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Chapter 15: Problem 54

Within an insulated system, \(418.6 \mathrm{kJ}\) of heat is conducted through a copper rod from a hot reservoir at \(+200.0^{\circ} \mathrm{C}\) to a cold reservoir at \(+100.0^{\circ} \mathrm{C} .\) (The reservoirs are so big that this heat exchange does not change their temperatures appreciably.) What is the net change in entropy of the system, in \(\mathrm{kJ} / \mathrm{K} ?\)

Short Answer

Expert verified
Answer: The net change in entropy of the system is \(0.237\,\mathrm{kJ/K}\).

Step by step solution

01

Convert temperatures to Kelvin

First, we need to convert the given temperatures to Kelvin. To do this, add 273.15 to each Celsius temperature: \(T_h = 200.0^\circ\mathrm{C} + 273.15 = 473.15\mathrm{K}\) \(T_c = 100.0^\circ\mathrm{C} + 273.15 = 373.15\mathrm{K}\)
02

Calculate the change in entropy for the hot reservoir

Use the formula \(ΔS_h = \dfrac{Q}{T_h}\) to find the change in entropy for the hot reservoir, where the heat transfer, Q, is negative because it loses heat: \(ΔS_h = \dfrac{-418.6\mathrm{kJ}}{473.15\mathrm{K}} = -0.885\,\mathrm{kJ/K}\)
03

Calculate the change in entropy for the cold reservoir

Use the same formula but with a positive heat transfer, as the cold reservoir gains heat: \(ΔS_c = \dfrac{418.6\mathrm{kJ}}{373.15\mathrm{K}} = 1.122\,\mathrm{kJ/K}\)
04

Calculate the net change in entropy for the system

Add the change in entropy for both reservoirs to find the net change in entropy of the system: \(ΔS_{net} = ΔS_h + ΔS_c = -0.885\,\mathrm{kJ/K} + 1.122\,\mathrm{kJ/K} = 0.237\,\mathrm{kJ/K}\) The net change in entropy of the system is \(0.237\,\mathrm{kJ/K}\).

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Most popular questions from this chapter

A heat pump is used to heat a house with an interior temperature of \(20.0^{\circ} \mathrm{C} .\) On a chilly day with an outdoor temperature of \(-10.0^{\circ} \mathrm{C},\) what is the minimum work that the pump requires in order to deliver \(1.0 \mathrm{kJ}\) of heat to the house?
Suppose 1.00 mol of oxygen is heated at constant pressure of 1.00 atm from \(10.0^{\circ} \mathrm{C}\) to \(25.0^{\circ} \mathrm{C} .\) (a) How much heat is absorbed by the gas? (b) Using the ideal gas law, calculate the change of volume of the gas in this process. (c) What is the work done by the gas during this expansion? (d) From the first law, calculate the change of internal energy of the gas in this process.
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