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