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Free Energy, Temperature
The way in which free energy depends on temperature shows the way in which the equilibrium constant depends on temperature.
The equilibrium constant can be evaluated from the relation ΔG˚ = - RT In K at any temperature for which free-energy data available. Frequently we calculate the equilibrium constant at 25˚C, the reference temperature for which most thermodynamic data are tabulated. Now an expression for the temperature dependence of the equilibrium constant is developed. It will allow us to extend an equilibrium constant value to other temperatures, and it will show on what the temperature variation of an equilibrium constant depends.
The free energy of each substance involved in a reaction depends on the temperature, as
(∂G/∂T)P = -S
We want to use this relation to deal with the temperature dependence of the equilibrium constant. This constant is related to free energies through the relation ΔG˚ = - RT In K.
Recall that the symbol ΔG and a corresponding ΔS symbol designate
ΔG = Gprod – Greact and ΔS = Sprod – Sreact
When equation is applied to each of the reagents, we have the relation
[(∂(ΔG))/∂T]P = - ΔS
A more convenient expression for ΔG results when ΔS is eliminated from equation
At any constant temperature the changes of free energy, enthalpy, and entropy for any reaction are related by
ΔG = ΔH – T ΔS or ΔS = (ΔH - ΔG)/T [T const]
The second expression can be used to eliminate ΔS from equation to give
[(∂(ΔG))/∂T]P = (ΔH + ΔG)/T = ΔH/T + ΔG/T
Or [(∂(ΔG))/∂T]P - ΔG/T = - ΔH/T
The two terms on the left side can be shown to be equilivalent to
T[(∂(ΔG))/∂T]P = T (T[∂(ΔG/∂T)]P - ΔG)/T = [(∂(ΔG))/∂T]P - ΔG/T
Now the left side of equation can be inserted in place of the left side of equation to give
T[(∂(ΔG/T))/∂T]P = ΔH/T
Finally standard states are indicated and relation ΔG˚ = - RT In K is inserted to give, on rearrangement
(d(In K))/dT = (ΔH˚)/(RT2 )
This important formula is the goal of the derivation. The rate of change of the equilibrium constant with temperature is seen to depend on the standard enthalpy change.
The change of In K or K can be obtained by an integration of this expression either with the assumption of a constant value of ΔH˚ or with the temperature dependence of this quantity expressed by the empirical expressions developed. Integrations can be carried out first rearranging to
(d(In K))/(d(1/T)) = - (ΔH˚)/R
The integrated form of these equations, on the assumption that ΔH˚ is temperature independent, is
In K = (ΔH˚)/RT + const
Both the integrated and differential forms show that a plot of In K versus 1/T should give a straight line with a slope equal to - ΔH˚/R. the linearity shown by good measurements can be judged by the straight line which has been drawn with the slope ΔH˚/R, with a value of ΔH˚.
Thus a measured value of ΔH˚ can be used to calculate the equilibrium constant at a temperatures other than that for which it is given. Conversely, it is possible to use measurements of the equilibrium constant at a number of temperatures to evaluate the standard enthalpy change for the reaction.
When much larger temperatures ranges are considered, the basis of the dependence of the equilibrium constant on temperature can be seen more clearly by returning to the expressions
ΔG˚ = ΔH˚ = T ΔS˚ and ΔG˚ = -RT In K
Or
RT In K = - ΔH˚ + T ΔS˚
This equation is valid at any temperature when values of ΔH˚ and T ΔS˚ appropriate to that temperature are used. Generally, the T ΔS˚ term, as might be expected from the presence of the explicit T factor, is the more temperature dependent. At high temperatures this term dominates the ΔH˚ term, to give an RT In K value is increasingly positive or negative, depending on whether Librium constant generally becomes increasingly smaller if ΔS˚ is negative.
In general, the more gas-phase molecular or atomic particles there are, the higher the entropy. This fact, and the overwhelming importance of the entropy of the system at high temperatures lead to the general breakup or dissociation of side of the equation with more gas-phase species will be dominant, a generalization that is not valid until the T ΔS˚ term dominates the ΔH˚ term is contributing to ΔG˚.
