A 32.5-g iron rod, initially at 22.7 °C, is submerged into an unknown mass of water at 63.2 °C, in an insulated container. The final temperature of the mixture upon reaching thermal equilibrium is 59.5 °C. What is the mass of the water?

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Identify the heat transfer process: The heat lost by the water is equal to the heat gained by the iron rod, since the system is insulated.

Use the formula for heat transfer: \( q = mc\Delta T \), where \( q \) is the heat transferred, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

Calculate the heat gained by the iron rod: \( q_{\text{iron}} = m_{\text{iron}} \times c_{\text{iron}} \times (T_{\text{final}} - T_{\text{initial, iron}}) \). Use \( m_{\text{iron}} = 32.5 \text{ g} \), \( c_{\text{iron}} = 0.449 \text{ J/g°C} \), and \( T_{\text{final}} = 59.5 \text{ °C} \), \( T_{\text{initial, iron}} = 22.7 \text{ °C} \).

Calculate the heat lost by the water: \( q_{\text{water}} = m_{\text{water}} \times c_{\text{water}} \times (T_{\text{final}} - T_{\text{initial, water}}) \). Use \( c_{\text{water}} = 4.18 \text{ J/g°C} \), \( T_{\text{final}} = 59.5 \text{ °C} \), \( T_{\text{initial, water}} = 63.2 \text{ °C} \).

Set the heat gained by the iron equal to the heat lost by the water and solve for \( m_{\text{water}} \): \( m_{\text{iron}} \times c_{\text{iron}} \times (T_{\text{final}} - T_{\text{initial, iron}}) = m_{\text{water}} \times c_{\text{water}} \times (T_{\text{final}} - T_{\text{initial, water}}) \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Thermal Equilibrium

Thermal equilibrium occurs when two objects at different temperatures come into contact and exchange heat until they reach the same temperature. In this scenario, the iron rod and water exchange heat until they stabilize at a final temperature of 59.5 °C. Understanding this concept is crucial for applying the principle of conservation of energy in thermal systems.

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 material has a unique specific heat capacity, which influences how much heat it can absorb or release. In this problem, the specific heat capacities of iron and water will be used to calculate the heat transfer and ultimately determine the mass of the water.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the heat lost by the iron rod must equal the heat gained by the water. This relationship allows us to set up an equation to solve for the unknown mass of water based on the heat transfer between the two substances.