Entropy Change of an Ideal Gas during Isothermal Expansion

Given:
- 1 mole of an ideal gas
- Isothermal expansion at 315 K
- Initial pressure: 5 atm
- Final pressure: 2.5 atm

Entropy Change:
The entropy change during a process can be calculated using the equation:
ΔS = nRln(V2/V1)
where ΔS is the entropy change, n is the number of moles, R is the gas constant, V1 is the initial volume, and V2 is the final volume.

Determining the Initial and Final Volumes:
To calculate the entropy change, we need to determine the initial and final volumes of the gas. Since the process is isothermal, we can use the ideal gas law equation:
PV = nRT

Initial Volume:
V1 = (nRT1) / P1
= (1 mol * 0.0821 L·atm/(mol·K) * 315 K) / 5 atm
= 4.951 L

Final Volume:
V2 = (nRT2) / P2
= (1 mol * 0.0821 L·atm/(mol·K) * 315 K) / 2.5 atm
= 9.902 L

Therefore, the initial volume (V1) is 4.951 L and the final volume (V2) is 9.902 L.

Calculating the Entropy Change:
Using the equation for entropy change, we can substitute the values:
ΔS = (1 mol * 0.0821 L·atm/(mol·K)) * ln(9.902 L / 4.951 L)
= 0.0821 L·atm/(mol·K) * ln(2)
≈ 0.057 L·atm/(mol·K)

Thus, the entropy change when 1 mole of an ideal gas expands isothermally from 5 atm to 2.5 atm at 315 K is approximately 0.057 L·atm/(mol·K).

Explanation:
During an isothermal expansion, the gas is in thermal equilibrium with its surroundings, so the temperature remains constant. As the gas expands, it does work on its surroundings, leading to an increase in entropy. The entropy change is positive because the system becomes more disordered as it expands.

The entropy change is calculated using the equation ΔS = nRln(V2/V1). By substituting the values of n, R, V1, and V2, we can determine the entropy change.

In this case, the initial volume is smaller than the final volume, indicating that the gas expands. As a result, the natural logarithm term in the entropy change equation is greater than 1, leading to a positive entropy change.

It's important to note that entropy is a state function, meaning it only depends on the initial and final states of the system. The specific pathway taken during the expansion does not affect the entropy change, as long as the temperature remains constant.

In summary, the entropy change during the isothermal expansion of an ideal gas from 5 atm to 2.5 atm at 315 K is