Open Library - an open library of educational information. Chemical potentials of ideal gases From two or more components
chemical potential.
It is important to note that for a system consisting of one substance, it is true:
Any extensive state function is a function of the amount of matter in the system.
For this reason, if the system consists of several components, then
where n i is the number of moles i-th component. Differentiate (62) with respect to n i
p, Т, n j≠i =const
Gibbs named the quantity chemical potential and denoted μ i It is also called Gibbs partial molar energy(partial thermodynamic Gibbs function)
We can give the following definition of chemical potential:
This change in the Gibbs energy of a homogeneous multicomponent system when 1 mole of a given component is added to it at constant pressure, temperature and composition of the system (ᴛ.ᴇ. addition must occur with infinitely large amounts of all components so that the composition of the system does not change).
The chemical potential, unlike, for example, the Gibbs energy G, is an intense quantity, ᴛ.ᴇ. it does not depend on the mass of the system, but depends on the nature of the system and its composition, temperature, and pressure. Generally speaking, m i depends on the strength of the chemical interaction of a given component with other components: the stronger this interaction, the smaller m i . The strength of the interaction depends on the concentration of the component, and the lower the concentration of the i-th component, the stronger the interaction, and the less m i . The substance tends to move from a state where its m is greater to a state where its m is less (ᴛ.ᴇ. to where the interaction of this component with other components is stronger).
Any energy characteristic is the product of an intensive factor by an extensive one. In our case, μ i is an intensive parameter, and n i is an extensive one. Then:
For T, p = const. (64)
The introduction of a certain amount of dn i moles of the i-th component with a constant amount of other components and constant Т and р will increase the value of the Gibbs energy by . Similar changes will be caused by the addition of other components. The total change in the Gibbs energy of the system when several components are added to it is:
or, in general,
This equation is called fundamental Gibbs equation.
We integrate relation (64) at a constant composition of the system (ᴛ.ᴇ. when m i = const):
Relation (67) is sometimes called Gibbs-Duhem equation(more often this equation is written as follows:
Where x i is the mole fraction of the i-th component.)
When p, T \u003d const for a chemical reaction, it is true:
Calculation of the chemical potential of an ideal gas:
If we have one pure component, then its chemical potential m is equal to the Gibbs molar energy:
(Here and are the molar volume and the molar entropy of the substance), then we get:
Let the ideal gas be at T = const, then
Let us integrate expression (71) from p 0 = 1 atm to any p and, accordingly, from m 0 to m; we get:
But for an ideal gas, the Mendeleev–Clapeyron law is fulfilled, which for 1 mole of gas has the form:
p= RT, hence = . (73)
Then we get:
If p 0 = 1 a tm, then
In equation (75), p is not the pressure itself, but dimensionless a value numerically equal to pressure expressed in atmospheres ().
m 0 - standard chemical potential, ᴛ.ᴇ. chemical potential at standard pressure p 0 = 1 atm;
If there is a mixture of gases, then for any i-th component of the mixture.
m i = m 0 i + RT ln (76)
Here, is a dimensionless quantity numerically equal to the partial pressure of the i-th component of the mixture (ᴛ.ᴇ. that part of the total pressure that falls on the i-th component), expressed in atmospheres ().
Since , where is the mole fraction of the i-th gas in the mixture, p is the total pressure in the system, then
chemical potential. - concept and types. Classification and features of the category "Chemical potential." 2017, 2018.
Thermodynamics of phase transitions. Definitions Consider the thermodynamics of systems in which phase transitions can take place. A thermodynamic system that can exchange matter with its environment is called an open system. - Thermodynamic ... .
; Chemical potential is not given in reference tables. It serves as evidence. The chemical potential of a substance in a solution depends on the concentration: m(X) = m°(X) + RTlnc(X) This is a heuristic equation proposed in a logical way for ideal solutions. With him... .
If the total pressure of the gas mixture is small, then each gas will exert its own pressure, moreover, such as if it alone occupied the entire volume. This pressure is called partial pressure. The total observed pressure p is equal to the sum of the partial pressures of each gas....
