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Chapter 19: Problem 46

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

Short Answer

Expert verified
The entropy change \(ΔS\) of \(n\) moles of an ideal gas when the temperature is changed from \(T_{1}\) to \(T_{2}\) at constant pressure is specifically expressed by the formula: \(ΔS = nC_{p} \ln (T_{2} / T_{1}) \).

Step by step solution

01

Formulate Basic Relation of Entropy Change

The formula for the change in entropy is given by: \(ΔS =∫ q_{rev}/T\). In an isobaric process, the reversible heat exchange \(q_{rev}\) is equal to \(nC_{p}dT\), where \(C_{p}\) is the heat capacity at constant pressure and \(n\) is the number of moles. So we substitute on the entropy change expression: \(ΔS=∫ (nC_{p}dT) / T\).
02

Calculate Integral

The integral bounds are from \(T_{1}\) to \(T_{2}\), so the integral becomes \(ΔS= nC_{p}∫ (dT/T) |_{T_{1}}^{T_{2}} \). The result of the integral is \(\ln (T_{2} / T_{1})\), so after integration it becomes \(ΔS = nC_{p} \ln (T_{2} / T_{1}) \).
03

Conclude the Result

Therefore, the entropy change \(ΔS\) of \(n\) moles of an ideal gas when the temperature is changed from \(T_{1}\) to \(T_{2}\) at constant pressure is: \(ΔS = nC_{p} \ln (T_{2} / T_{1}) \).

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Most popular questions from this chapter

Consider a gas containing an even number \(N\) of molecules, distributed among the two halves of a closed box. Find expressions for (a) the total number of microstates and (b) the number of microstates with half the molecules on each side of the box. (You can either work out a formula, or explore the term "combinations" in a math reference source.) (c) Use these results to find the ratio of the probability that all the molecules will be found on one side of the box to the probability that there will be equal numbers on both sides. (d) Evaluate for \(N=4\) and \(N=100\).

A heat pump extracts energy from groundwater at \(10^{\circ} \mathrm{C}\) and transfers it to water at \(70^{\circ} \mathrm{C}\) to heat a building. Find (a) its COP and (b) its electric power consumption if it supplies heat at the rate of \(20 \mathrm{kW}\). (c) Compare the pump's hourly operating cost with that of an oil furnace if electricity costs \(15.5 \notin / \mathrm{kWh}\) and oil costs \(\$ 2.60 /\) gallon and releases about \(30 \mathrm{kWh} / \mathrm{gal}\) when burned.

Why do refrigerators and heat pumps have different definitions of COP?

A refrigerator maintains an interior temperature of \(4^{\circ} \mathrm{C}\) while its exhaust temperature is \(30^{\circ} \mathrm{C} .\) The refrigerator's insulation is imperfect, and heat leaks in at the rate of 340 W. Assuming the refrigerator is reversible, at what rate must it consume electrical energy to maintain a constant \(4^{\circ} \mathrm{C}\) interior?

Use energy-flow diagrams to show that the existence of a perfect heat engine would permit the construction of a perfect refrigerator, thus violating the Clausius statement of the second law.
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