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Chapter 19: Problem 51

The temperature of \(n\) moles of ideal gas is changed from \(T_{1}\) to \(T_{2}\) at constant volume. Show that the corresponding entropy change is \(\Delta S=n C_{V} \ln \left(T_{2} / T_{1}\right)\).

Short Answer

Expert verified
\(\Delta S = nC_v \ln (T2 / T1)\)

Step by step solution

01

Definition of entropy

Recall the differential of entropy, \(dS\), for a reversible process, which is defined as: \(dS = \frac{dq_{rev}}{T}\) where dq_{rev} is the infinitesimal heat transferred in a reversible process and T is the absolute temperature.
02

Expression of heat in terms of specific heat

The amount of heat, \(dq_{rev}\), for an ideal gas at constant volume, is given by \(dq_{rev} = nC_v dT\). Here, \(C_v\) is the molar specific heat at constant volume, dT is the infinitesimal change in temperature and n is the number of moles of gas.
03

Plug \(dq_{rev}\) into the entropy equation

Substitute \(dq_{rev} = nC_v dT\) into the equation from step 1 to get: \(dS = \frac{nC_v dT}{T}\)
04

Integrate both sides to find the entropy change

Integrating both sides from \(T1\) to \(T2\) gives the total change in entropy, \(\Delta S = nC_v \int_{T1}^{T2} \frac{dT}{T}\). Solving the integral on the right side yields: \(\Delta S = nC_v \ln (T2 / T1)\)

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Most popular questions from this chapter

A Carnot engine absorbs \(950 \mathrm{~J}\) of heat each cycle and provides \(360 \mathrm{~J}\) of work. (a) What's its efficiency? (b) How much heat is rejected each cycle? (c) If the engine rejects heat at \(10^{\circ} \mathrm{C}\), what's its maximum temperature?

In an adiabatic free expansion, \(6.36\) mol of ideal gas at \(305 \mathrm{~K}\) expands 15 -fold in volume. How much energy becomes unavailable to do work?

A box containing 10 coins lying heads up is found to have only 5 coins heads up after shaking. Is there any positive entropy change?

A cosmic heat engine might operate between the Sun's \(5800 \mathrm{~K}\) surface and the \(2.7 \mathrm{~K}\) temperature of intergalactic space. What would be its maximum efficiency?

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