legendre transform Wednesday, June 24, 2015 9:12 AM Pasted from Q is a very diff
ID: 1288887 • Letter: L
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legendre transform
Wednesday, June 24, 2015 9:12 AM Pasted from Q is a very difficult variable to work with, because it is not a state variable, so the total derivative is a very tricky concept. Hence, physical chemist prefer to use Enthalpy, H=U+PV which is a state variable, when referring to heat. By the means of a Legendre transform, how the change in enthalpy of a system is related to the heat absorbed or expelled. i. State your assumptions. ii. Explain why your assumptions in (i) is valid for most chemical reactions (Hint: most chemical reactions take place at room temperature and pressure and are rather quasi-static) Also, show using a Legendre Transform that delta F, where F is the Helmholtz Free energy, F = U - TS is a measure of all forms of work done on the system. State your assumptions.Explanation / Answer
The first law of thermodynamics is a statement of the conservation of energy. Specifically, the first law addresses the interconvertability of work and heat as forms of energy.
In words the law is
Internal energy = heat + work
or in integral form as
DU = q + w
Calculating the work is possible starting with the definition
work = force x distance
dw = -F dr
This can be recast by dividing the work by an area and by multiplying the displacement by an area
dw = -(F/A) d(rA) or dw = -P dV
The heat can be calculated using the constant volume heat capacity.
heat = heat capacity x temperature change
dq = Cv dT
The second law of thermodynamics tells us that the entropy of system is
dS = dq/T for a reversible process.
For an irreversible process the heat is exhanged between the system and surroundings, dqirr will be less than TdS such that
dS > dqirr/T
For a reversible process we can write a combined expression for the first and second laws of thermodynamics
dU <= TdS - PdV
Free energy functions
At constant volume, dV = 0, and we have
dU <= TdS
or
dU - TdS <= 0
Since T and V are held constant, we can write this expression as
d(U - TS) <= 0
Thus, we can define a new state function, F = U - TS.
dF < = 0
The quantity F is called the Helmholtz free energy. In a system held at constant T and V, the Helmholtz energy will decrease until all possible spontaneous processes have occurred at which time the system will be in equilibrium.
The internal energy is a natural function of entropy and volume, U(S,V). The Helmholtz free energy is a natural function of temperature and volume, F(T,V). We can also consider the enthalpy, H as a natural function of entropy and pressure
and the Gibbs free energy as a natural function of temperature and pressure, G(T,P). The transformation of variables between these two sets of functions is known as a LeGendre transform.
H = U + PV
In differential form we have
dH = dU + PdV + VdP = TdS + VdP
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