient at in the first term on the right side is equal to

Calculation of the activity of the solute from the solidification points of the solution. Just as it was done when deriving formula (10.52), we use the Gibbs-Duhem equation; we will apply it for a binary solution, but, unlike the derivation of formula (10.52), we will not pass from the number of moles to mole fractions. Then we get

Combining this with (10.56), we find

Further, we will keep in mind a solution containing the indicated numbers of moles in a solvent having a molecular weight. In this case, we note that for solutions in water, the coefficient at in (10.58) turns out to be equal. To integrate (10.58), following Lewis, an auxiliary quantity is introduced

(For solutions not in water, but in some other solvent, instead of 1.86, the corresponding value of the cryoscopic constant is substituted.) The result is [A - 16]

The activity of a radionuclide is the amount of radioactive material expressed as the number of decays of atomic nuclei per unit time.

The activity of a radionuclide in a source A p is defined as the ratio of the number dN 0 of spontaneous (spontaneous) nuclear transformations occurring in the source (sample) over a time interval dt:

BUT R = dN 0 / dt (5.12)

The unit of radionuclide activity is the becquerel (Bq). A becquerel is equal to the activity of a radionuclide in a source (sample) in which one spontaneous nuclear transformation occurs in 1 s.

Radionuclide activity BUT R(t) or the number of radioactive atoms of the nuclide N(t), decreases in time t according to the exponential law

BUT R(t)= BUT R 0exp(-λt)= BUT R 0exp(-0.693t/T 1/2) (5.13)

N(t)=N 0 exp(-λt)=N 0 exp(-0.693t/T 1/2) (5.14)

where BUT R 0, N 0 – activity of the radionuclide and the number of radioactive atoms of the nuclide in the source at the initial time t=0, respectively; λ – decay constant – the ratio of the fraction of nuclei dN/N of the radionuclide decaying over a time interval dt to this time interval: λ=-(1/N)(dN/dt); T 1/2 - half-life of a radionuclide - the time during which the number of nuclei of a radionuclide decreases by half as a result of radioactive decay; 0.693=ln2.

From the above definitions it follows that the activity of the radionuclide BUT R is related to the number of radioactive atoms in the source at a given time by the relation

BUT R\u003d λN \u003d 0.693N / T 1/2 (5.15)

Let us relate the mass m of the radionuclide in grams (excluding the mass of the inactive carrier) with its activity BUT R in becquerels. The number of radioactive atoms N corresponding to activity is determined from formula (5.15), where T 1/2 is expressed in seconds; the mass of one atom in grams m a \u003d A / N A, where A is the atomic mass, N A is the Avogadro constant.

m = Nm a =( BUT R T 1/2 / 0.693) * (A / N A) \u003d 2 * 40 * 10 -24 AT 1/2 BUT R (5.16)

From formula (5.16), one can also express the activity in becquerels of a radionuclide with mass m in grams:

BUT R\u003d 4.17 * 10 23 m / (A * T 1/2) (5.17).

Calculation of effective equivalent dose

Different organs or tissues have different sensitivity to radiation. It is known, for example, that with the same equivalent dose of radiation, the occurrence of cancer in the lungs is more likely than in the thyroid gland, and irradiation of the gonads (sex glands) is especially dangerous due to the risk of genetic damage. Therefore, in recent years, for cases of uneven irradiation of different organs or tissues of the human body, the concept of effective equivalent dose H E has been introduced.

Effective equivalent dose

Н Е = ∑ w i Н i , (5.18)

where H i is the average equivalent dose in the i-th organ or tissue; w i is a weighting factor, which is the ratio of the stochastic risk of death as a result of irradiation of the i-th organ or tissue to the risk of death from uniform irradiation of the body at the same equivalent doses (Table 5.11). Thus, w i determines the significant contribution of a given organ or tissue to the risk of adverse effects for the body under uniform irradiation:

∑w i = 1 (5.19)

Table 5.11 Weighting factors

Organ or tissue	Disease
	hereditary defects
mammary gland
red bone marrow	Leukemia

Thyroid
Bone surface	Malignant neoplasms
All other organs