Question

A \(150.0 -\mathrm{g}\) sample of a metal at \(75.0^{\circ} \mathrm{C}\) is added to \(150.0 \mathrm{g} \mathrm{H}_{2} \mathrm{O}\) at \(15.0^{\circ} \mathrm{C}\). The temperature of the water rises to \(18.3^{\circ} \mathrm{C}\). Calculate the specific heat capacity of the metal, assuming that all the heat lost by the metal is gained by the water.

Short answer

The specific heat capacity of the metal can be calculated using the formula \( c_m = \frac{m_w \cdot c_w \cdot ΔT_w}{m_m \cdot (T_{m_i} - T_{m_f})} \) Plugging in the known values: \( c_m = \frac{150.0\,\text{g} \cdot 4.18\,\text{J/g°C} \cdot (18.3\,°\text{C} - 15.0\,°\text{C})}{150.0\,\text{g} \cdot (75.0\,°\text{C} - 18.3\,°\text{C})} \) Solving the equation, we find that the specific heat capacity of the metal is approximately \(0.22\,\text{J/g°C}\).

Step-by-step solution

Recognizing heat transfer concept

Since the amount of heat lost by the metal sample is gained by the water sample, we can express the heat transfer using the following formula: \(Q_{lost} = Q_{gained}\) We also know that the formula for the heat transfer is \(Q = m \cdot c \cdot ΔT\) where \(Q\) represents the heat transfer, \(m\) represents the mass, \(c\) represents the specific heat capacity, and \(ΔT\) represents the temperature difference. Tag_titleStep 2: Identify the known values

The information given in the problem is as follows: Mass of the metal sample (\(m_m\)) = 150.0 g Initial temperature of the metal sample (\(T_{m_i}\)) = 75.0°C Mass of the water sample (\(m_w\)) = 150.0 g Initial temperature of the water sample (\(T_{w_i}\)) = 15.0°C Final temperature of the water sample (\(T_{w_f}\)) = 18.3°C

Step 3: Calculate the temperature changes for the metal and water samples

We need to find the temperature difference for both the metal and water samples: For the metal: \(ΔT_m = T_{m_i} - T_{m_f}\) For the water: \(ΔT_w = T_{w_f} - T_{w_i}\)

Step 4: Determine the heat gained by the water

We'll first find the heat gained by the water using the formula \(Q = m \cdot c \cdot ΔT\), where the specific heat capacity of water (\(c_w\)) is 4.18 J/g°C: \(Q_{gained} = m_w \cdot c_w \cdot ΔT_w\)

Step 5: Calculate the specific heat capacity of the metal

Now, using the fact that heat lost is equal to heat gained (\(Q_{lost} = Q_{gained}\)), and the formula \(Q_{lost} = m_m \cdot c_m \cdot ΔT_m\), we can calculate the specific heat capacity of the metal: \( m_m \cdot c_m \cdot (T_{m_i} - T_{m_f}) = m_w \cdot c_w \cdot ΔT_w \)

Solving for \(c_m\), the specific heat capacity of the metal, remembering that \(ΔT_m = T_{m_i} - T_{m_f}\), we get: \( c_m = \frac{m_w \cdot c_w \cdot ΔT_w}{m_m \cdot (T_{m_i} - T_{m_f})} \) Finally, plug in the known values and solve for the specific heat capacity of the metal.

Key concepts

Understanding Heat Transfer

Heat transfer is a fundamental concept that involves the movement of energy from one body or substance to another due to a temperature difference. When two objects at different temperatures come into contact, energy will transfer from the hotter object to the cooler one until thermal equilibrium is reached. There are three main methods of heat transfer: conduction, convection, and radiation.
In the context of the exercise, heat transfer occurs when the metal, which is initially hot, is placed into the cooler water. The heat loss from the metal is absorbed by the water, causing its temperature to rise. This is a direct application of the principle of conservation of energy, where the heat lost by the metal (heat release) is equal to the heat gained by the water. This interchanging of energy is crucial to understanding calorimetry calculations. By writing this energy balance equation as:

• \(Q_{lost} = Q_{gained}\)

we set the basis for calculating unknown variables such as specific heat capacity.

Analyzing Temperature Change

Temperature change is vital in calculating heat transfer because it reflects how much the temperature of a substance has shifted due to gaining or losing heat. The temperature change (\(ΔT\)) is calculated as the difference between the final and initial temperatures of the substance. This is crucial because the temperature change provides insight into the amount of heat energy transferred.
In the given problem, you have separate temperature changes for the metal and the water:

• For the metal, which cools down, \(ΔT_m = T_{m_i} - T_{m_f}\).
• For the water, which heats up, \(ΔT_w = T_{w_f} - T_{w_i}\).

These calculations help inform us how much heat each has either lost or gained. Understanding temperature change is necessary for setting up the equations needed to solve for unknown properties, like the specific heat capacity of the metal.
Without the precise understanding of these temperature changes, you wouldn't be able to accurately apply the heat transfer formula to complete the problem.

Exploring Calorimetry

Calorimetry is an experimental method employed to measure the heat exchanged in a chemical or physical process. This technique involves a calorimeter, an apparatus that isolates a system thermally to measure the energy changes within that system.
In solving the problem, calorimetry is applied when we calculate the specific heat capacity of the metal using the heat transfer between the metal and water. This calculation depends on understanding how calorimetry functions to keep track of energy changes.
Here, the equation relating the energy changes involves the known specific heat of water and the calculated changes in temperature. By knowing the mass, specific heat, and temperature change of water, you can understand how much heat it absorbed, allowing calculation of the metal's unknown specific heat capacity. Calorimetry allows for precise energy transfer measurements, vital for deducing the characteristics like specific heat. In the exercise, the calorimeter is essentially the environment where the metal and water exchange heat, showcasing the direct application of calorimetry principles.