What quantity must be integrated to find the entropy change between two equilibrium states?

For a reversible heat transfer \(Q_{\mathrm{rev}}\) at constant temperature \(T\), what is \(\Delta S\)?

A thermal reservoir at \(300\,\mathrm{K}\) absorbs \(1200\,\mathrm{J}\) of heat. Find its entropy change.

A thermal reservoir at \(500\,\mathrm{K}\) releases \(750\,\mathrm{J}\) of heat. Find its entropy change.

Find the entropy change when \(0.020\,\mathrm{kg}\) of ice melts reversibly at \(273\,\mathrm{K}\). Use \(L_f=3.34\times10^5\,\mathrm{J\,kg^{-1}}\).

Find the entropy change of \(0.010\,\mathrm{kg}\) of water when it vaporizes reversibly at \(373\,\mathrm{K}\). Use \(L_v=2.26\times10^6\,\mathrm{J\,kg^{-1}}\).

One mole of ideal gas expands isothermally from \(V_i\) to \(2V_i\). Find \(\Delta S\).

Two moles of ideal gas are compressed isothermally from \(3.0\,\mathrm{L}\) to \(1.0\,\mathrm{L}\). Find \(\Delta S\).

A body with constant heat capacity \(C=80\,\mathrm{J\,K^{-1}}\) is warmed reversibly from \(300\,\mathrm{K}\) to \(360\,\mathrm{K}\). Find \(\Delta S\).

A monatomic ideal gas with \(n=2.0\,\mathrm{mol}\) is heated at constant volume from \(300\,\mathrm{K}\) to \(450\,\mathrm{K}\). Find \(\Delta S\). Use \(C_V=3R/2\).

A monatomic ideal gas with \(n=1.0\,\mathrm{mol}\) is heated at constant pressure from \(300\,\mathrm{K}\) to \(600\,\mathrm{K}\). Find \(\Delta S\). Use \(C_P=5R/2\).

A \(500\,\mathrm{J}\) heat transfer occurs directly from a \(400\,\mathrm{K}\) reservoir to a \(250\,\mathrm{K}\) reservoir. Find \(\Delta S_{\mathrm{univ}}\) and decide whether the process is reversible.

An ideal gas freely expands into vacuum in an insulated container until its volume doubles. For \(n=1.5\,\mathrm{mol}\), find \(Q\), \(W\), \(\Delta U\), and \(\Delta S\).

A system undergoes a process with \(\Delta S_{\mathrm{sys}}=-18\,\mathrm{J\,K^{-1}}\). It dumps heat into a large reservoir at \(300\,\mathrm{K}\). What is the minimum heat the reservoir must receive for the process to be allowed by the second law?

An ideal gas changes from \((T_i,V_i)\) to \((2T_i,3V_i)\). Derive \(\Delta S\) in terms of \(n\), \(C_V\), and \(R\).

A hot reservoir at \(T_h\) transfers heat \(Q\) directly to a cold reservoir at \(T_c\), where \(T_h>T_c\). Derive \(\Delta S_{\mathrm{univ}}\) and explain why the result is positive.

Two identical objects of heat capacity \(C\) are initially at \(T_h\) and \(T_c\), with \(T_h>T_c\). They are placed in thermal contact and isolated from the surroundings. Derive the final temperature and the total entropy change, assuming constant \(C\).

An ideal gas has the same initial and final states for two processes: reversible isothermal expansion from \(V_i\) to \(V_f\), and free expansion into vacuum from \(V_i\) to \(V_f\). Compare \(Q\), \(W\), \(\Delta U\), and \(\Delta S\) for the two processes, and explain what this shows about entropy.

A process takes an ideal gas from state \(i\) to state \(f\). You know \(T_i,T_f,V_i,V_f,n,\) and \(C_V\), but the actual path is irreversible. Explain how to calculate \(\Delta S\), and derive the expression you would use.

A system interacts only with a single thermal reservoir at temperature \(T_0\). During a cycle, the system returns to its initial state and receives net heat \(Q\) from the reservoir. Use entropy to determine the allowed sign of \(Q\), and interpret the result physically.