Standard Deviation We are now ready to calculate the risk of asset A. To do this, the standard deviation is calculated, which serves as a measure of the spread of a set of numbers. Calculations can be done manually, but this is too tedious. They are usually produced using

Standard deviation

Example: copying multiple files to standard output

Example: Copying Multiple Files to Standard Output Program 2.3 illustrates the use of standard input/output devices and demonstrates how to improve error control and improve user experience. This

5.26. Variance and standard deviation

5.26. Dispersion and Standard Deviation Dispersion is a measure of the "scatter" of values ​​in a set. (Here we do not distinguish between biased and unbiased estimators.) The standard deviation, usually denoted by the letter ?, is equal to the square root of the variance. Data = (1)

where
- molar Gibbs free energy at standard pressure, J/mol; is the enthalpy of formation of a substance at T\u003d 0 K from simple chemical elements:

is a state function and depends only on temperature.

Take the derivative of () with respect to temperature at p=const:

(2)

In equation (2), the derivative of the Gibbs energy with respect to temperature is

, (3)

and the value is by definition equal to

(4)

Substituting (3) and (4) into (2) we get

(5)

(6)

The first derivative of the reduced Gibbs energy with respect to temperature gives the excess enthalpy. For practical problems, it is much more convenient to take the derivative with respect to the temperature logarithm, given that dT=Td ln T. Then we have

(7)

We write expression (6) as
(8)

The second derivative of by temperature at R=const gives heat capacity

=
(9)

or
(10)

Dependencies (6), (7), (9) and (10) for (
)/T and are used to obtain temperature approximations of the thermodynamic properties of individual substances. The molar entropy at standard pressure is also expressed in terms of the reduced Gibbs energy:

(11)

Representation of the thermodynamic properties of individual substances in the reference literature

In the reference book edited by V.P. Glushko for the 1st mole of each individual substance in the standard state, depending on the temperature, tables of values ​​are given in the interval t 0 from 100K to 6000K:

- isobaric heat capacity, J/molK;

is the reduced Gibbs energy, J/molK;

- entropy, J/molK;

- excess enthalpy, kJ/mol;

, where K 0 is the equilibrium constant of XP decay of a given substance AT into gaseous atoms, a dimensionless quantity. Substance decay formula:
, where - number of atoms in a molecule of matter AT.

For example:
.

Values ​​are given:

- thermal effect of the decomposition reaction of substance B into gaseous atoms at T 0 = 0K, kJ / mol;

- enthalpy of formation of a substance from pure chemical elements (heat effect of formation) at T 0 =0K, kJ/mol;

- enthalpy of formation of a substance at T 0 =298.15K, kJ/mol;

M - relative molecular weight, dimensionless quantity;

- the nuclear component of the entropy of a substance, which depends on the isotopic composition of the substance and does not change during XP, J/molK. The value does not affect Practical functions are given in the handbook without taking into account .

The handbook provides approximations of the reduced Gibbs energy as a function of temperature in the form of a polynomial for each individual substance.

Approximation ( T) depending on the temperature is represented as a polynomial:

where x = T 10 -4 K; φ , φ n (n=-2, -1, 0, 1, 2, 3) – approximation coefficients for the temperature range T min  T T max ,( T min = 500K, T max =6000K).

Using the approximation coefficients φ , φ n the excess enthalpy and heat capacity of a substance can be calculated:

as well as the molar entropy:
For a complete assignment of all thermodynamic properties of individual substances of chemically reacting systems at a temperature T for calculations on a computer when choosing T 0 = 298.15K, you must enter the following values:

those. 13 parameters in total, where .

When choosing T 0 = 0K value
and
should be excluded from the list. Then there are 11 parameters left:
(7 odds)



. Thus, in thermodynamic calculations of rocket and aircraft engines, it is advisable to choose the temperature of the enthalpy reference point T 0 = 0K.