Consider a thermodynamic system that is an ideal gas. The chemical potential of an ideal gas is: , where is the Gibbs molar energy (isobaric potential of 1 mole of an ideal gas). Since, then, where is the molar volume of an ideal gas (the volume of 1 mol of gas). ... .
Enthalpy, the Gibbs thermodynamic function, If, when a certain amount of heat is transferred to a gas, it expands isobarically, then the first law of thermodynamics for an elementary process in this case can be written as: . The value under the sign ... .
Partial derivatives of extensive properties with respect to n at constant P. T, V, n are called partial quantities. Depending on the units in which the mass of the component is expressed, molar and specific partial quantities are distinguished. Thus, &... .
Electrode processes. The concept of potential jumps and electromotive force (EMF). Electrochemical circuits, galvanic cells. Standard hydrogen electrode, standard electrode potential. Classification of electrochemical circuits and electrodes. LECTURE... .
Free energy of the system (Gibbs energy G and Helmholtz energy F) depends on external conditions:
This dependence is complete for the simplest systems consisting of one component.
A thermodynamic system can consist of one or more components. Obviously, the value of the free energy of a multicomponent system will depend both on external conditions ( T,R, or V), and on the nature and quantity of substances that make up the system, i.e., free energy, like any thermodynamic function, is an extensive property of the system. If the composition of the system changes with time (i.e., a chemical reaction occurs in the system), it is necessary to take into account the effect of the change in composition on the value of the free energy of the system.
Consider a thermodynamic system consisting of k components. Let n 1 , n 2 , …, n k- the number of moles of the 1st, 2nd, ..., k th components. Then the Gibbs free energy is a function of the following variables:
G = f(p, T, n 1 , n 2 , …, n k )
Differentiate with respect to all variables:
(2.1)
Let's introduce the notation:
………………….
,
where μ 1 , μ 2 , …, μ k– chemical potentials of the 1st, 2nd, …, k th components, respectively.
In general
.
We obtain a similar expression for the Helmholtz free energy:
F = f(V, T, n 1 , n 2 , …, n k )
.
Thus, chemical potential is the partial derivative of free energy with respect to the number of moles i-th component with the constancy of the corresponding external parameters and the number of moles of all other components. Thus, the chemical potential is the partial molar Gibbs energy (at R, T = const):
.
Free energy is a general property of the system, the chemical potential characterizes the properties of an individual component included in the system. The chemical potential is an intensive property of the system, since does not depend on the mass of the system.
At p, T =const equation (2.1) has the form:
.
(2.2)
In this case, the change in the Gibbs energy, that is, the useful work of the system, is due only to a change in the composition of the system as a result of a chemical reaction or due to the exchange of matter between the system and the environment.
Equation (2.2) expresses the relationship between the general property of the system and the properties of each of its components. Assuming that the chemical potential is a constant value, we integrate equation (2.2):
.
The integration constant is zero because if all n i= 0, the Gibbs energy is also equal to zero.
For an individual substance
,
those. the chemical potential of an individual substance is equal to the Gibbs molar energy.
At equilibrium dG = 0 and equation (2.1) takes the form:
. (2.3)
The resulting equation is a general equilibrium condition in a system with variable composition at R,T =const.
Chemical potential of an individual ideal gas.
Consider a thermodynamic system that is an ideal gas. The chemical potential of an ideal gas is:
,
where 
is the Gibbs molar energy (isobaric potential of 1 mole of an ideal gas).
,
then 
,
where 
is the molar volume of an ideal gas (the volume of 1 mol of gas).
If the process is running at T =const, the partial derivative can be replaced by the total. Then
The equation of state for 1 mole of an ideal gas is:
;
,
Let us integrate in the interval from R 0 before R:
where μ 0 , R 0 is the chemical potential and pressure of an ideal gas in the standard state.
The standard state is taken to be the state of an ideal gas at R 0 = 1 atm.
.
(2.3)
The standard chemical potential is the chemical potential at a gas pressure of 1 atm. Chemical potential in the standard state μ 0 depends only on temperature and does not depend on pressure, i.e. μ 0 = f(T). Equation (2.3) shows that under the sign of the logarithm is a dimensionless quantity equal to the pressure ratio R to standard pressure R 0 = 1 atm.
