Give step-by-step solution with explanation and final answer:Extremum conditions
1. Helmholtz free energy for the ideal gas
166 Extremum Principles
From the expression for the Helmholtz free energy of the ideal gas, calculate
AF for an isothermal expansion from volume Vj to Vp when the system is in
equilibrium with a thermal reservoir at temperature Tp. Compare this to the
work done on the system during the same expansion.
2. Gibbs free energy minimum principle
Consider a composite system with a property X that can be varied externally.
[For example, X could be the volume of a subsystem, as discussed in class.]
Prove that at equilibrium,
IG
(5%),
X) pn
and
le )
“v5 >0
(ax 2) ren
3. Let dW be the work done on the system by manipulating the property X from
outside the system.
Prove that dG = dW, when the system is in contact with a reservoir that fixes
the temperature at Tz and the pressure at Pp.
Question:
Give step-by-step solution with explanation and final answer:Extremum conditions
1. Helmholtz free energy for the ideal gas
166 Extremum Principles
From the expression for the Helmholtz free energy of the ideal gas, calculate
AF for an isothermal expansion from volume Vj to Vp when the system is in
equilibrium with a thermal reservoir at temperature Tp. Compare this to the
work done on the system during the same expansion.
2. Gibbs free energy minimum principle
Consider a composite system with a property X that can be varied externally.
[For example, X could be the volume of a subsystem, as discussed in class.]
Prove that at equilibrium,
IG
(5%),
X) pn
and
le )
“v5 >0
(ax 2) ren
3. Let dW be the work done on the system by manipulating the property X from
outside the system.
Prove that dG = dW, when the system is in contact with a reservoir that fixes
the temperature at Tz and the pressure at Pp.
This is AI generated Response.
Asked by: Adapaka Jagadeesh
Created at: 2025-07-30 09:59:11
Final Answer:
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Let's tackle each part step by step:
---
### **1. Helmholtz Free Energy for the Ideal Gas (Isothermal Expansion)**
#### **Step 1: Helmholtz Free Energy Expression**
For an ideal gas, the Helmholtz free energy \( F \) is:
\[
F = U - TS
\]
where:
- \( U \) = internal energy
- \( T \) = temperature
- \( S \) = entropy
For an ideal gas, \( U = nC_vT \) (but for isothermal processes, \( U \) doesn't change), and the entropy change is:
\[
\Delta S = nR \ln\left(\frac{V_B}{V_A}\right)
\]
#### **Step 2: Change in Helmholtz Free Energy (\(\Delta F\))**
For an isothermal expansion from \( V_A \) to \( V_B \) at temperature \( T_R \):
\[
\Delta F = \Delta U - T_R \Delta S
\]
But for an ideal gas in an isothermal process, \( \Delta U = 0 \), so:
\[
\Delta F = -T_R \Delta S = -T_R nR \ln\left(\frac{V_B}{V_A}\right)
\]
#### **Step 3: Work Done by the System**
For isothermal expansion, the work done **by the gas** is:
\[
W = \int_{V_A}^{V_B} P dV = nRT_R \int_{V_A}^{V_B} \frac{dV}{V} = nRT_R \ln\left(\frac{V_B}{V_A}\right)
\]
#### **Step 4: Compare \(\Delta F\) and \(W\)**
- \( \Delta F = -nRT_R \ln\left(\frac{V_B}{V_A}\right) \)
- \( W = nRT_R \ln\left(\frac{V_B}{V_A}\right) \)
So,
\[
\Delta F = -W
\]
#### **Final Answer:**
**The change in Helmholtz free energy for an isothermal expansion of an ideal gas from \( V_A \) to \( V_B \) at temperature \( T_R \) is:**
\[
\boxed{\Delta F = -nRT_R \ln\left(\frac{V_B}{V_A}\right)}
\]
**This is equal in magnitude but opposite in sign to the work done by the system during the same expansion:**
\[
\Delta F = -W
\]
---
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Extremum conditions
1. Helmholtz free energy for the ideal gas
166 Extremum Principles
From the expression for the Helmholtz free energy of the ideal gas, calculate
AF for an isothermal expansion from volume Vj to Vp when the system is in
equilibrium with a thermal reservoir at temperature Tp. Compare this to the
work done on the system during the same expansion.
2. Gibbs free energy minimum principle
Consider a composite system with a property X that can be varied externally.
[For example, X could be the volume of a subsystem, as discussed in class.]
Prove that at equilibrium,
IG
(5%),
X) pn
and
le )
“v5 >0
(ax 2) ren
3. Let dW be the work done on the system by manipulating the property X from
outside the system.
Prove that dG = dW, when the system is in contact with a reservoir that fixes
the temperature at Tz and the pressure at Pp.