Thermodynamics/Thermodynamic Network

Introduction
We now want to combine the first and second laws. The approach we will use was developed by J. Willard Gibbs in the late nineteenth century.

Fundamental Equation
The first law of thermodynamics is $$dU=dQ+dW$$

For a reversible process with only PV work $$dW=-PdV$$

The Clausis Inequality gives for a reversible process $$dQ=TdS$$

Combining these equations gives $$dU=TdS-PdV$$

This is called the Fundamental Equation of Thermodynamics

Importance
Let us look at the this equation in more detail. We derived the equation using reversible processes. However, let us look at each of the terms in the fundamental equation. Every term is in fact a state function. As explained in that section, state functions are path independent. Therefore, the fundamental equation applies to all closed, simple systems.

We will now make a rather bold statement: All of thermodynamics can be derived from the fundamental equation. (That is why it is called the fundamental equation)

Thermodynamic Network
We now wish to make some relations between the thermodynamic properties (P, V, T, U, S, and H) using the fundamental equation. These relations are called the Thermodynamic Network.

Legendre Transformation
If we look at the fundamental equation we see that the independent variables are S and V. However, in most applications these are not in fact the independent variables.

To interchange the independent and dependent variables we use the technique called a Legendre Transformation.

Let us consider a function F1 = F1(x,y,z,…)

The total differential is then

$$dF_1=Xdx+Ydy+Zdz+\cdots\qquad (1)$$

where

$$X=\frac{\partial F}{\partial x},\ldots$$

Reminder: x, y, z, ... are the independent variables and X, Y, Z, ... are dependent variables.

We now introduce the following function

$$F_2=F_1+Xx$$

then

$$dF_2=dF_1-Xdx-xdX$$

inserting this into equation (1) gives

$$dF_2=-xdX+Ydy+Zdz+\cdots$$

We now have the function F2 = F2(X,y,z,…)

Thus X is now the independent variable and x the dependent variable.

Enthalpy
Now we are ready to look at the thermodynamic network. Let us start by considering systems where the independent variables are S and P. We need to change the fundamental equation so that P, instead of V, is the independent variable. Therefore we will apply the Legendre transformation to PdV:

$$PdV=d(PV)-VdP$$

Then the fundamental equation becomes

$$dU=TdS-d(PV)+VdP$$

or

$$d(U+PV)=TdS+VdP$$

on the left-side is the enthalpy, therefore

$$dH=TdS+VdP$$

Helmholtz Energy
Now let us consider a system with T and V as the independent variables. This time we will apply the Legendre Transformation to TdS:

$$TdS = d(TS)-SdT$$

and the fundamental equation becomes

$$dU = d(TS)-SdT-PdV$$ $$d(U-TS) = -SdT-PdV$$ The quantity U - TS is called the Helmholtz Energy, A

$$A=U-TS$$

Therefore,

$$dA=-SdT-PdV$$

Gibbs Energy
By far the most important systems used in the application of thermodynamics are those with T and P as the independent variables. In this case we need to apply the Legendre Transformation to both TdS and PdV:

$$TdS=d(TS)-SdT$$ $$PdV=d(PV)-VdP$$

and the fundamental equation becomes

$$dU=d(TS)-SdT-d(PV)+VdP$$ $$d(U-TS+PV)=-SdT+VdP$$

The quantity U - TS + PV is called the Gibbs Energy, G

$$G=U-TS+PV$$

Which can also be written

$$G=H-TS$$ or $$G=A+PV$$

Therefore,

$$dG=-SdT+VdP$$

The quantities U, H, A, and G are known as Thermodynamic Potentials

Maxwell Equations
Remember this theorem concerning exact differentials?

The equation

$$df=M(x,y)dx+N(x,y)dy$$

is an exact differential if and only if

$$\left (\frac{\partial M}{\partial y}_x\right ) = \left (\frac{\partial N}{\partial x}_y\right ) dy$$

Now since all linear combinations of state functions are state functions, H, A, and G must be state functions. Therefore, we can use this theorem to develop some relationships.

Consider the fundamental equation

$$dU=TdS-PdV$$

Comparing this to above we can set M = T, N = -P, x = S, and y = V.

Now using the theorem gives

$$\left (\frac{\partial T}{\partial V}\right )_S=-\left (\frac{\partial P}{\partial S}\right )_V$$

Doing the same method with $$dH=TdS+VdP$$ gives

$$\left (\frac{\partial T}{\partial P}\right )_S=\left (\frac{\partial V}{\partial S}\right )_P$$

From $$dA=-SdT-PdV$$

$$\left (\frac{\partial S}{\partial V}\right )_T=\left (\frac{\partial P}{\partial T}\right )_V$$

From $$dG=-SdT+VdP$$

$$\left (\frac{\partial S}{\partial P}\right )_T=-\left (\frac{\partial V}{\partial T}\right )_P$$

These four relations are known as the Maxwell Equations

For reference a table of the above is given HERE.

Please, do not memorize these. Instead, you should be able to derive any of them if necessary.

Derivatives of Thermodynamic Quantities
Let us look at the derivatives which involve only P, V, and T.

There are three such derivatives. Two of them are given special names and symbols:

$$\alpha\equiv\frac{1}{V}\left (\frac{\partial V}{\partial T}\right )_P$$

$$\kappa_T\equiv -\frac{1}{V}\left (\frac{\partial V}{\partial P}\right )_T$$

&alpha; is called the volume expansivity or coefficient of expansion

&kappa; is called the isothermal compressibility

These are often found in tables.

Note: Different sources use different (and often confusing) notation.

The remaining partial derivative is $$\left(\frac{\partial P}{\partial T}\right)_V$$

Using the properties of partial derivatives we can write this as $$\left (\frac{\partial P}{\partial T} \right )_V=\frac{-\left (\frac{\partial P}{\partial V} \right )_T}{\left (\frac{\partial T}{\partial V} \right )_P}=\frac{\alpha}{\kappa_T}$$

The Network
If we consider all of the thermodynamic variables (P, V, T, S, U, H, A, G) we have 336 possible derivatives.

Amazingly, we can write all of these in terms of just three derivatives: &alpha;, &kappa;T, and CP!

This is in addition a very important result. We cannot directly measure S, U, H, A, or G. However, we can get &alpha; and &kappa;T from PVT data. Using this and CP data we can get S, etc.

Derivatives of Entropy
If we take the derivatives of S with respect to P, V, and T we get six derivatives:

$$\left ( \frac{\partial S}{\partial P} \right )_{T} \; \left ( \frac{\partial S}{\partial V} \right )_{T} \; \left ( \frac{\partial S}{\partial T} \right )_{P} \; \left ( \frac{\partial S}{\partial T} \right )_{V} \; \left ( \frac{\partial S}{\partial P} \right )_{V} \; \left ( \frac{\partial S}{\partial V} \right )_{P} $$

For the first two of these we can use the Maxwell equations:

$$\left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{\partial T}\right)_{P}=-\alpha V$$

$$\left(\frac{\partial S}{\partial V}\right)_{T}=-\left(\frac{\partial P}{\partial T}\right)_{V}=\frac{\alpha}{\kappa_{T}}$$

The two derivatives of S with respect to T are related to the heat capacities:

$$\left(\frac{\partial S}{\partial T}\right)_{P}=\frac{C_{P}}{T}$$

$$\left(\frac{\partial S}{\partial P}\right)_{T}=\frac{C_{V}}{T}$$

CP is related to CV by:

$$C_{V}=C_{P}-\frac{TV\alpha^{2}}{\kappa_{T}}$$