Fundamental Property Relations: Screencasts

Reviews exact differentials and partial derivatives using a geometric example. These topics are important for deriving Maxwell relations in thermodynamics. 

We suggest you list the important points in this screencast as a way to increase retention.

Uses a general form of an exact differential to relate state functions to Maxwell relationships.

We suggest you list the important points in this screencast as a way to increase retention.

Important Equations:

Fundamental Property Relation for \(dU\):

\[dU = TdS – PdV\]

where \(U\) is internal energy, \(T\) is absolute temperature, \(S\) is entropy, \(P\) is pressure, and \(V\) is volume.

\[dH = TdS +VdP\]

where \(H\) is enthalpy.

\[dG = -SdT + VdP\]

where \(G\) is the Gibbs free energy.

\[\left( \frac{\partial x}{\partial y} \right) _z = \frac{1}{ \left( \frac{\partial y}{\partial x} \right)} _z \]

\[\left( \frac{\partial x}{\partial y} \right) _x = 0 \hspace{5mm} and \hspace{5mm} \left( \frac{\partial x}{\partial y} \right) _y = \infty\]

\[\left( \frac{\partial x}{\partial x} \right) _y = 1\]

Triple product rule

\[\left( \frac{\partial x}{\partial y} \right) _F \left( \frac{\partial y}{\partial F} \right) _x \left( \frac{\partial F}{\partial x} \right) _y = -1\]

where \(x, y, z,\) and \(F\) are state functions.

Chain rule

\[\left( \frac{\partial x}{\partial y} \right) _F = \left( \frac{\partial x}{\partial z} \right) _F \left( \frac{\partial z}{\partial y} \right) _F\]

Expansion rule

\[\left( \frac{\partial F}{\partial w} \right) _z = \left( \frac{\partial F}{\partial x} \right) _y \left( \frac{\partial x}{\partial w} \right) _z + \left( \frac{\partial F}{\partial y} \right) _x \left( \frac{\partial y}{\partial w} \right) _z\]

Exact differential

\[dF = \left( \frac{\partial F}{\partial x} \right) _y dx + \left( \frac{\partial F}{\partial y} \right) _x dy\]

\[M \equiv \left( \frac{\partial F}{\partial x} \right) _ y \hspace{5mm}N \equiv \left( \frac{\partial F}{\partial y} \right) _ x\]

To derive Maxwell relations

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

Maxwell relations

\[dU = TdS – PdV \Rightarrow – \left( \frac{\partial P}{\partial S} \right) _V = \left( \frac{\partial T}{\partial V} \right) _S\]

\[dH = TdS +PdV \Rightarrow \left( \frac{\partial V}{\partial S} \right) _P = \left( \frac{\partial T}{\partial P} \right) _S\]

\[dA = -SdT -PdV \Rightarrow \left( \frac{\partial P}{\partial T} \right) _V = \left( \frac{\partial S}{\partial V} \right) _T\]

where \(A\) is the Hemholtz free energy.

\[dG = -SdT +VdP \Rightarrow – \left( \frac{\partial V}{\partial T} \right) _P = \left( \frac{\partial S}{\partial P} \right) _T\]

\[dS(T,V) = \frac{C_V}{T} dT+ \left( \frac{\partial P}{\partial T} \right) _V dV\]

where \(C_V\) is the heat capacity at constant volume.

\[dS(V,P) = \frac{C_P \left( \frac{\partial T}{\partial V} \right) _T}{T} dV + \frac{C_V \left( \frac{\partial T}{\partial P} \right)_V}{T} dP \]

where \(C_P\) is the heat capacity at constant pressure.

\[dH(T,P) = C_PdT + \left[ V – T \left( \frac{\partial V}{\partial T} \right) _P \right] dP\]

\[dU(T,V) = C_VdT + \left[T \left( \frac{\partial P}{\partial T} \right) _V -P \right] dV\]

\(C_P\) and \(C_V\) relations

\[\left( \frac{\partial S}{\partial T} \right) _V = \frac{C_V}{T}\]

\[\left( \frac{\partial S}{\partial T} \right) _P = \frac{C_P}{T}\]