Because the μ
=
, can be written
where 
is the Gibbs standard molar energy.
Chemical potential of a component of ideal solutions.
If the total pressure of the gas mixture is small, then each gas will exert its own pressure, moreover, such as if it alone occupied the entire volume. This pressure is called partial. Total observed pressure R is equal to the sum of the partial pressures of each gas (Dalton's law):
.
The chemical potential of a component of an ideal gas mixture is:
,
where R i is the partial pressure of the gas.
Expressing the partial pressure of a gas R i through the total pressure and the mole fraction of the gas x i, get the expression for the dependence of the chemical potential i-th component from the mole fraction:
where 
is the chemical potential of an ideal gas at x i= 1 (i.e. in the individual state) at pressure R and temperature T;
depends on both temperature and pressure.
For ideal liquid solutions applicable equation
,
where 
is the standard chemical potential of an individual component in the liquid state ( 
) depends on temperature and pressure; x i is the mole fraction of the component.
Chemical potential of a component of real solutions .
For real solutions, all the dependences considered are inapplicable. Chemical potential of the component real gas solution calculated using the Lewis method. In this case, to preserve the form of thermodynamic equations, instead of partial pressure, they introduce a fictitious quantity f i, which is called partial fugacity, or volatility. Then
,
where 
is the chemical potential of a component of a real gas mixture in the standard state.
The ratio of volatility to the partial pressure of a real gas solution is called the volatility coefficient:
;
Similarly, for liquid real solutions the actual concentration is replaced by the corresponding fictitious value - activity a i :
,
where 
is the chemical potential of a component of a real liquid solution in the standard state.
Activity is related to concentration through the activity coefficient:
,
where γ i is the activity coefficient.
Depending on the method of expressing the concentration of a solution, rational, molar and molal activity coefficients are distinguished:
The activity coefficient depends on the concentration of the solution. In infinitely dilute solutions γ → 1, a i and f i → c i and p i respectively.
Let us rewrite the equation for the chemical potential in the form
,
therefore, thermodynamic activity is the work of transfer of 1 mol i th component from the standard solution to the given real solution.
There are two main ways to choose standard condition- symmetrical and asymmetrical.
symmetrical way. The same standard state is chosen for the solvent and the solute - the state of the pure component at the temperature of the solution. Then in the standard state x i = 1, a i = 1and γ i = 1. This method is more often used for non-electrolyte solutions.
asymmetric way. A different standard state is selected for the solvent and the solute. For the solvent - as in the symmetrical method: x i → 1, a i → 1and γ i → 1. For a solute the standard state is the state of matter in an infinitely dilute solution: x i → 0, a i → x i and γ i → 1. The method is most often used in the thermodynamics of electrolyte solutions.
Energy transformations occurring in systems during processes under various conditions are described using the corresponding thermodynamic functions U,H,G,A. It should be noted that these functions were introduced for an ideal process in which the amount of each substance was considered constant and equal to one mole. However, their values should depend on the amount of a given substance in the system, which can change during the process. For example, in a closed system, during a chemical reaction, the amount of initial substances decreases and the amount of products increases while maintaining the total mass of the substance (the qualitative and quantitative composition of the system changes). To take into account the influence of this circumstance on the values of thermodynamic functions, the concept of chemical potential was introduced.
The increase in the internal energy of the system with an increase in the amount of a given substance under conditions of constancy of the entropy of the system and its volume, with a constancy of the amounts of other substances, is called chemical potential of i-th substance:
It can be shown that the value of the chemical potential i-th substance is determined by a change in the thermodynamic function when the amount of this substance changes by one mole in processes occurring at a constant corresponding parameters and a constant amount of other substances:
.
In isobaric-isothermal processes, the change in the Gibbs energy with a change in the amount i th substance will be determined by the expression dG=m i ×dn i. When a chemical reaction occurs, the quantities of all substances participating in the reaction change, therefore dG=Sm i ×dn i .