9.4. Calculation of the Gibbs free energy and entropy of matter at a pressure different from the pressure under standard conditions

Molar enthalpy , heat capacity
and internal energy depend only on temperature:

Molar entropy , Gibbs free energy , Helmholtz free energy depend on temperature and pressure.

Let's establish a connection between the quantities:
and their values ​​in the standard state
which are determined using reference materials.

Let us first obtain an expression for the Gibbs free energy. From the combined expression of the 1st and 2nd laws of thermodynamics for a simple, closed TS and for reversible processes for 1 mole of a substance, we have:

At T= const( dT= 0) we get
, where
. Where after integration for the final process in the pressure range from R 0 to R we have

, or
(1)

where
-molar Gibbs free energy at R 0 \u003d 1 physical atm,
- the same with pressure
. Dependence (1) is valid for gaseous and condensed substances at T= const.

For an ideal gas,
. Consequently,
and the integral in (1) will be equal to
. Denoting through
dimensionless pressure; where R 0 = 101325Pa; ~ tilde, we obtain for an ideal gas a formula for calculating the Gibbs free energy at pressure p≠p 0:

If the substance is in a gas mixture, then for i th component of a mixture of ideal gases, we have:

where is the normalized partial pressure and normalized mixture pressure related by the ratio
, taking into account the molar fraction
i-th gas,
, and the pressure of the gas mixture is determined by the Dalton law
.To get the formula for calculation , expressed in terms of mole fractions, we represent formula (3) as:

Let us denote the Gibbs molar free energy i th gas at mixture pressure. Then we get

The molar Gibbs free energies of condensed substances do not depend on pressure, since their volumes can be neglected in comparison with the volumes of gaseous components. Then the formula for calculating
condensed substances will take the form:

where X i– mole fraction i-th substance relative to the phase in which it is located (to the number of moles of its phase),
is the molar Gibbs free energy of a pure condensed matter, at p= p 0 =101325Pa.

The effect of pressure on entropy can be determined from the expression for the Gibbs molar free energy for i th component of an ideal gas at pressure p ≠p 0

from which it follows that

(7)

After substituting (8) into (7) and taking into account that
, we get:

For the i-th component of the condensed matter, by analogy with expression (9), one can obtain a formula for calculating the entropy at p ≠p 0

Value - taken from the handbook R 0 =101325 Pa.

9.5. Calculation of Gibbs free energy for real gases and solutions. Volatility and activity

When calculating the Gibbs molar free energy for real gases and solutions, you can use the formulas obtained for ideal gases and solutions. At the same time, partial pressures p i are replaced by the value of volatility f i [Pa] and mole fractions x i- on activity a i. Volatility is the pressure, determined from the equation of state for real gases, which has the same effect on the system as in the case of an ideal gas. Actually f i is the corrected pressure, which characterizes the deviation of the thermodynamic system from the ideal state described by the equation of state for an ideal gas.

Thus, for real gases, the value of the Gibbs molar free energy will be determined by the expression

where
,
composition). As the state of a real gas approaches the state of an ideal gas, volatility tends to partial pressure . For an ideal gas f i = p i(at low pressures).

Activity a i(dimensionless quantity) is the corrected mole fraction x i, which characterizes the deviation of the condensed system from the ideal state. As the real solution approaches the ideal state, the activity a i tends to the molar fraction x i. For weak solutions a i =x i . Thus, for real solutions

The described method for calculating the Gibbs free energy was proposed by the American physical chemist Lewis G.N. (1875-1946).

In thermodynamics, the concepts of fugacity coefficients are also used
and activities
. For ideal gases and solutions
.

9.6. Third law of thermodynamics, and by isothermal expansion of the working fluid, since the working fluid ceases to give off heat to the environment, because states Calculation of pressure drop in system ship's gas outlet, when using a shore gas outlet for ...

