Here’s a breakdown and explanation of the questions and tasks from the images: --- ### 1. **Helmholtz Free Energy for the Ideal Gas** **Task:** Using the expression for the Helmholtz free energy \( F \) of the ideal gas, calculate \( \Delta F \) for an isothermal expansion from volume \( V_A \) to \( V_B \) when the system is in equilibrium with a thermal reservoir at temperature \( T_R \). Compare this to the work done on the system during the same expansion. **Hints for Solution:** - The Helmholtz free energy for an ideal gas: \[ F = U - TS \] - For an isothermal process (\(T = T_R\)), \[ \Delta F = -k_B T_R \ln \left( \frac{V_B}{V_A} \right) \] for 1 mole (more generally, multiply by \( n \) for \( n \) moles). - The work done by the gas (reversible isothermal expansion): \[ W = nRT_R \ln \left( \frac{V_B}{V_A} \right) \] Note the sign convention: if you want the work *done on* the system, it will be negative. - **Compare:** \( \Delta F \) equals the maximum work extractable in an isothermal process. --- ### 2. **Gibbs Free Energy Minimum Principle** **Task:** For a composite system with a variable property \( X \) (e.g., volume of a subsystem), prove that at equilibrium: \[ \left( \frac{\partial G}{\partial X} \right)_{T,P,N} = 0 \] and \[ \left( \frac{\partial^2 G}{\partial X^2} \right)_{T,P,N} > 0 \] **Hints for Solution:** - At equilibrium, the Gibbs free energy is minimized (for constant \( T \), \( P \), \( N \)), so the first derivative with respect to \( X \) vanishes. - The second derivative must be positive for a minimum (stability condition). --- ### 3. **Work Done via Manipulating Property \( X \)** **Task:** Let \( \delta \bar{W} \) be the work done on the system by manipulating property \( X \) externally. Prove: \[ dG = \delta \bar{W} \] when the system is in contact with a reservoir at fixed \( T_R \) and \( P_R \). **Hints for Solution:** - The differential of the Gibbs free energy: \[ dG = -S dT + V dP + \left( \frac{\partial G}{\partial X} \right)_{T,P,N} dX \] - At constant \( T \) and \( P \), \( dG = \left( \frac{\partial G}{\partial X} \right)_{T,P,N} dX \). - Work done by varying \( X \) is \( \delta \bar{W} \), so \( dG = \delta \bar{W} \). --- If you’d like detailed step-by-step solutions for any of these, let me know which one to start with!