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The equilibrium constant can be evaluated from the relation ΔG˚ = - RT In K at any temperature for which free-energy data available. Frequently we calculate the equilibrium constant at 25˚C, the reference temperature for which most thermodynamic data are tabulated. Now an expression for the temperature dependence of the equilibrium constant is developed. It will allow us to extend an equilibrium constant value to other temperatures, and it will show on what the temperature variation of an equilibrium constant depends.
The free energy of each substance involved in a reaction depends on the temperature, as
(∂G/∂T)P = -S
We want to use this relation to deal with the temperature dependence of the equilibrium constant. This constant is related to free energies through the relation ΔG˚ = - RT In K.
Recall that the symbol ΔG and a corresponding ΔS symbol designate
ΔG = Gprod – Greact and ΔS = Sprod – Sreact
When equation is applied to each of the reagents, we have the relation
[(∂(ΔG))/∂T]P = - ΔS
A more convenient expression for ΔG results when ΔS is eliminated from equation
At any constant temperature the changes of free energy, enthalpy, and entropy for any reaction are related by
ΔG = ΔH – T ΔS or ΔS = (ΔH - ΔG)/T [T const]
The second expression can be used to eliminate ΔS from equation to give
[(∂(ΔG))/∂T]P = (ΔH + ΔG)/T = ΔH/T + ΔG/T
Or [(∂(ΔG))/∂T]P - ΔG/T = - ΔH/T
The two terms on the left side can be shown to be equilivalent to
T[(∂(ΔG))/∂T]P = T (T[∂(ΔG/∂T)]P - ΔG)/T = [(∂(ΔG))/∂T]P - ΔG/T
Now the left side of equation can be inserted in place of the left side of equation to give
T[(∂(ΔG/T))/∂T]P = ΔH/T
Finally standard states are indicated and relation ΔG˚ = - RT In K is inserted to give, on rearrangement
(d(In K))/dT = (ΔH˚)/(RT2 )
This important formula is the goal of the derivation. The rate of change of the equilibrium constant with temperature is seen to depend on the standard enthalpy change.
The change of In K or K can be obtained by an integration of this expression either with the assumption of a constant value of ΔH˚ or with the temperature dependence of this quantity expressed by the empirical expressions developed. Integrations can be carried out first rearranging to
(d(In K))/(d(1/T)) = - (ΔH˚)/R
The integrated form of these equations, on the assumption that ΔH˚ is temperature independent, is
In K = (ΔH˚)/RT + const
Both the integrated and differential forms show that a plot of In K versus 1/T should give a straight line with a slope equal to - ΔH˚/R. the linearity shown by good measurements can be judged by the straight line which has been drawn with the slope ΔH˚/R, with a value of ΔH˚.
Thus a measured value of ΔH˚ can be used to calculate the equilibrium constant at a temperatures other than that for which it is given. Conversely, it is possible to use measurements of the equilibrium constant at a number of temperatures to evaluate the standard enthalpy change for the reaction.
When much larger temperatures ranges are considered, the basis of the dependence of the equilibrium constant on temperature can be seen more clearly by returning to the expressions
ΔG˚ = ΔH˚ = T ΔS˚ and ΔG˚ = -RT In K
Or
RT In K = - ΔH˚ + T ΔS˚
This equation is valid at any temperature when values of ΔH˚ and T ΔS˚ appropriate to that temperature are used. Generally, the T ΔS˚ term, as might be expected from the presence of the explicit T factor, is the more temperature dependent. At high temperatures this term dominates the ΔH˚ term, to give an RT In K value is increasingly positive or negative, depending on whether Librium constant generally becomes increasingly smaller if ΔS˚ is negative.
In general, the more gas-phase molecular or atomic particles there are, the higher the entropy. This fact, and the overwhelming importance of the entropy of the system at high temperatures lead to the general breakup or dissociation of side of the equation with more gas-phase species will be dominant, a generalization that is not valid until the T ΔS˚ term dominates the ΔH˚ term is contributing to ΔG˚.
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