Common abbreviations

d - gas, gaseous state of matter

g - liquid, liquid state of matter

t - solid state of matter (in this manual, t is equivalent to the crystalline state, since the non-crystalline state of a solid is not considered within the program)

aq is the dissolved state, and the solvent is water (from the word aqueous- water)

EMF - electromotive force

Comments

Standard state in thermodynamics. The standard states are as follows:

for a gaseous substance, pure or in a gas mixture, the hypothetical state of a pure substance in the gas phase, in which it has the properties of an ideal gas and standard pressure R°. In this manual, it is accepted R° \u003d 1.01325 × 10 5 Pa (1 atm).

for a pure liquid or solid phase, as well as for a liquid solution solvent - the state of a pure substance in the corresponding state of aggregation under standard pressure R°.

for a solute in a solid or liquid solution, the hypothetical state of that substance in a solution with a standard concentration FROM°, which has the properties of an infinitely dilute solution (for a given substance) under standard pressure R°. Standard concentration accepted FROM° \u003d 1 mol / dm 3.

Choice of stoichiometric coefficients. The stoichiometric coefficients of a chemical reaction show the molar ratio in which these substances react with each other. For example, in the reaction A + B \u003d Z, the stoichiometric coefficients of the reactants are equal (in absolute value), from which it follows that 1 mol A reacts without residue with 1 mol B to form 1 mol Z. The meaning of this entry will not change if choose any other equal coefficients. For example, the equation 2A + 2B = 2Z corresponds to the same stoichiometric ratio between the reactants. Therefore, in the general case, the coefficients n i any reaction are defined up to an arbitrary common factor. However, in different sections of physical chemistry, different conventions are adopted regarding the choice of this factor.

In thermochemistry, in the reactions of the formation of substances from simple substances, the coefficients are chosen so that the coefficient 1 stands in front of the formed substance. For example, for the formation of hydrogen iodide:

1/2H 2 + 1/2I 2 = HI

In chemical kinetics, the coefficients are chosen to match, if possible, the reaction orders for the respective reactants. For example, the formation of HI is first order in H 2 and first order in I 2 . Therefore, the reaction is written as:

H 2 + I 2 ® 2HI

In the thermodynamics of chemical equilibria, the choice of coefficients is generally arbitrary, but depending on the type of reaction, preference may be given to one or another choice. For example, to express the equilibrium constant of acid dissociation, it is customary to choose the coefficient in front of the acid symbol equal to 1. In particular, for the acid dissociation of hydrogen iodide, choose

HI H + + I –

(coefficient before HI is 1).

Concentration designations. With the same symbol, the concentration or content of a component in a mixture can have a different meaning. The concentration can be equilibrium (one that is reached at equilibrium), current (one that exists at a given time or at a given stage of the process) and gross or "analytical". These concentrations may vary. For example, if you prepare a solution of acetic anhydride (CH 3 CO) 2 O in water, taking 1 mol of 100% acetic anhydride and diluting it with water to 1 liter, then the resulting solution will have a gross or analytical concentration FROM\u003d 1 mol / l (CH 3 CO) 2 O. In fact, acetic anhydride undergoes irreversible hydrolysis to acetic acid (CH 3 CO) 2 O + H 2 O ® 2CH 3 COOH, therefore its current concentration decreases from 1 mol / l to initial time to an equilibrium concentration of approximately 0 mol/l at the end of the reaction. On the other hand, based on the complete hydrolysis of the anhydride, we can say that the total concentration of the solution is 2 mol/l CH 3 COOH (regardless of the stage of the hydrolysis process). However, the reaction product is subject to acid dissociation of CH 3 COOH CH 3 COO - + H +, so that the real concentrations in the solution, including the real concentration of CH 3 COOH, are not equal to any of the gross ones. The real concentrations of CH 3 COOH, CH 3 COO - and H + at equilibrium are called equilibrium. Chemists often use the same notation FROM for all these kinds of concentrations, assuming that the meaning of the designation is clear from the context. If you want to emphasize the difference, then the following notation is usually used for molar concentrations : FROM is the gross or analytical concentration, [A] is the current or equilibrium concentration of component A, and (sometimes) [A]e is the equilibrium concentration of component A. This index makes the writing of equilibrium constants, such as

The standard thermodynamic state was introduced as a common origin of volatility for all gases.

