Work done per mole in an isothermal ? change is (A) RT log e(v2 / v1) (B) RT log10(v2 / v1) (C)

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Work done per mole in an isothermal ? change is (A) RT log...

Work done per mole in an isothermal ? change is (A) RT log $_{e}\left(\mathrm{v}_{2} / \mathrm{v}_{1}\right)$ (B) $\mathrm{RT} \log _{10}\left(\mathrm{v}_{2} / \mathrm{v}_{1}\right)$ (C) RT $\log _{10}\left(\mathrm{v}_{1} / \mathrm{v}_{2}\right)$ (D) RT $\log _{\mathrm{e}}\left(\mathrm{v}_{1} / \mathrm{v}_{2}\right)$

Work done per mole in an isothermal ? change is
(A) RT log $_{e}\left(\mathrm{v}_{2} / \mathrm{v}_{1}\right)$
(B) $\mathrm{RT} \log _{10}\left(\mathrm{v}_{2} / \mathrm{v}_{1}\right)$
(C) RT $\log _{10}\left(\mathrm{v}_{1} / \mathrm{v}_{2}\right)$
(D) RT $\log _{\mathrm{e}}\left(\mathrm{v}_{1} / \mathrm{v}_{2}\right)$

Key Concepts

Isothermal Process

An isothermal process is one in which the temperature of the system remains constant throughout the process. For an ideal gas, this means that the internal energy does not change with volume variations, and the work done is entirely due to pressure–volume changes under constant temperature.

Ideal Gas Law

The ideal gas law, given by PV = nRT, relates the pressure, volume, and temperature of an ideal gas. In the context of an isothermal process, since the temperature (T) is constant, any change in volume (V) results in an inversely proportional change in pressure (P), which is crucial for determining the work done during expansion or compression.

Work Done in an Isothermal Process

The work done by (or on) a gas during an isothermal expansion or compression is determined by integrating the pressure with respect to volume. Using the ideal gas law to substitute for pressure, this integration leads to a natural logarithmic relationship, typically expressed as nRT ln(V2/V1), indicating that the work depends logarithmically on the ratio of the final to initial volumes.

Logarithmic Integration

The key mathematical step in calculating work in an isothermal process involves integrating the function 1/V with respect to V, which results in the natural logarithm, ln(V). This is why the expression for work uses the natural logarithm rather than logarithms of other bases, ensuring the correct derivation of the formula nRT ln(V2/V1).

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