Question

One mole of a monoatomic ideal gas starting from state A, goes through B and C to state D, as shown in the figure. Total change in entropy (in J K\(^{-1}\)) during this process is ............... 

Hint:

For isothermal processes, use the formula \( \Delta S = nR \ln \left( \frac{V_f}{V_i} \right) \) to calculate entropy changes based on volume changes. Remember that no entropy change occurs for constant pressure processes with no volume change.

Answer 1

The total change in entropy \( \Delta S \) for an ideal gas during a process can be calculated using the formula: \[ \Delta S = nR \ln \left( \frac{V_f}{V_i} \right) \] where:
- \( n \) is the number of moles of the gas,
- \( R \) is the gas constant (8.314 J mol\(^{-1}\) K\(^{-1}\)),
- \( V_f \) and \( V_i \) are the final and initial volumes, respectively.
We can calculate the entropy change for each part of the process (A to B, B to C, C to D):
1. A to B: Isothermal compression (constant temperature)
The volume changes from 20 dm\(^3\) to 10 dm\(^3\), so the change in entropy is: \[ \Delta S_{AB} = nR \ln \left( \frac{V_B}{V_A} \right) = 1 \times 8.314 \times \ln \left( \frac{10}{20} \right) \] \[ \Delta S_{AB} = 8.314 \times \ln \left( 0.5 \right) = -8.314 \times 0.6931 = -5.76 \, \text{J K}^{-1} \] 2. B to C: Isothermal expansion (constant temperature)
The volume increases from 10 dm\(^3\) to 20 dm\(^3\), so the change in entropy is: \[ \Delta S_{BC} = 1 \times 8.314 \times \ln \left( \frac{V_C}{V_B} \right) = 8.314 \times \ln \left( \frac{20}{10} \right) \] \[ \Delta S_{BC} = 8.314 \times \ln \left( 2 \right) = 8.314 \times 0.6931 = 5.76 \, \text{J K}^{-1} \] 3. C to D: Isobaric process (constant pressure)
There is no volume change, so the entropy change is zero: \[ \Delta S_{CD} = 0 \] Step 1: Total entropy change.
The total change in entropy is the sum of the individual changes: \[ \Delta S_{\text{total}} = \Delta S_{AB} + \Delta S_{BC} + \Delta S_{CD} \] \[ \Delta S_{\text{total}} = -5.76 + 5.76 + 0 = 0 \, \text{J K}^{-1} \] Final Answer: \[ \boxed{0} \]