Question

An ideal gas at a given state expands to a fixed final volume first at constant pressure and then at constant temperature. For which case is the work done greater?

Short answer

Answer: Without specific numerical values for pressure, temperature, and volume, we cannot definitively say which process results in greater work done. However, if the initial and final pressures and temperatures are the same in both processes, the isothermal process typically results in greater work done, as the logarithmic term takes into account the initial and final volumes, whereas the isobaric process only considers the difference in addition to non-varying pressure.

Step-by-step solution

Recall the Ideal Gas Law

The Ideal Gas Law is given by the equation PV=nRT, where P represents pressure, V represents volume, n represents the number of moles of gas, R represents the ideal gas constant, and T represents temperature.

Understand the isobaric and isothermal processes

In an isobaric (constant pressure) process, the pressure remains constant while volume and temperature can change. In an isothermal (constant temperature) process, the temperature remains constant while pressure and volume can change.

Calculate work done during isobaric process

In an isobaric process, work done is given by the formula: W_isobaric = P * (V_final - V_initial)

Calculate work done during isothermal process

In an isothermal process, the work done by an ideal gas is given by the formula: W_isothermal = nRT * ln(V_final/V_initial)

Compare the work done in both processes

To determine which process results in a greater amount of work done, we need to compare W_isobaric and W_isothermal. Since the final volume (V_final) is the same in both cases, we need to examine how the different processes affect work done. In the isobaric case, work done is proportional to the change in volume (V_final - V_initial) and the constant pressure P. In the isothermal case, work done is proportional to the natural logarithm of the ratio of final volume to initial volume (ln(V_final/V_initial)) and the constant temperature T. Without specific numerical values for pressure, temperature, and volume, we cannot definitively say which process results in greater work done. However, we can say that if the initial and final pressures and temperatures are the same in both processes, the isothermal process typically results in greater work done as the logarithmic term takes into account the initial and final volumes, whereas the isobaric process only considers the difference in addition to a non varying P.