Answer
🔧 Thermodynamics: Isothermal Compression
🧪 Problem Overview
A gas at 22°C (295 K) is compressed isothermally from a pressure of 120 kPa and volume of 4.2 m³, with a final volume that is one-third of the original.
- R = 0.288 kJ/kg·K
- Cv = 0.712 kJ/kg·K
- Volume ratio:
V₁ : V₂ = 3 : 1
📐 Calculations
1. Final Volume (V₂)
V₂ = V₁ / 3 = 4.2 / 3 = 1.4 m³
2. Mass of the Gas (m)
m = (P₁ × V₁) / (R × T)
m = (120 × 4.2) / (0.288 × 295) ≈ 5.93 kg
3. Final Pressure (P₂)
P₂ = (P₁ × V₁) / V₂ = (120 × 4.2) / 1.4 = 360 kPa
4. Work Done (W)
W = m × R × T × ln(V₁ / V₂)
W = 5.93 × 0.288 × 295 × ln(3) ≈ 556.3 kJ
5. Change in Internal Energy (ΔU)
ΔU = 0 kJ (isothermal process)
6. Heat Transferred (Q)
Q = ΔU + W = 0 + 556.3 = 556.3 kJ
7. Entropy Change (ΔS)
ΔS = m × R × ln(V₁ / V₂)
ΔS = 5.93 × 0.288 × ln(3) ≈ −1.88 kJ/K (entropy decreases)
✅ Final Results
| Quantity | Value |
|---|---|
| Final Volume (V₂) | 1.4 m³ |
| Mass (m) | 5.93 kg |
| Final Pressure (P₂) | 360 kPa |
| Work Done (W) | 556.3 kJ |
| Change in Internal Energy (ΔU) | 0 kJ |
| Heat Transferred (Q) | 556.3 kJ |
| Entropy Change (ΔS) | −1.88 kJ/K |
📘 Conclusion
In an isothermal process:
- Internal energy remains constant (ΔU = 0).
- Work done by the gas equals the heat exchanged.
- Entropy decreases in compression due to reduced volume.