The equilibrium condition for a chemical reaction occurring under isobaric-isothermal conditions, D r G=0, therefore , sm i ×dn i= 0. For the reaction n a A+n b B=n with C+n d D the equilibrium condition will be Sm i × n i= 0,
(m c × n C+ m d × n D)–(m a × n A+ m b × n B)=0.
Obviously, the chemical potential i th substance will depend on its quantity per unit volume - on the concentration of the substance. This dependence can be obtained by considering the change in the Gibbs energy during isobaric-isothermal mixing of two ideal gases.
Let two ideal gases under standard conditions be separated by a partition and occupy volumes V 1 and V 2 respectively (fig.1.5). The amount of the first gas is equal to one mole (n 1 =1), and the second n 2. If the partition is removed, the gases mix as a result of mutual diffusion. Each gas will occupy the entire volume of the system, and the volume of each will be V 1 +V 2. In this case, the concentration of each gas (the amount of substance per unit volume) will decrease. Each gas will do the work of expansion at constant pressure and temperature. Obviously, as a result of this process, the Gibbs energy of the system will decrease by the value of the perfect work of expansion.
 |
Rice. 1.5. Mixing of two ideal gases under isobaric-isothermal conditions
as a result of mutual diffusion
The change in the Gibbs energy as a result of a decrease in the concentration of the first gas will be equal to its work of expansion. The expansion work of the first gas is defined as follows:
dA=p 0 × dV, given that p× V=n× R× T and n 1 =1,
® A=–R× T×ln .
Since equal volumes of ideal gases contain the same number of moles of a substance,
,
where X 1 – mole fraction of the 1st gas; p 1 – partial pressure of the 1st gas; R 0 \u003d 1.013 × 10 5 Pa - standard pressure; FROM 1 – molar concentration of the 1st gas; FROM 0 =1 mol/l standard concentration.
Thus, the Gibbs energy of the 1st gas will change by the value D G 1 =R× T×ln X one . Since n 1 \u003d 1 mol, then, obviously, D f G i T=D f G 0 i T + R× T×ln X i .
Thus, the chemical potential of a substance depends on its concentration in the mixture:
m i=m i 0 + R× T×lnX i, m i=m i 0 + R× T×ln , m i=m i 0 + R× T×ln .
It should be noted that these concentration dependences of the chemical potential characterize ideal gases and solutions. Intermolecular interactions in real gases and solutions lead to a deviation of the calculated chemical potentials from the values obtained for ideal systems. To take this into account, the concepts of fugacity and activity are introduced.
Fugacity f(volatility) is a thermodynamic quantity used to describe the properties of real gas mixtures. It allows one to apply equations expressing the dependence of the chemical potential of an ideal gas on temperature, pressure, and the composition of the system. In this case, the partial pressure of the component of the gas mixture pi replaced by its fugacity fi. Intermolecular interaction leads to a decrease in the value of the effective partial pressure of the gas mixture component. To take this into account, the value of partial pressure is multiplied by the fugacity coefficient (g i<1).Очевидно, что при pi®0 g i®1 and fi® pi.
Unlike ideal solutions, in real solutions there are intermolecular interactions and interactions between ions formed as a result of electrolytic dissociation. This leads to the fact that the effective concentration of molecules and ions in real solutions decreases. Therefore, when calculating the chemical potential, instead of the concentration, we use FROM value activity a. Activity and molar concentration i th component are related by the relation a i=g i× C i, where g i is the molar activity coefficient (g i<1). Очевидно, что при C i®0 g i®1 and a i® C i.
The chemical potential is called the partial derivatives of the thermodynamic potential with respect to the number of moles of the component with the constancy of the natural variables and the composition of the system.
Chemical potential of ideal and real gas.
PV=nRT => V=
G=nRT R Ϭ , R Ϭ - constant integral
G=RTLnP+ R Ϭ (1)
G 0 =RTLnP 0 + R Ϭ (2)
Subtract 1ur from 2nd
G-G 0 =RT(LnP-LnP 0) => G-G 0 =RTLn
P 0 =1atm => G-G 0 =RTLnP => G=G 0 +RTLnP
G=ϻ i => ϻ i = ϻ i 0 +RTLnP
This is ur for determining the chemical potential of an ideal gas mixture.
