A laboratory technician drops a $0.0850$-kg sample of unknown solid material, at $100.0$°C, into a calorimeter. The calorimeter can, initially at $19.0$°C, is made of $0.150$ kg of copper and contains $0.200$ kg of water. The final temperature of the calorimeter can and contents is $26.1$°C. Compute the specific heat of the sample.

Verified step by step guidance

1. Identify the principle of conservation of energy, which states that the heat lost by the sample will be equal to the heat gained by the calorimeter and its contents.
2. Write the equation 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.
3. Calculate the heat gained by the copper calorimeter using its specific heat capacity \( c_{\text{Cu}} = 0.385 \text{ J/g°C} \). Use the formula: \( Q_{\text{Cu}} = m_{\text{Cu}} \cdot c_{\text{Cu}} \cdot (T_{\text{final}} - T_{\text{initial, Cu}}) \).
4. Calculate the heat gained by the water using its specific heat capacity \( c_{\text{water}} = 4.186 \text{ J/g°C} \). Use the formula: \( Q_{\text{water}} = m_{\text{water}} \cdot c_{\text{water}} \cdot (T_{\text{final}} - T_{\text{initial, water}}) \).
5. Set up the equation for the heat lost by the sample: \( Q_{\text{sample}} = m_{\text{sample}} \cdot c_{\text{sample}} \cdot (T_{\text{initial, sample}} - T_{\text{final}}) \). Solve for \( c_{\text{sample}} \) using the equation: \( Q_{\text{sample}} = Q_{\text{Cu}} + Q_{\text{water}} \).

Key Concepts

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is a material-specific property and is crucial in determining how a substance responds to heat energy. In this problem, it helps calculate the heat absorbed or released by the unknown material.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In calorimetry, this principle implies that the heat lost by the hot object (the unknown sample) is equal to the heat gained by the cooler objects (the calorimeter and water).

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. It involves using a calorimeter to measure the heat exchanged between substances. In this scenario, calorimetry is used to determine the specific heat of the unknown sample by analyzing the temperature changes in the calorimeter system.