Question

Objects in thermal contact. Suppose two objects \(A\) and \(B\), with heat capacities \(C_{A}\) and \(C_{B}\) and initial temperatures \(T_{A}\) and \(T_{B}\), are brought into thermal contact. If \(C_{A} \gg C_{B}\), is the equilibrium temperature \(T\) closer to \(T_{A}\) or to \(T_{B}\) ?

Short answer

The equilibrium temperature \(T\) is closer to \(T_{A}\).

Step-by-step solution

- Understand the situation

Two objects, A and B, with heat capacities \(C_{A}\) and \(C_{B}\) respectively, are brought into thermal contact. Initial temperatures are \(T_{A}\) for object A and \(T_{B}\) for object B. The goal is to find the equilibrium temperature \(T\) and determine if it is closer to \(T_{A}\) or \(T_{B}\).

- Write the heat transfer equation

When objects A and B are brought into thermal contact, heat will flow from the hotter object to the cooler one until thermal equilibrium is reached. The heat lost by the hotter object is equal to the heat gained by the cooler object: \[ C_{A} (T_{A} - T) = C_{B} (T - T_{B}) \]

- Solve for the equilibrium temperature T

Reorganize the equation to solve for T: \[ C_{A} T_{A} - C_{A} T = C_{B} T - C_{B} T_{B} \] Combining like terms gives: \[ C_{A} T_{A} + C_{B} T_{B} = T (C_{A} + C_{B}) \] Finally, solve for the equilibrium temperature T: \[ T = \frac{C_{A} T_{A} + C_{B} T_{B}}{C_{A} + C_{B}} \]

- Evaluate the limits of the equilibrium equation

Given that \(C_{A} \gg C_{B}\), analyze the formula in this limit. When \(C_{A} \gg C_{B}\), \(C_{A} + C_{B} \approx C_{A}\). Thus, \[ T = \frac{C_{A} T_{A} + C_{B} T_{B}}{C_{A} + C_{B}} \approx \frac{C_{A} T_{A}}{C_{A}} = T_{A} \] Therefore, the equilibrium temperature \(T\) will be very close to \(T_{A}\).

Key concepts

heat capacity

Heat capacity, denoted by the symbol \(C\), represents the amount of heat energy required to change an object's temperature by a certain amount. It’s a measure of how much heat an object can absorb or release without undergoing a large temperature change.

To put it simply, objects with larger heat capacities can absorb more heat before their temperatures rise significantly, compared to objects with smaller heat capacities.

• Larger Heat Capacity: Means the object can absorb a lot of heat energy without much change in temperature.
• Smaller Heat Capacity: Means the object will quickly change temperature, even with a small amount of absorbed or released heat.

When two objects with different heat capacities are brought into contact, the object with the larger heat capacity will have a more significant influence on the final equilibrium temperature. This is because it can absorb more heat without a substantial change in its temperature.

In the given problem, object A has a much larger heat capacity (\(C_A\)) compared to object B (\(C_B\)). Therefore, it dictates the final equilibrium temperature since it can absorb or release a greater amount of heat.

thermal contact

Thermal contact occurs when two objects can exchange heat energy due to a temperature difference. This is an essential concept in understanding how heat is transferred between objects until they reach thermal equilibrium.

When objects are in thermal contact:

• Heat flows from the hotter object to the cooler object.
• This process continues until both objects reach the same temperature, known as equilibrium temperature.

In the problem, objects A and B are brought into thermal contact. Object A, with an initial temperature \(T_A\), will transfer heat to object B, with an initial temperature \(T_B\), if \(T_A\) is greater than \(T_B\).

The heat transfer will keep happening until the temperatures of the two objects equalize, achieving thermal equilibrium. The rate and direction of heat flow depend on the initial temperatures and heat capacities of the objects involved.

equilibrium temperature

The equilibrium temperature is the final temperature reached when two objects in thermal contact no longer exchange heat. At this point, both objects have the same temperature.

To find this equilibrium temperature, we use the principle of conservation of energy. The heat lost by the hotter object equals the heat gained by the cooler object. In the provided exercise, this involves setting up the equation:

\[ C_{A} (T_{A} - T) = C_{B} (T - T_{B}) \]

The formula for the equilibrium temperature \(T\) can be derived as:

\[ T = \frac{C_{A} T_{A} + C_{B} T_{B}}{C_{A} + C_{B}} \]

Given the condition \(C_{A} \gg C_{B}\), we see that the equilibrium temperature \(T\) will be closer to \(T_A\). This is because the large heat capacity of object A means it dominates the heat transfer process.

In simpler terms:

• If an object with high heat capacity and high initial temperature comes into contact with an object with low heat capacity and lower initial temperature, the final shared temperature will be much closer to the initial temperature of the object with the higher heat capacity.

Understanding equilibrium temperature is crucial in many practical scenarios, such as when designing heating and cooling systems, or when exploring natural processes like the Earth's climate system.