A gas undergoes an isothermal expansion against a constant external pressure in stepwise manner such that the external pressure is equal to the final pressure at the end of each step. The magnitude of work done by the gas will be largest when the expansion is carried out in

eight steps

The process described involves an isothermal expansion, which means the temperature of the gas remains constant throughout the expansion. The work done by the gas during such a process depends on the external pressure and the manner in which the expansion occurs. When a gas undergoes expansion in such a manner, the work done is determined by the integration of the pressure-volume relationship over the expansion steps.

In a stepwise expansion, the gas expands in increments where the external pressure is adjusted to match the internal pressure at the end of each step. This stepwise process allows the gas to do more work compared to a single-step process because the external pressure opposes the gas expansion more gradually and effectively.

As the number of steps increases, the external pressure at each step becomes closer to the internal pressure of the gas. This reduces the inefficiencies associated with abrupt changes in pressure, thereby increasing the magnitude of work done by the gas. Mathematically, this is because the work done (W) in an isothermal process is given by:

W = nRT ln(Vf/Vi)

where Vf is the final volume, Vi is the initial volume, and nRT remains constant for isothermal conditions. However, in stepwise expansion, the effective work is maximized as the pressure gradient is minimized.

In the given options, eight steps represent the largest number of steps. Therefore, the magnitude of work done by the gas is greatest when the expansion is carried out in eight steps.

Thus, the correct answer is d) eight steps.