Since the properties of all gases are different, in real conditions they cannot have common points on the curve f=f(P). Consequently, the state common to all gases can only be imaginary.

It is most convenient to assume that all the properties of various gases will coincide if they turn (imaginatively!) into ideal gases.

Historically, for decades the unit of pressure has been atmosphere(atm.) , where 1 atm is equal to 1.01325×10 5 Pa. It is easy to understand that in the standard state the gas must be at this pressure.

Although the system of units changed in subsequent years, the pressure of an ideal gas in the standard state remained the same, i.e. equal to 1 atm.

The definition of the standard thermodynamic state for gases is:

The standard thermodynamic state of a gas at a given temperature is an imaginary state in the form of an ideal gas at a pressure of 1.01325×10 5 Pa.

Let us consider the process of gas transition from the standard state to the given state, which corresponds to the volatility f.

We will adhere to the following mandatory condition:

All quantities related to the standard state or counted from it are denoted by the symbol o, which is placed at the top right of the determined value..

For this reason, in the standard state, the pressure and the volatility equal to it will be denoted as follows: f o = P o =1.01325×10 5 Pa.

The first stage of the transition from the standard state to the given state of the gas involves the expansion of the gas. Since in the standard state it is endowed with the properties of an ideal gas, then its expansion (we should not forget that we are talking about an isothermal function) must occur along the isotherm of an ideal gas to a very small pressure P* or fugacity f*. The change in the Gibbs energy at this stage is

At very low pressures, the properties of a real gas actually coincide with those of an ideal gas. Therefore, there is no difference between the ideal gas isotherms and real gas isotherms under these conditions. In this regard, the transition from the ideal gas isotherm to the real gas isotherm will not cause any changes in the system. Consequently, for the second stage of the process, the change in the Gibbs energy will be equal to zero.

The third stage is the isothermal compression of a real gas from fugacity f* to fugacity in a given state f. The change in the Gibbs energy at this stage is

The total change in the Gibbs energy as a result of all stages is

The main thermodynamic functions used in metallurgical calculations are the internal energy u, enthalpy H, entropy S, as well as their most important combinations: isobaric-isothermal G = H - TS and isochoric-isothermal F=U-TS potentials, reduced potential F \u003d -G / T.

According to the Nernst theorem for entropy the natural reference point is zero degrees on the Kelvin scale, at which the entropies of crystalline substances are equal to zero. Therefore, from a formal standpoint, in principle, one can always measure or calculate the absolute value of entropy and use it for quantitative thermodynamic estimates. That is, entropy does not introduce any difficulties into the practice of performing numerical thermodynamic calculations.

But internal energy has no natural origin, and its absolute value simply does not exist. The same is true for all other thermodynamic functions or potentials, because they are linearly related to internal energy:

H = U + PV;

F = U - TS;

G = H - TS = U - TS + PV;

F= -G/T = S - H/T = S -(U+PV)/T.

Therefore, the values U, H, F, G and F thermodynamic system due to the uncertainty of the reference point can only be established up to constants. This fact does not lead to fundamental complications, because for solving all applied problems enough to knowchange quantities thermodynamic functions when changing temperature, pressure, volume, during the passage of phase and chemical transformations.

But in order to be able to carry out real calculations, it was necessary to adopt certain agreements (standards) on the unambiguous choice of certain constants and establish uniform rules for calculating the initial values ​​of thermodynamic functions for all substances found in nature. Due to the linear dependence of the thermodynamic functions H, F, G, F from internal energy U this is enough do for only one of these functions. was real unified origin of valuesenthalpy . Made it giving zero value to the enthalpies of certain substances in certain states under precisely specified physical conditions, which bear the name standard substances, standard conditions and standard states.

