A 2.85-g lead weight, initially at 10.3 °C, is submerged in 7.55 g of water at 52.3 °C in an insulated container. What is the final temperature of both substances at thermal equilibrium?

Verified step by step guidance

1. Identify the concept of thermal equilibrium, where the heat lost by the hotter substance (water) equals the heat gained by the cooler substance (lead).
2. Use the formula for heat transfer: \( q = mc\Delta T \), where \( q \) is the heat absorbed or released, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.
3. Set up the equation for thermal equilibrium: \( m_{\text{lead}}c_{\text{lead}}(T_f - T_{\text{initial, lead}}) = -m_{\text{water}}c_{\text{water}}(T_f - T_{\text{initial, water}}) \).
4. Substitute the known values: \( m_{\text{lead}} = 2.85 \text{ g} \), \( c_{\text{lead}} = 0.128 \text{ J/g°C} \), \( T_{\text{initial, lead}} = 10.3 \text{ °C} \), \( m_{\text{water}} = 7.55 \text{ g} \), \( c_{\text{water}} = 4.18 \text{ J/g°C} \), \( T_{\text{initial, water}} = 52.3 \text{ °C} \).
5. Solve the equation for \( T_f \), the final temperature, which will be the same for both substances at thermal equilibrium.

Key Concepts

Thermal Equilibrium

Thermal equilibrium occurs when two substances at different temperatures come into contact and exchange heat until they reach the same temperature. In this scenario, the lead weight and water will transfer heat between each other 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 a unit mass of a substance by one degree Celsius. Each material has a unique specific heat capacity, which influences how much its temperature changes when heat is added or removed. In this problem, the specific heat capacities of lead and water will be essential for calculating the final temperature.

Heat Transfer Calculation

Heat transfer calculations involve determining the amount of heat lost or gained by a substance during a temperature change. This is typically expressed using the formula Q = mcΔT, where Q is the heat transferred, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. In this case, the heat lost by the water will equal the heat gained by the lead weight, allowing for the calculation of the final equilibrium temperature.