A 31.1-g wafer of pure gold, initially at 69.3 °C, is submerged into 64.2 g of water at 27.8 °C in an insulated container. What is the final temperature of both substances at thermal equilibrium?

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Identify the specific heat capacities for gold and water. The specific heat capacity of gold is approximately 0.129 J/g°C, and for water, it is approximately 4.18 J/g°C.

Set up the heat transfer equation assuming no heat is lost to the surroundings. The heat lost by gold will equal the heat gained by water. Use the formula: \(m_{gold} \cdot c_{gold} \cdot (T_{final} - T_{initial, gold}) = m_{water} \cdot c_{water} \cdot (T_{final} - T_{initial, water})\).

Substitute the masses and specific heat capacities of gold and water into the equation. Replace \(m_{gold}\) with 31.1 g, \(c_{gold}\) with 0.129 J/g°C, \(T_{initial, gold}\) with 69.3 °C, \(m_{water}\) with 64.2 g, and \(c_{water}\) with 4.18 J/g°C, and \(T_{initial, water}\) with 27.8 °C.

Solve the equation for \(T_{final}\), the final temperature. This will involve algebraic manipulation to isolate \(T_{final}\) on one side of the equation.

Check the physical plausibility of your answer. The final temperature should be between the initial temperatures of the gold and the water, and closer to the initial temperature of the substance with the higher heat capacity (water in this case).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Thermal Equilibrium

Thermal equilibrium occurs when two substances at different temperatures are brought into contact, resulting in a transfer of heat until they reach the same temperature. In this scenario, the gold wafer and water will exchange heat until they stabilize at a common final temperature, which can be calculated using the principle of conservation of energy.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to raise the temperature of one gram of a substance by one degree Celsius. Each substance has a unique specific heat capacity, which influences how much heat it can absorb or release. For this problem, the specific heat capacities of gold and water will be essential in determining how much heat each substance exchanges during the thermal equilibrium process.

Heat Transfer Equation

The heat transfer equation, often expressed as Q = mcΔT, relates the heat gained or lost (Q) to the mass (m), specific heat capacity (c), and change in temperature (ΔT) of a substance. In this case, the heat lost by the gold will equal the heat gained by the water, allowing us to set up an equation to solve for the final equilibrium temperature.

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