The following is the most common set of conventions under discussion as recommended by the International Commission on Thermodynamics of the International Union of Pure and Applied Chemistry (IUPAC). This set can be called thermodynamic standards, as practically established in the modern literature on chemical thermodynamics.

Standard Conditions

According to Nernst's theorem, for entropy, the natural reference point, or natural standard temperature, is zero degrees on the Kelvin scale, at which the entropies of substances are zero. In some reference books, published mainly in the USSR, the standard temperature is 0 K. Despite the great logic from the physical and mathematical points of view, this temperature is not widely used as a standard. This is due to the fact that at low temperatures the dependence of heat capacity on temperature is very complex, and it is not possible to use sufficiently simple polynomial approximations for it.

Standard physical conditions correspond to a pressure of 1 atm(1 physical atmosphere = 1.01325 bar)and temperature 298.15 K(25° FROM). It is believed that such conditions are most consistent with the actual physical conditions in chemical laboratories in which thermochemical measurements are carried out.

Standard Substances

In nature, all isolated, independent substances, called in thermodynamics individual , consist of pure elements of the table of D.I. Mendeleev, or are obtained by chemical reactions between them. That's why sufficient condition to establish a reference frame for thermodynamic quantities is the choice of enthalpies only for chemical elements as simple substances. It is accepted that the enthalpies of all elements in their standard states are zero under standard conditions – temperature and pressure. Therefore, the chemical elements in thermodynamics are also called standard substances.

All other substances are considered as compounds obtained by chemical reactions between standard substances (chemical elements in the standard state) They are called " individual substances ". The starting point for the enthalpies for chemical compounds (as well as for elements in non-standard states) is the value of the enthalpy of the reaction of their formation from standard substances, as if carried out under standard conditions. In fact, of course, the thermal effect (enthalpy) of the reaction is determined experimentally under real conditions, and then recalculated to standard conditions. This value is taken as standard enthalpy of formation chemical compound as an individual substance.

In practical calculations, it should be remembered that in thermochemistry the following is accepted as a standard sign rule to characterize the enthalpy. If, during the formation of a chemical compound, heat stands out, the sign ” is selected minus” - heat is lost to the system during the isothermal process. If heat is needed to form a chemical compound absorbed, the sign ” is selected a plus” - heat is supplied to the system from the environment to maintain isothermality.

Standard States

For such a state, the equilibrium state is chosen, i.e. most stable form of existence (aggregate state, molecular form) chemical element under standard conditions For example, these are elements in the solid state - lead,  carbon in the form of graphite, in liquid - mercury and bromine, diatomic molecules of gaseous nitrogen or chlorine, monatomic noble gases, etc.

Standard notations

To denote any thermodynamic property calculated at standard pressure from a standard value and therefore called standard property, the upper right index 0 (zero) of the character is used. That the property is counted down from the selected standard, indicated by the “” sign in front of the algebraic symbol of the thermodynamic function. The temperature corresponding to the value of the function is often given as a right subscript. For example, standard enthalpy substances at 298.15 K is denoted as

The standard enthalpies of individual substances are taken to be the heats of their formation by chemical reactions from standard substances in the standard state. Therefore, thermodynamic functions are sometimes denoted using the index f(from English formation- education):

Unlike enthalpy, for entropy its absolute value is calculated at any temperature. Therefore, there is no “” sign in the designation of entropy:
standard entropy substances at 298.15 K, standard entropy at temperature T.

Standard properties of substances under standard conditions, i.e. standard thermodynamic functions summarized in tables of thermochemical quantities and published as handbooks of thermochemical quantities of individual substances.

Isobaric processes are most often encountered in reality, since technological processes tend to be carried out in devices that communicate with the atmosphere. Therefore, reference books of thermochemical data for the most part contain, as necessary and sufficient information for calculating any thermodynamic function, quantity

If the values ​​of the standard absolute entropy and enthalpy of formation are known, as well as dependence of heat capacity on temperature, it is possible to calculate the values ​​or changes in the values ​​of all other thermodynamic functions.