Lewis proposed to use fugacity (volatility) for real gases, since so far only such systems have been considered whose gaseous parts are ideal, i.e. obey the equation PV = nRT for non-ideal systems, all the above thermodynamic relationships are not true.
Lewis' proposal remains valid for non-ideal systems, if we replace the actual partial pressures in them with effective pressures (volatility).
ϻ i = ϻ i 0 +RTLnf, f-volatility
The volatility coincides with the pressure if the latter is so small that the gas becomes ideal.
14.Thermodynamic law of acting masses. Derivation of the expression for the standard thermodynamic constant. Factors affecting the constant.
The basic quantitative law of chemical equilibrium is the law of mass action (LMA) applied to homogeneous equilibrium systems. Let us consider in general a homogeneous chemical reaction.
aA+bBómM+nN (1)
Gibbs energy change for a given chemical reaction
∆G=m(G M)+n(G N)-a(G A)-b(G B) (2)
Change for total i-th change
dG(i)=VdP(i)-S i dT(3)
PV=RT => V=RT/P
dG i =RT =RTlnP i (5)
let us integrate ur (5) within the standard state of the gas at the beginning of the reaction
(G 0 (i) ,P 0 i) (G (i) ,P 0(i))
(i) = lnP (i) (6)
G (i) -G 0 i =RTln (7)
The gas pressure in the standard state is 1 atm.
G (i) =G 0 i +RTlnP 0(i) (8)
P 0(i) - partial pressure of the i-th participant in the reaction at the beginning of the reaction. Given ur(8), ur(2) would look like
∆G=m(G 0 M)+m RTlnP 0(M) +n(G 0 N)+ nRTlnP 0(N) -a(G 0 A)-a RTlnP 0(A) -b(G 0 B) - bRTlnP 0(B) (9)
∆G=m(G 0 M)+n(G 0 N)-a(G 0 A)-b(G 0 B) +RTln (10)
∆G=∆G 0 +RTln (11)
If there is an equilibrium state in a chemical reaction then ∆G will be equal to 0, but the partial pressures will have equilibrium values
∆G 0 \u003d -RTln (12) K p (13) (equilibrium constant)
Thus, the equilibrium constant of any homogeneous chemical reaction occurring in the gas phase is equal to the product of the equilibrium partial pressures of the reaction products divided by the product of the equilibrium partial pressures of the starting materials in powers of the corresponding stoichiometric coefficients of the participants in the reactions.
Taking into account ur (13) ur (12), we can write ∆G 0 =-RTlnК р (14)
Taking into account ur (14), ur (11) will take the form ∆G=-RTln (15)
Ur (11), (15) are called van't Hoff isotherm ur, it is also written for an ideal gas in terms of partial pressures
∆G=-RTlnК r +RTlnП r"
Conclusion: The value of the equilibrium constant depends only on the temperature and the nature of the participants in the reaction. From ∆G, depends on initial concentrations (activities) or pressures (volatility).
Energy transformations occurring in systems during processes under various conditions are described using the corresponding thermodynamic functions U,H,G,A. It should be noted that these functions were introduced for an ideal process in which the amount of each substance was considered constant and equal to one mole. However, their values should depend on the amount of a given substance in the system, which can change during the process. For example, in a closed system, during a chemical reaction, the amount of initial substances decreases and the amount of products increases while maintaining the total mass of the substance (the qualitative and quantitative composition of the system changes). To take into account the influence of this circumstance on the values of thermodynamic functions, the concept of chemical potential was introduced.
The increase in the internal energy of the system with an increase in the amount of a given substance under conditions of constancy of the entropy of the system and its volume, with a constancy of the amounts of other substances, is called chemical potential of i-th substance:
It can be shown that the value of the chemical potential i-th substance is determined by a change in the thermodynamic function when the amount of this substance changes by one mole in processes occurring at a constant corresponding parameters and a constant amount of other substances:
.
In isobaric-isothermal processes, the change in the Gibbs energy with a change in the amount i th substance will be determined by the expression dG=m i ×dn i. When a chemical reaction occurs, the quantities of all substances participating in the reaction change, therefore dG=Sm i ×dn i .
