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One mole of an ideal monoatomic gas expa...

One mole of an ideal monoatomic gas expands isothermally against constant external pressure of 1 atm from initial volume of 1l to a state where its final pressure becomes equal to external pressure. If initial temperature of gas is 300K then total entropy change of system in the above process is: [R =0.082L atm `mol^(-1)K^(-1)= 8.3J mol^(-1) K^(-1)`]

A

0

B

R ln (24.6)

C

R ln (2490)

D

`(3)/(2)` R ln (24.6)

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will calculate the total entropy change of the system during the isothermal expansion of the ideal monoatomic gas. ### Step 1: Identify the Given Values - Number of moles (n) = 1 mole - Initial volume (V_initial) = 1 L - External pressure (P_external) = 1 atm - Initial temperature (T) = 300 K - Ideal gas constant (R) = 0.082 L atm K⁻¹ mol⁻¹ ### Step 2: Calculate the Final Volume (V_final) Using the ideal gas law, we can find the final volume when the gas expands isothermally against a constant external pressure. \[ V_{final} = \frac{nRT}{P} \] Substituting the known values: \[ V_{final} = \frac{1 \text{ mol} \times 0.082 \text{ L atm K}^{-1} \text{ mol}^{-1} \times 300 \text{ K}}{1 \text{ atm}} = 24.6 \text{ L} \] ### Step 3: Calculate the Change in Entropy (ΔS) The change in entropy for an isothermal process can be calculated using the formula: \[ \Delta S_{system} = nR \ln\left(\frac{V_{final}}{V_{initial}}\right) \] Substituting the values we have: \[ \Delta S_{system} = 1 \text{ mol} \times 0.082 \text{ L atm K}^{-1} \text{ mol}^{-1} \times \ln\left(\frac{24.6 \text{ L}}{1 \text{ L}}\right) \] Calculating the logarithm: \[ \ln\left(24.6\right) \approx 3.201 \] Now substituting this back into the equation for ΔS: \[ \Delta S_{system} = 0.082 \times 3.201 \approx 0.262 \text{ L atm K}^{-1} \text{ mol}^{-1} \] ### Step 4: Convert to Joules Since 1 L atm = 101.325 J, we convert the entropy change into Joules: \[ \Delta S_{system} \approx 0.262 \times 101.325 \approx 26.6 \text{ J K}^{-1} \] ### Final Answer The total entropy change of the system is approximately **26.6 J K⁻¹**. ---
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