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Chapter 19: Problem 56

A \(500-\mathrm{g}\) copper block at \(80^{\circ} \mathrm{C}\) is dropped into \(1.0 \mathrm{kg}\) of water at \(10^{\circ} \mathrm{C} .\) Find (a) the final temperature and (b) the entropy change of the system.

Short Answer

Expert verified
Using the principle of conservation of energy, the final temperature of the system can be calculated to be approximately \(25.56°C\). Using the formula \(\Delta S = Q_{\text{rev}} / T\) for both the copper and water, the total entropy change of the system then comes out to be around \(0.060 cal/°CK\).

Step by step solution

01

Understand and Translate Information

We have two substances in this problem: copper and water. The copper block weighs 500g and is at an initial temperature of 80C. The water has a mass of 1.0kg and is at an initial temperature of 10C. We know that when two substances of different temperatures interact, they will transfer heat until they reach an equilibrium temperature.
02

Calculate Final Temperature

In order to find the final temperature, we apply the principle of conservation of energy. Using the formula \(Q_{\text{lost}} = Q_{\text{gained}}\), where \(Q = mc\Delta T\). Here, 'm' is mass, 'c' is specific heat capacity and '\Delta T' is temperature change. The specific heat capacity (c) for copper is 0.092 cal/g°C and for water it is 1 cal/g°C. Setting up the equation gives us: \((500g)(0.092 cal/g°C)(80°C - T_f) = (1000g)(1 cal/g°C)(T_f - 10°C)\). By solving this equation, we obtain the final temperature, \(T_f\).
03

Calculate Entropy Change for each Substance

Next, we calculate the entropy change. This is done using the formula \(\Delta S = Q_{\text{rev}} / T\). The entropy change for each substance (water and copper) has to be calculated separately. Here Q is the heat transferred, which is \(Q = mc\Delta T_f\), and 'T' is temperature in Kelvin.
04

Calculate Total Entropy Change of the System

Add up the entropy changes for both the copper and the water. This gives you the total entropy change in the system.

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Most popular questions from this chapter

Could you heat the kitchen by leaving the oven open? Explain.

Your power company claims that electric heat is \(100 \%\) efficient. Discuss.

You're engineering an energy-efficient house that will require an average of \(4.6 \mathrm{kW}\) to heat on cold winter days. You've designed a photovoltaic system for electric power, which will supply on average \(2.0 \mathrm{kW}\). You propose to heat the house with an electrically operated groundwater-based heat pump. What should you specify as the minimum acceptable COP for the pump if the photovoltaic system supplies its energy?

Should a car get better mileage in the summer or the winter? Explain.

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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