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The principle of heat exchange is an important concept in understanding how heat is transferred between two objects. It states that when two bodies at different temperatures come into contact, heat will flow from the hotter object to the cooler one until thermal equilibrium is reached.
This principle means that the amount of heat lost by the hotter object is exactly equal to the amount of heat gained by the cooler object. In our problem, this principle is used to calculate the specific heat capacity of the metal by equating the heat lost by the metal to the heat gained by the water. Understanding this principle is key to solving many thermodynamics problems, as it provides a conservation law for energy exchange between systems.

To calculate the heat gained by water, we use the formula: \[ q_{\text{water}} = m_{\text{water}} \cdot c_{\text{water}} \cdot (T_f - T_{i, \text{water}}) \]In this equation:

• \( q_{\text{water}} \) is the heat gained by the water.
• \( m_{\text{water}} \) is the mass of the water. In our problem, it's 250 grams.
• \( c_{\text{water}} \) is the specific heat capacity of water, a constant at 4.18 J/g°C.
• \( T_f \) is the final temperature, and \( T_{i, \text{water}} \) is the initial temperature of the water.

Substitute the known values to find the heat gained by the water. This calculation helps us understand how much energy the water absorbs when heated by the metal.

The heat lost by the metal can be determined using the principle of heat exchange, which tells us it is equal in magnitude and opposite in sign to the heat gained by the water. Therefore, \[ q_{\text{metal}} = -q_{\text{water}} \] This formula demonstrates the idea that any energy gained by the water is lost by the metal. As both systems reach equilibrium, the energy exchange adjusts their temperatures. To find \( q_{\text{metal}} \), just calculate \( q_{\text{water}} \) and switch the sign. This is crucial for our next steps, as it provides the value needed to solve for the specific heat of the metal.

Once you have the heat lost by the metal, you can calculate its specific heat capacity. The formula is:\[ c_{\text{metal}} = \frac{q_{\text{metal}}}{m_{\text{metal}} \cdot (T_i, \text{metal} - T_f)} \] Here:

• \( q_{\text{metal}} \) is the heat lost by the metal.
• \( m_{\text{metal}} \) is the mass of the metal. In this problem, it's 50 grams.
• \( T_i, \text{metal} \) is the initial temperature of the metal.
• \( T_f \) is the final equilibrium temperature.

By substituting the known values from the problem, you can calculate the specific heat of the metal, which tells us how much heat per gram per degree the metal can absorb before its temperature changes. This is an intrinsic property of the metal, unique to its material.

Temperature equilibrium is the state reached when the metal and water achieve the same final temperature. Initially, the metal is at a higher temperature, while the water starts cooler. Over time, the heat transfer from the metal warms the water, and the cooling of the metal continues until they both reach a balanced final temperature of 19.4°C. At this point, no more heat transfer occurs, as the system has reached thermal stability. This concept is important because the final equilibrium temperature is used in both the heat gained and lost equations to determine the specific heat capacity of the metal. Understanding equilibrium simplifies thermodynamics by setting a constant final state to calculate changes in energy. This state confirms successful energy exchange between the substances involved.