Gibbs Helmholtz equation, Vant Hoff isotherm and isochore derivation in detail with each and every steps

Gibbs Helmholtz equation, Vant Hoff isotherm and isochore derivation in detail with each and every steps.

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Sure! Let me provide detailed derivations and explanations of the Gibbs-Helmholtz equation, van’t Hoff isotherm, and isochore step-by-step.

1. Gibbs-Helmholtz Equation

Purpose: Relates the temperature dependence of the Gibbs free energy GG to the enthalpy HH.

Starting point:

Gibbs free energy is defined as G=H−TSG = H – TS

Differentiating GG with respect to temperature TT at constant pressure PP: (∂G∂T)P=(∂H∂T)P−(S+T∂S∂T)P\left(\frac{\partial G}{\partial T}\right)_P = \left(\frac{\partial H}{\partial T}\right)_P – \left( S + T\frac{\partial S}{\partial T} \right)_P

Using the thermodynamic relation (∂G∂T)P=−S\left(\frac{\partial G}{\partial T}\right)_P = -S

and recognizing (∂H∂T)P=CP\left(\frac{\partial H}{\partial T}\right)_P = C_P (heat capacity at constant pressure),

We rewrite the Gibbs free energy as: G=H−TS  ⟹  GT=HT−SG = H – TS \implies \frac{G}{T} = \frac{H}{T} – S

Differentiating GT\frac{G}{T} with respect to TT gives: ∂∂T(GT)=1T∂G∂T−GT2=1T(−S)−GT2=−ST−GT2\frac{\partial}{\partial T} \left( \frac{G}{T} \right) = \frac{1}{T} \frac{\partial G}{\partial T} – \frac{G}{T^2} = \frac{1}{T}(-S) – \frac{G}{T^2} = -\frac{S}{T} – \frac{G}{T^2}

Recall G=H−TS  ⟹  S=H−GTG = H – TS \implies S = \frac{H-G}{T}, substitute into above: ∂∂T(GT)=−1T⋅H−GT−GT2=−HT2+GT2−GT2=−HT2\frac{\partial}{\partial T} \left( \frac{G}{T} \right) = -\frac{1}{T} \cdot \frac{H-G}{T} – \frac{G}{T^2} = -\frac{H}{T^2} + \frac{G}{T^2} – \frac{G}{T^2} = -\frac{H}{T^2}

Final Gibbs-Helmholtz equation: (∂∂T(GT))P=−HT2\boxed{ \left( \frac{\partial}{\partial T} \left( \frac{G}{T} \right) \right)_P = – \frac{H}{T^2} }

2. van’t Hoff Isotherm Derivation

The van’t Hoff equation relates the change in the equilibrium constant KK with temperature.

Start from Gibbs free energy change at equilibrium: ΔG=ΔG∘+RTln⁡K\Delta G = \Delta G^\circ + RT \ln K

At equilibrium, ΔG=0\Delta G = 0, so: 0=ΔG∘+RTln⁡K  ⟹  ΔG∘=−RTln⁡K0 = \Delta G^\circ + RT \ln K \implies \Delta G^\circ = -RT \ln K

Since ΔG∘=ΔH∘−TΔS∘\Delta G^\circ = \Delta H^\circ – T \Delta S^\circ

We substitute: −RTln⁡K=ΔH∘−TΔS∘  ⟹  ln⁡K=−ΔH∘RT+ΔS∘R-RT \ln K = \Delta H^\circ – T \Delta S^\circ \implies \ln K = -\frac{\Delta H^\circ}{RT} + \frac{\Delta S^\circ}{R}

Taking derivative w.r.t temperature TT at constant pressure: dln⁡KdT=ΔH∘RT2\frac{d \ln K}{dT} = \frac{\Delta H^\circ}{RT^2}

This is the van’t Hoff equation: dln⁡KdT=ΔH∘RT2\boxed{ \frac{d \ln K}{dT} = \frac{\Delta H^\circ}{RT^2} }

3. Isochore Derivation

An isochore is a path of constant volume in a thermodynamic system. The derivation involves relating pressure PP, temperature TT, and internal energy UU at constant volume VV.

From the first law of thermodynamics: dU=δQ−PdVdU = \delta Q – PdV

At constant volume (dV=0dV = 0): dU=δQVdU = \delta Q_V

Define heat capacity at constant volume: CV=(∂U∂T)VC_V = \left(\frac{\partial U}{\partial T}\right)_V

For an ideal gas, internal energy depends only on temperature: U=U(T)U = U(T)

Use thermodynamic identity for pressure change at constant volume: (∂P∂T)V=ακT\left(\frac{\partial P}{\partial T}\right)_V = \frac{\alpha}{\kappa_T}

where α\alpha is the coefficient of thermal expansion and κT\kappa_T is the isothermal compressibility.

For an ideal gas: PV=nRT  ⟹  P=nRTVP V = nRT \implies P = \frac{nRT}{V}

At constant VV: (∂P∂T)V=nRV\left(\frac{\partial P}{\partial T}\right)_V = \frac{nR}{V}

Thus the isochore is described by: P=constant×T(for ideal gases)\boxed{ P = \text{constant} \times T \quad \text{(for ideal gases)} }

Summary (≈300 words)

The Gibbs-Helmholtz equation relates the temperature dependence of Gibbs free energy to enthalpy. Starting from the definition G=H−TSG = H – TS, differentiating G/TG/T with respect to TT yields (∂(G/T)∂T)P=−H/T2\left(\frac{\partial (G/T)}{\partial T}\right)_P = -H/T^2. This relation helps predict how the Gibbs free energy changes with temperature, essential in chemical thermodynamics.

The van’t Hoff isotherm relates the temperature dependence of the equilibrium constant KK to the reaction enthalpy. By expressing the Gibbs free energy at equilibrium, ΔG∘=−RTln⁡K\Delta G^\circ = -RT \ln K, and differentiating, one obtains dln⁡KdT=ΔH∘RT2\frac{d \ln K}{dT} = \frac{\Delta H^\circ}{RT^2}. This equation predicts how equilibrium shifts with temperature changes and is widely used to analyze reaction spontaneity.

The isochore represents a process at constant volume. Using the ideal gas law P=nRT/VP = nRT/V, the pressure increases linearly with temperature when volume is fixed. Heat capacity at constant volume CVC_V relates to internal energy changes since U=U(T)U = U(T) for ideal gases. Isochore paths are crucial in thermodynamic cycles and processes involving confined gases.

These derivations underscore core thermodynamic principles linking energy, entropy, and equilibrium, vital for understanding chemical and physical systems