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Chapter 11: Problem 64

Three liquids are at temperatures of \(10^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}\), and \(30^{\circ} \mathrm{C}\), respectively. Equal masses of the first two liquids are mixed, and the equilibrium temperature is \(17^{\circ} \mathrm{C}\). Equal masses of the second and third are then mixed, and the equilibrium temperature is \(28^{\circ} \mathrm{C}\). Find the equilibrium temperature when equal masses of the first and third are mixed.

Short Answer

Expert verified
The equilibrium temperature when equal masses of the first and third liquids are mixed is \(19^{\circ} C\).

Step by step solution

01

Calculate specific heat capacity of Liquid 2

From the first condition, equal masses of liquids 1 and 2 are mixed and the equilibrium temperature is \(17^{\circ} C\). Therefore, heat gained by liquid 2 is equal to the heat lost by liquid 1. Mathematically it can be stated as: \( mc_1(17-10) = mc_2(20-17) \). Simplify to find \( c_2 = 7c_1/3 = 2.33c_1 \)
02

Calculate specific heat capacity of Liquid 3

From the second condition, equal masses of liquids 2 and 3 are mixed and the equilibrium temperature is \(28^{\circ} C\). Therefore, heat gained by liquid 3 is equal to the heat lost by liquid 2. Mathematically it can be stated as: \( mc_2(28-20) = mc_3(30-28) \). Replace \( c_2 \) with the expression from step 1 and simplify to find \( c_3 = 8c_1/5 = 1.6c_1 \)
03

Find the equilibrium temperature when equal masses of the first and third are mixed.

Now, when equal masses of liquids 1 and 3 are mixed, the heat gained by liquid 3 will be equal to the heat lost by liquid 1. Let's denote the equilibrium temperature as \( T \). Therefore, \( mc_1(T-10) = mc_3(30-T) \). Replace \( c_1 \) and \( c_3 \) with the expressions from step 1 and 2 respectively, and simplify to find \( T = 19^{\circ} C \)

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Most popular questions from this chapter

A steam pipe is covered with \(1.50-\mathrm{cm}\)-thick insulating material of thermal conductivity \(0.200 \mathrm{cal} / \mathrm{cm} \cdot{ }^{\circ} \mathrm{C} \cdot \mathrm{s}\). How much energy is lost every second when the steam is at \(200^{\circ} \mathrm{C}\) and the surrounding air is at \(20.0^{\circ} \mathrm{C}\) ? The pipe has a circumference of \(800 \mathrm{~cm}\) and a length of \(50.0 \mathrm{~m}\). Neglect losses through the ends of the pipe.

A \(1.5-\mathrm{kg}\) copper block is given an initial speed of \(3.0 \mathrm{~m} / \mathrm{s}\) on a rough horizontal surface. Because of friction, the block finally comes to rest. (a) If the block absorbs \(85 \%\) of its initial kinetic energy as internal energy, calculate its increase in temperature. (b) What happens to the remaining energy?

An ice-cube tray is filled with \(75.0 \mathrm{~g}\) of water. After the filled tray reaches an equilibrium temperature \(20.0^{\circ} \mathrm{C}\), it is placed in a freezer set at \(-8.00^{\circ} \mathrm{C}\) to make ice cubes. (a) Describe the processes that occur as energy is being removed from the water to make ice. (b) Calculate the energy that must be removed from the water to make ice cubes at \(-8.00^{\circ} \mathrm{C}\).

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A \(50-\mathrm{g}\) ice cube at \(0{ }^{\circ} \mathrm{C}\) is heated until \(45 \mathrm{~g}\) has become water at \(100^{\circ} \mathrm{C}\) and \(5.0 \mathrm{~g}\) has become steam at \(100^{\circ} \mathrm{C}\). How much energy was added to accomplish the transformation?
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