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Chapter 14: Problem 96

A piece of gold of mass \(0.250 \mathrm{kg}\) and at a temperature of \(75.0^{\circ} \mathrm{C}\) is placed into a \(1.500-\mathrm{kg}\) copper pot containing \(0.500 \mathrm{L}\) of water. The pot and water are at $22.0^{\circ} \mathrm{C}$ before the gold is added. What is the final temperature of the water?

Short Answer

Expert verified
Answer: The final temperature of the water after heat exchange with the gold and the copper pot is approximately \(46.0 ^\circ C\).

Step by step solution

01

Write down the given information

: We are provided with the following information: Mass of gold: \(m_g = 0.250 \, kg\) Initial temperature of gold: \(T_g = 75.0 \, ^\circ C\) Mass of copper pot: \(m_p = 1.50 \, kg\) Initial temperature of pot: \(T_p = 22.0 \, ^\circ C\) Mass of water: \(m_w = 0.5 \, L\), which is equivalent to \(0.5 \, kg\), since water has a density of \(1\, \frac{kg}{L}\) Initial temperature of water: \(T_w = 22.0 \, ^\circ C\) We also need the specific heat capacities of gold, copper, and water: Specific heat of gold: \(c_g = 129 \, \frac{J}{kg \cdot K}\) Specific heat of copper: \(c_p = 385 \, \frac{J}{kg \cdot K}\) Specific heat of water: \(c_w = 4186 \, \frac{J}{kg \cdot K}\)
02

Apply the heat exchange principle

: The total heat gained by the water and the copper pot is equal to the heat lost by the gold. This can be expressed as: \((m_g \cdot c_g \cdot (T_g - T)) + (m_p \cdot c_p \cdot (T_p - T)) = (m_w \cdot c_w \cdot (T - T_w))\)
03

Solve for the final temperature T

: Now we need to solve for T, the final temperature of the water after heat exchange: \((0.250 \cdot 129 \cdot (75.0 - T)) + (1.50 \cdot 385 \cdot (22.0 - T)) = (0.5 \cdot 4186 \cdot (T - 22.0))\) Rearranging the equation and applying the distributive property, we get: \(4833.75 - 32.25T - 866.5T = 2093T - 46183\) Combining like terms, we get: \(-898.75T = -41349.75\) Now, we divide both sides by -898.75 to solve for T: \(T = \frac{-41349.75}{-898.75} \approx 46.0 ^\circ C\)
04

Present the final answer

: The final temperature of the water after the heat exchange with the gold and the copper pot is approximately \(46.0 ^\circ C\).

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