Understanding Isothermal Process Work Done

An isothermal process is a thermodynamic process where the temperature of a system remains constant (T = constant) throughout the process. For an ideal gas undergoing such a process, we can determine the work done.

Derivation of Work Done Formula

The work done ($W$) during any thermodynamic process is generally defined by the integral:

$W = \int_{V_1}^{V_2} P \, dV$

Where:

• $P$ is the pressure of the gas.
• $V_1$ is the initial volume.
• $V_2$ is the final volume.
• $dV$ is the infinitesimal change in volume.

For an ideal gas, the relationship between pressure, volume, and temperature is given by the ideal gas law:

$PV = nRT$

Where:

• $n$ is the number of moles of the gas.
• $R$ is the ideal gas constant.
• $T$ is the absolute temperature.

Since the process is isothermal, the temperature ($T$) is constant. We can rearrange the ideal gas law to express pressure ($P$) in terms of volume ($V$):

$P = \frac{nRT}{V}$

Now, substitute this expression for $P$ into the work integral:

$W = \int_{V_1}^{V_2} \frac{nRT}{V} \, dV$

Because $n$, $R$, and $T$ are constants during this isothermal process, we can pull them out of the integral:

$W = nRT \int_{V_1}^{V_2} \frac{1}{V} \, dV$

The integral of $\frac{1}{V}$ with respect to $V$ is $\ln V$ (natural logarithm of V). Evaluating this from $V_1$ to $V_2$:

$W = nRT [\ln V]_{V_1}^{V_2}$

$W = nRT (\ln V_2 - \ln V_1)$

Using the properties of logarithms ($\ln a - \ln b = \ln \frac{a}{b}$), we get the final formula for the work done in an isothermal process:

$W = nRT \ln \frac{V_2}{V_1}$

Analysis of Options

• Option 1: $W = \frac{PV}{T_2-T_1}$ is dimensionally incorrect for work.
• Option 2: $W=nRTln\frac{P_1}{P_2}$ is also a correct formula for isothermal work, derived using $\frac{V_2}{V_1} = \frac{P_1}{P_2}$ from Boyle's Law (applicable to isothermal processes).
• Option 3: $W=nRTln\frac{V_2}{V_1}$ is the standard formula derived directly from integrating $P\,dV$ using the ideal gas law for a constant temperature process. This matches the derivation.
• Option 4: $W=PV(T_2-T_1)$ is not the correct formula for work done in an isothermal process.

Conclusion

The work done by an ideal gas during an isothermal process is correctly represented by the formula involving the natural logarithm of the ratio of final to initial volumes (or initial to final pressures).