A unit mass of an ideal gas at temperature T undergoes a...

A unit mass of an ideal gas at temperature $T$ undergoes a reversible isothermal process from pressure $P_{1}$ to pressure $P_{2}$ while losing heat to the surroundings at temperature $T$ in the amount of $q .$ If the gas constant of the gas is $R,$ the entropy change of the gas $\Delta s$ during this process is $(a) \Delta s=R \ln \left(P_{2} / P_{1}\right)$ $(b) \Delta s=R \ln \left(P_{2} / P_{1}\right)-q / T$ $(c) \Delta s=R \ln \left(P_{1} / P_{2}\right)$ $(d) \Delta s=R \ln \left(P_{1} / P_{2}\right)-q / T$ $(e) \Delta s=0$

A unit mass of an ideal gas at temperature $T$ undergoes a reversible isothermal process from pressure $P_{1}$ to pressure $P_{2}$ while losing heat to the surroundings at temperature $T$ in the amount of $q .$ If the gas constant of the gas is $R,$ the entropy change of the gas $\Delta s$ during this process is
$(a) \Delta s=R \ln \left(P_{2} / P_{1}\right)$
$(b) \Delta s=R \ln \left(P_{2} / P_{1}\right)-q / T$
$(c) \Delta s=R \ln \left(P_{1} / P_{2}\right)$
$(d) \Delta s=R \ln \left(P_{1} / P_{2}\right)-q / T$
$(e) \Delta s=0$

Key Concepts

Reversible Isothermal Process

A reversible isothermal process is one in which the system remains in thermal equilibrium with its surroundings throughout the process at constant temperature. In such a process, any change occurs infinitely slowly, ensuring that the system passes through a continuous series of equilibrium states. This idealized process allows for the direct application of thermodynamic formulas, such as calculating the work done or the heat exchanged using integrals, and is a key concept when analyzing entropy changes in systems undergoing heat exchange.

Ideal Gas Behavior

The ideal gas law, PV = RT, is fundamental in relating the pressure, volume, and temperature of an ideal gas. In an isothermal process involving an ideal gas, the constancy of temperature implies that the product of pressure and volume remains constant. This relationship is crucial in deducing how volume and pressure changes relate to one another, and it is used to reformulate expressions for work done and entropy change during the process.

Entropy Change in Thermodynamics

Entropy, a measure of disorder or randomness in a system, changes when heat is exchanged in a reversible process. The change in entropy is given by the integral of the infinitesimal heat transfer divided by the temperature, ?s = ?(dQ_rev)/T. For an isothermal process in an ideal gas, this leads to a logarithmic relationship in terms of pressure or volume changes. Understanding this concept is essential to correctly determine the entropy change during processes where heat is transferred between a gas and its surroundings.

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