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Chapter 5: Problem 23

A 1 -kg piece of iron is heated to \(100^{\circ} \mathrm{C}\), and then submerged in \(1 \mathrm{~kg}\) of water initially at \(0^{\circ} \mathrm{C}\). The iron cools and the water warms until they are at the same temperature (in thermal equilibrium). Assuming there is no other transfer of heat involved, is the final temperature closer to \(0^{\circ} \mathrm{C}, 50^{\circ} \mathrm{C}\), or \(100^{\circ} \mathrm{C} ?\) Why?

Short Answer

Expert verified
Provide a brief explanation. Answer: The final temperature of the system is closer to 0°C. This is because water has a much higher specific heat capacity compared to iron, allowing it to absorb more heat energy while experiencing a smaller change in temperature. As a result, the final temperature is closer to the initial temperature of the water, which was 0°C.

Step by step solution

01

Identify the given information

We are given: 1. Mass of iron (m_1) = 1 kg 2. Initial temperature of iron (T1_initial) = \(100^{\circ} \mathrm{C}\) 3. Mass of water (m_2) = 1 kg 4. Initial temperature of water (T2_initial) = \(0^{\circ} \mathrm{C}\) We need to find the final temperature (T_final) when the iron and water are in thermal equilibrium.
02

Determine the specific heat capacities

We need to know the specific heat capacities of iron and water to proceed. 1. Specific heat capacity of iron (c_1) = 449 J/(kg·\(^{\circ} \mathrm{C}\)) 2. Specific heat capacity of water (c_2) = 4186 J/(kg·\(^{\circ} \mathrm{C}\))
03

Apply conservation of energy and the heat transfer equation

At thermal equilibrium, the total heat lost by the iron equals the total heat gained by the water: \(m_1 \cdot c_1 \cdot (T_{1_{initial}} - T_{final}) = m_2 \cdot c_2 \cdot (T_{final} - T_{2_{initial}})\)
04

Plug in the given values and solve for T_final

Substitute the given values into the equation: \((1 \, \text{kg}) \cdot (449 \, \text{J/(kg·\)^{\circ} \mathrm{C}\()}) \cdot (100^{\circ} \mathrm{C} - T_{final}) = (1 \, \text{kg}) \cdot (4186 \, \text{J/(kg·\)^{\circ} \mathrm{C}\()}) \cdot (T_{final} - 0^{\circ} \mathrm{C})\) Simplify and solve for T_final: \(449(100 - T_{final}) = 4186(T_{final})\) Upon solving for T_final, we find that the final temperature is approximately \(21.7^{\circ} \mathrm{C}\).
05

Compare the final temperature with the given options

The final temperature is closer to \(0^{\circ} \mathrm{C}\) than \(50^{\circ} \mathrm{C}\) or \(100^{\circ} \mathrm{C}\), so the answer is closer to \(0^{\circ} \mathrm{C}\). The reason is that the specific heat capacity of water is much higher (4186 J/(kg·\(^{\circ} \mathrm{C}\))) compared to the specific heat capacity of iron (449 J/(kg·\(^{\circ} \mathrm{C}\))). This means that water is able to absorb more heat energy while having a smaller change in temperature than the iron. Therefore, the final temperature is closer to the initial temperature of the water (\(0^{\circ} \mathrm{C}\)).

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