Derivation of Maxwell's Equations

What are Maxwell's Equations?

Maxwell's Equations are useful in replacing unmeasurable quantities appearing in thermodynamic equations by measurable quantities.


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Maxwell's Equations derivation is as follows

Maxwell \(1^{st}\) equation derivation

• Fundamental property relation equation for Internal energy as given below.
\[dU = TdS - PdV -------- (1)\]
• Differenting equation no. (1) with respect to \(S\) keeping \(V\) constant.
\[\left ( \frac{\partial U}{\partial S}\right )_V=T\]
• Differenting above equation with respect to \(V\) keeping \(S\) constant.
\[\frac{\partial^2 U}{\partial V \; \partial S}= \left( \frac{\partial T}{\partial V}\right)_S ---------- (2)\]
• Differenting equation no. (1) with respect to \(S\) keeping \(V\) constant.
\[\left ( \frac{\partial U}{\partial V}\right )_S= -P\]
• Differenting above equation with respect to \(S\) keeping \(V\) constant.
\[\frac{\partial^2 U}{\partial S \;\partial V}= - \left( \frac{\partial P}{\partial S}\right)_V ---------- (3)\]
• Comparing equation no. (2) and (3).
\[\boxed{\left( \frac{\partial T}{\partial V}\right)_S = - \left( \frac{\partial P}{\partial S}\right)_V} ---------- (4)\]


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Maxwell \(2^{nd}\) equation derivation

• Fundamental property relation equation for Enthalpy as given below.
\[dH = TdS + VdP -------- (5)\]
• Differenting equation no. (5) with respect to \(S\) keeping \(P\) constant.
\[\left ( \frac{\partial H}{\partial S}\right )_P=T\]
• Differenting above equation with respect to \(P\) keeping \(S\) constant.
\[\frac{\partial^2 H}{\partial P \; \partial S}= \left( \frac{\partial T}{\partial P}\right)_S ---------- (6)\]
• Differenting equation no. (5) with respect to \(P\) keeping \(S\) constant.
\[\left ( \frac{\partial H}{\partial P}\right )_S= V\]
• Differenting above equation with respect to \(S\) keeping \(P\) constant.
\[\frac{\partial^2 H}{\partial S \;\partial P}= \left( \frac{\partial V}{\partial S}\right)_P ---------- (7)\]
• Comparing equation no. (6) and (7).
\[\boxed{\left( \frac{\partial T}{\partial P}\right)_S = \left( \frac{\partial V}{\partial S}\right)_P} ---------- (8)\]


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Maxwell \(3^{rd}\) equation derivation

• Fundamental property relation equation for Helmholtz free energy as given below.
\[dA = -SdT - PdV -------- (9)\]
• Differenting equation no. (9) with respect to \(T\) keeping \(V\) constant.
\[\left ( \frac{\partial A}{\partial T}\right )_V=-S\]
• Differenting above equation with respect to \(V\) keeping \(T\) constant.
\[\frac{\partial^2 A}{\partial V \; \partial T}= -\left( \frac{\partial S}{\partial V}\right)_T ---------- (10)\]
• Differenting equation no. (9) with respect to \(V\) keeping \(T\) constant.
\[\left ( \frac{\partial A}{\partial V}\right )_T= -P\]
• Differenting above equation with respect to \(T\) keeping \(V\) constant.
\[\frac{\partial^2 A}{\partial T \;\partial V}= -\left( \frac{\partial P}{\partial T}\right)_V ---------- (11)\]
• Comparing equation no. (10) and (11).
\[\boxed{\left( \frac{\partial S}{\partial V}\right)_T = \left( \frac{\partial P}{\partial T}\right)_V} ---------- (12)\]


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Maxwell \(4^{th}\) equation derivation

• Fundamental property relation equation for Gibbs free energy as given below.
\[dG = -SdT + VdP -------- (13)\]
• Differenting equation no. (13) with respect to \(T\) keeping \(P\) constant.
\[\left ( \frac{\partial G}{\partial T}\right )_P=-S\]
• Differenting above equation with respect to \(P\) keeping \(T\) constant.
\[\frac{\partial^2 G}{\partial P \; \partial T}= -\left( \frac{\partial S}{\partial P}\right)_T ---------- (14)\]
• Differenting equation no. (13) with respect to \(P\) keeping \(T\) constant.
\[\left ( \frac{\partial G}{\partial P}\right )_T= V\]
• Differenting above equation with respect to \(T\) keeping \(P\) constant.
\[\frac{\partial^2 G}{\partial T \;\partial P}= \left( \frac{\partial V}{\partial T}\right)_P ---------- (15)\]
• Comparing equation no. (14) and (15).
\[\boxed{-\left( \frac{\partial S}{\partial P}\right)_T = \left( \frac{\partial V}{\partial T}\right)_P} ---------- (16)\]

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