The equilibrium condition for a chemical reaction occurring under isobaric-isothermal conditions, D r G=0, therefore , sm i ×dn i= 0. For the reaction n a A+n b B=n With C+n d The equilibrium condition will be Sm i × n i= 0,
(m c × n C+ m d × n D)–(m a × n A+ m b × n B)=0.
Obviously, the chemical potential i th substance will depend on its quantity per unit volume - on the concentration of the substance. This dependence can be obtained by considering the change in the Gibbs energy during isobaric-isothermal mixing of two ideal gases.
Let two ideal gases under standard conditions be separated by a partition and occupy volumes V 1 and V 2 respectively (fig.5.5).
Rice. 5‑5 Mixing of two ideal gases under isobaric-isothermal conditions as a result of mutual diffusion
The amount of the first gas is equal to one mole (n 1 =1), and the second n 2. If the partition is removed, the gases mix as a result of mutual diffusion. Each gas will occupy the entire volume of the system, and the volume of each will be V 1 +V 2. In this case, the concentration of each gas (the amount of substance per unit volume) will decrease. Each gas will do the work of expansion at constant pressure and temperature. Obviously, as a result of this process, the Gibbs energy of the system will decrease by the value of the perfect work of expansion.
The change in the Gibbs energy as a result of a decrease in the concentration of the first gas will be equal to its work of expansion. The expansion work of the first gas is defined as follows:
dA=p 0 × dV, given that p× V=n× R× T and n 1 =1,
® A=–R× T×ln .
Since equal volumes of ideal gases contain the same number of moles of a substance,
,
where X 1 – mole fraction of the 1st gas; p 1 – partial pressure of the 1st gas; R 0 \u003d 1.013 × 10 5 Pa - standard pressure; FROM 1 – molar concentration of the 1st gas; FROM 0 =1 mol/l standard concentration.
Thus, the Gibbs energy of the 1st gas will change by the value D G 1 =R× T×ln X one . Since n 1 \u003d 1 mol, then, obviously, D f G i=D f G 0 i + R× T×ln X i .
Thus, the chemical potential of a substance depends on its concentration in the mixture:
m i=m i 0 + R× T×ln X i, m i=m i 0 + R× T×ln , m i=m i 0 + R× T×ln .
It should be noted that these concentration dependences of the chemical potential characterize ideal gases and solutions. Intermolecular interactions in real gases and solutions lead to a deviation of the calculated chemical potentials from the values obtained for ideal systems. To take this into account, the concepts of fugacity and activity are introduced.
Fugacity f(volatility) is a thermodynamic quantity used to describe the properties of real gas mixtures. It allows one to apply equations expressing the dependence of the chemical potential of an ideal gas on temperature, pressure, and the composition of the system. In this case, the partial pressure of the component of the gas mixture pi replaced by its fugacity fi. Intermolecular interaction leads to a decrease in the value of the effective partial pressure of the gas mixture component. To take this into account, the value of partial pressure is multiplied by the fugacity coefficient (g i<1).Очевидно, что при pi®0 g i®1 and fi® pi.
Unlike ideal solutions, in real solutions there are intermolecular interactions and interactions between ions formed as a result of electrolytic dissociation. This leads to the fact that the effective concentration of molecules and ions in real solutions decreases. Therefore, when calculating the chemical potential, instead of the concentration, we use FROM value activity a. Activity and molar concentration i th component are related by the relation a i=g i× C i, where g i is the molar activity coefficient (g i<1). Очевидно, что при C i®0 g i®1 and a i® C i.
Test questions.
1. Thermodynamic system, parameters and state functions. thermodynamic process.
2. The first law of thermodynamics. Internal energy and enthalpy.
3. Thermal effect of a chemical reaction. Enthalpy of formation of a substance.
4. Temperature dependence of enthalpy.
5. Entropy. The second law of thermodynamics.
7. Temperature dependence of the Gibbs energy.
8. Concentration dependence of the Gibbs energy. activity and fugacity.
9. Thermodynamic calculations of the thermal effect of a chemical reaction.
10 Assessment of the thermodynamic possibility of a chemical reaction.
