In an isothermal expansion of an ideal gas, the mechanical work done can be expressed using the following equation:
W = nRT ln(Vf/Vi)
where: W is the mechanical work done by the gas during the expansion. n is the number of moles of the gas. R is the ideal gas constant (8.314 J/(mol·K) or 0.0821 L·atm/(mol·K)). T is the temperature of the gas in Kelvin. Vf is the final volume of the gas. Vi is the initial volume of the gas.
This equation is derived from the First Law of Thermodynamics for an isothermal process, which states that the change in internal energy (ΔU) of a system is zero. Since ΔU = Q - W, and Q (heat) is assumed to be zero for an isothermal process, the work done by the gas is equal in magnitude but opposite in sign to the heat transferred.
The equation for work in an isothermal expansion shows that the work done depends on the initial and final volumes of the gas. If the gas expands (Vf > Vi), the natural logarithm term will be positive, indicating positive work done by the gas. Conversely, if the gas is compressed (Vf < Vi), the natural logarithm term will be negative, indicating negative work done on the gas.
It's important to note that this equation assumes an ideal gas and an isothermal process, where the temperature of the gas remains constant throughout the expansion.