Question

Calculate the work done when 1 mole of an ideal gas is compressed reversibly from 1.0 bar to 4.00 bar at a constant temperature of 300 K:

• A. 3.46 kJ
• B. -8.20 kJ
• C. 18.02 kJ
• D. -14.01 kJ

Answer 1

Answer: (A)

3.46 kJ

Answer 2

The problem asks to calculate the work done when 1 mole of an ideal gas is compressed reversibly and isothermally.

1. Identify the process: The gas is compressed reversibly at a constant temperature (isothermal process).

2. Formula for reversible isothermal work: For an ideal gas undergoing a reversible isothermal process, the work done (W) can be calculated using the formula: $W = -nRT \ln \left(\frac{V_2}{V_1}\right)$ Since it's an isothermal process, according to Boyle's Law ($P_1V_1 = P_2V_2$), we have $\frac{V_2}{V_1} = \frac{P_1}{P_2}$. Substituting this into the work equation: $W = -nRT \ln \left(\frac{P_1}{P_2}\right)$ Alternatively, using the property of logarithms $\ln(a/b) = -\ln(b/a)$, we can write: $W = nRT \ln \left(\frac{P_2}{P_1}\right)$ For compression, work is done on the system, so the work done (W) should be positive according to the IUPAC convention in chemistry. The second form of the equation directly yields a positive value for compression ($P_2 > P_1$).

3. Given values:

   • Number of moles (n) = 1 mole
   • Initial pressure (P₁) = 1.0 bar
   • Final pressure (P₂) = 4.00 bar
   • Temperature (T) = 300 K
   • Gas constant (R) = 8.314 J mol⁻¹ K⁻¹ (since the answer is required in kJ, using J mol⁻¹ K⁻¹ is appropriate, and then converting to kJ)

4. Calculation: Substitute the values into the formula $W = nRT \ln \left(\frac{P_2}{P_1}\right)$: $W = (1 \text{ mol}) \times (8.314 \text{ J mol}^{-1} \text{ K}^{-1}) \times (300 \text{ K}) \times \ln \left(\frac{4.00 \text{ bar}}{1.0 \text{ bar}}\right)$ $W = (1 \times 8.314 \times 300) \times \ln(4)$ $W = 2494.2 \times 1.38629$ (approximately, as $\ln(4) \approx 1.38629$) $W = 3456.98 \text{ J}$

5. Convert to kilojoules (kJ): $W = \frac{3456.98}{1000} \text{ kJ}$ $W = 3.45698 \text{ kJ}$

6. Round to appropriate significant figures: $W \approx 3.46 \text{ kJ}$

Since it's compression, work is done on the gas, and the positive sign indicates work done on the system.