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

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

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