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Chapter 19: Problem 61

Find an expression for the entropy gain when hot and cold water are irreversibly mixed. A corresponding reversible process you can use to calculate this change is to bring each water sample slowly to their common final temperature \(T_{\mathrm{f}}\) and then mix them. Express your answer in terms of the initial temperatures \(T_{\mathrm{h}}\) and \(T_{\mathrm{c}} .\) Assume equal masses of hot and cold water, with constant specific heat \(c .\) What's the sign of your answer?

Short Answer

Expert verified
The total entropy gain for the irreversible process of mixing hot and cold water can be calculated using the formula: \( \Delta S_{\mathrm{total}} = c \ln{\frac{T_{\mathrm{f}}}{T_{\mathrm{h}}}} + c \ln{\frac{T_{\mathrm{f}}}{T_{\mathrm{c}}}} \). The sign of the entropy gain is positive, indicating an increase in disorder.

Step by step solution

01

Calculate Entropy Change for the Cooling Process

To find the entropy change, ΔS, for the cooling process of hot water, we use the following expression: \( \Delta S_{\mathrm{h}} = c \ln{\frac{T_{\mathrm{f}}}{T_{\mathrm{h}}}} \) Here, \( T_{\mathrm{h}} \) is the initial temperature of hot water, \( T_{\mathrm{f}} \) is the final common temperature and c is the specific heat. We must ensure that all temperatures are measured in Kelvin.
02

Calculate Entropy Change for the Heating Process

Next, we calculate the entropy change, ΔS, for the heating process of cool water using the following expression: \( \Delta S_{\mathrm{c}} = c \ln{\frac{T_{\mathrm{f}}}{T_{\mathrm{c}}}} \) Here, \( T_{\mathrm{c}} \) is the initial temperature of cold water.
03

Calculate Total Entropy Gain

The total entropy gain for this irreversible process of mixing is the sum of the entropy changes due to cooling of hot water and heating of cold water: \( \Delta S_{\mathrm{total}} = \Delta S_{\mathrm{h}} + \Delta S_{\mathrm{c}} \).
04

Interpret the Sign of the Answer

The sign of the total entropy gain, ΔS, can be positive, negative or zero. If ΔS is positive, the disorder of the system is increased; if ΔS is negative, the order of the system is increased; if ΔS equals zero, there is no change in the order or disorder of the system. In this case, because heat is transferred from a high temperature source (hot water) to a low temperature source (cold water), the total entropy gain would be positive, reflecting an increase in disorder.

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Most popular questions from this chapter

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?

The molar specific heat at constant pressure for a certain gas is given by \(C_{p}=a+b T+c T^{2},\) where \(a=33.6 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\), \(b=2.93 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2},\) and \(c=2.13 \times 10^{-5} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{3} .\) Find the entropy change when 2 moles of this gas are heated from \(20^{\circ} \mathrm{C}\) to \(200^{\circ} \mathrm{C}\).

You operate a store that's heated by an oil furnace supplying 30 kWh of heat from each gallon of oil. You're considering switching to a heat-pump system. Oil costs \(\$ 1.75 /\) gallon, and electricity costs \(16.5 \notin / \mathrm{kWh} .\) What's the minimum heat-pump COP that will reduce your heating costs?

A Carnot engine extracts heat from a block of mass \(m\) and specific heat \(c\) initially at temperature \(T_{\mathrm{h} 0}\) but without a heat source to maintain that temperature. The engine rejects heat to a reservoir at constant temperature \(T_{\mathrm{c}} .\) The engine is operated so its mechanical power output is proportional to the temperature difference \(T_{\mathrm{h}}-T_{\mathrm{c}}\) : $$P=P_{0} \frac{T_{\mathrm{h}}-T_{\mathrm{c}}}{T_{\mathrm{h} 0}-T_{\mathrm{c}}}$$ where \(T_{\mathrm{h}}\) is the instantaneous temperature of the hot block and \(P_{0}\) is the initial power. (a) Find an expression for \(T_{\mathrm{h}}\) as a function of time, and (b) determine how long it takes for the engine's power output to reach zero.

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 \(\Delta S=n C_{V} \ln \left(T_{2} / T_{1}\right)\).
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