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The **Maxwell relations** are a set of equations in thermodynamics derived from the equality of mixed partial derivatives of thermodynamic potentials. They establish connections between partial derivatives of state variables (such as pressure (P), volume (V), temperature (T), and entropy (S)), allowing difficult-to-measure quantities to be expressed in terms of easily measurable ones (e.g., compressibility, thermal expansion). ### **Mathematical Basis** They arise from **Schwarz’s theorem** (Clairaut’s theorem), which states that for a state function ( Phi ) with exact differential ( dPhi ), the mixed second-order partial derivatives are equal: [ frac{partial}{partial y} left( frac{partial Phi}{partial x} right) = frac{partial}{partial x} left( frac{partial Phi}{partial y} right) ] ### **The Four Primary Maxwell Relations** Each relation corresponds to a thermodynamic potential: 1. **From internal energy ((U = U(S, V))):** [ left( frac{partial T}{partial V} right)_S = -left( frac{partial P}{partial S} right)_V ] *Derived from:* ( dU = T dS – P dV ). 2. **From enthalpy ((H = H(S, P))):** [ left( frac{partial T}{partial P} right)_S = left( frac{partial V}{partial S} right)_P ] *Derived from:* ( dH = T dS + V dP ). 3. **From Helmholtz free energy ((F = F(T, V))):** [ left( frac{partial S}{partial V} right)_T = left( frac{partial P}{partial T} right)_V ] *Derived from:* ( dF = -S dT – P dV ). 4. **From Gibbs free energy ((G = G(T, P))):** [ left( frac{partial S}{partial P} right)_T = -left( frac{partial V}{partial T} right)_P ] *Derived from:* ( dG = -S dT + V dP ). ### **Key Significance** – **Convert “unmeasurable” to “measurable”:** For example, (left( frac{partial S}{partial P} right)_T) (entropy change with pressure) is related to (-left( frac{partial V}{partial T} right)_P) (thermal expansion coefficient), which is experimentally accessible. – **Simplify thermodynamic analyses:** Used in deriving equations of state, analyzing phase transitions, and calculating properties like heat capacities or compressibilities. – **Validity:** Applicable to **simple compressible systems** in equilibrium (constant composition). ### **Mnemonic Aid: The Thermodynamic Square** A visual tool to recall the relations: “` S ─────── T │ │ │ │ V ─────── P “` – **Rule:** Start at a corner, move to adjacent corners for derivatives. *Example (Helmholtz: (F(T,V))):* From (S) to (V) ((partial S/partial V)) equals from (P) to (T) ((partial P/partial T)), with a sign if paths oppose. ### **Example Application** To find (left( frac{partial S}{partial V} right)_T) for an ideal gas: 1. Use the Helmholtz relation: (left( frac{partial S}{partial V} right)_T = left( frac{partial P}{partial T} right)_V). 2. Substitute (P = frac{nRT}{V}): (left( frac{partial P}{partial T} right)_V = frac{nR}{V}). 3. Result: (left( frac{partial S}{partial V} right)_T = frac{nR}{V}). The Maxwell relations are foundational in thermodynamics, bridging theoretical state functions with empirical data. For systems with variable composition (e.g., chemical reactions), additional terms involving chemical potential ((mu_i)) extend these relations.
