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Chapter 5: Problem 50

Two solid objects, A and \(\mathrm{B},\) are placed in boiling water and allowed to come to the temperature of the water. Each is then lifted out and placed in separate beakers containing 1000 \(\mathrm{g}\) water at \(10.0^{\circ} \mathrm{C} .\) Object A increases the water temperature by \(3.50^{\circ} \mathrm{C} ; \mathrm{B}\) increases the water temperature by \(2.60^{\circ} \mathrm{C}\) . (a) Which object has the larger heat capacity? (b) What can you say about the specific heats of \(\mathrm{A}\) and \(\mathrm{B} ?\)

Short Answer

Expert verified
Object A has a higher heat capacity as it produces a larger change in water temperature (\(ΔT_A = 3.5^{\circ}C\), \(ΔT_B = 2.6^{\circ}C\)). As for the specific heats, without knowing the objects' masses, we cannot make definitive conclusions. However, if both objects have equal masses, then the specific heat of object A is higher than object B due to its larger heat capacity. If the object with a larger heat capacity has a smaller mass, its specific heat will also be higher.

Step by step solution

01

Identify the Variables Given

We have the following information: - Mass of water in both beakers, m_water = 1000 g - initial water temperature, T_initial = 10.0 °C - final water temperatures, T_final_A = 13.5 °C, T_final_B = 12.6 °C From this, we can calculate the change in temperature for both beakers: ΔT_A = 13.5 °C - 10.0 °C = 3.5 °C ΔT_B = 12.6 °C - 10.0 °C = 2.6 °C
02

Heat Gained by the Water

Let Q_A and Q_B be the amount of heat gained by the water in beakers A and B, respectively. We can calculate these using the given mass, specific heat of water (c_water = 4.18 J/g°C), and change in temperature for each beaker. For object A: \(Q_A = 1000 * 4.18 * 3.5\) For object B: \(Q_B = 1000 * 4.18 * 2.6\)
03

Heat Lost by Objects A and B

Since the heat gained by the water is equal to the heat lost by the objects, we can determine the heat capacities, \(C_A\) and \(C_B\), for objects A and B. \(C_A = Q_A/ΔT_A\) \(C_B = Q_B/ΔT_B\)
04

Compare Heat Capacities

Compare the heat capacities \(C_A\) and \(C_B\) to determine which object has a larger heat capacity. (a) If \(C_A > C_B\), then object A has a larger heat capacity. (b) If \(C_B > C_A\), then object B has a larger heat capacity.
05

Conclusions About Specific Heats

We cannot determine the specific heats of objects A and B without knowing their masses. However, since the heat capacity is directly proportional to specific heat (C = mc), we can make some general observations: - If the object with the larger heat capacity has a smaller mass than the object with the smaller heat capacity, it implies that its specific heat is higher as well. - If both objects have the same mass, the object with the larger heat capacity also has a larger specific heat.

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Most popular questions from this chapter

Diethyl ether, \(\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{O}(l),\) a flammable compound that was once used as a surgical anesthetic, has the structure $$\mathrm{H}_{3} \mathrm{C}-\mathrm{CH}_{2}-\mathrm{O}-\mathrm{CH}_{2}-\mathrm{CH}_{3}$$ The complete combustion of 1 mol of \(\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{O}(l)\) to \(\mathrm{CO}_{2}(g)\) and \(\mathrm{H}_{2} \mathrm{O}(l)\) yields \(\Delta H^{\circ}=-2723.7 \mathrm{kJ}\) . (a) Write a balanced equation for the combustion of 1 \(\mathrm{mol}\) of \(\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{O}(l) .\) (b) By using the information in this problem and data in Table \(5.3,\) calculate \(\Delta H_{f}^{\circ}\) for diethyl ether.

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