Question

If two objects, \(A\) and \(B\) , of different temperature come into direct contact, what is the relationship between the heat lost by one object and the heat gained by the other? What is the relationship between the temperature changes of the two objects? (Assume that the two objects do not lose any heat to anything else.)

Short answer

The heat lost by the hotter object is equal to the heat gained by the cooler object. The relationship between the temperature changes of the two objects depends on their specific heat capacities and masses.

Step-by-step solution

Identify the Law of Conservation of Energy

When two objects at different temperatures come into contact, the Law of Conservation of Energy states that energy cannot be created or destroyed in an isolated system. Since no heat is lost to the environment, this means that the total amount of heat before and after the objects come into contact must remain constant.

Understand Heat Transfer

Heat transfer will occur from the hotter object (say object A) to the cooler one (object B) until thermal equilibrium is reached (both objects have the same temperature). The heat lost by object A will be equal to the heat gained by object B.

Quantify the Heat Transfer

The quantity of heat transferred is proportional to the temperature change and the heat capacity of each object. If we use the symbols Q for heat, T for temperature, and C for specific heat capacity, and the subscripts A and B for the respective objects, we can represent this as: \(Q_A = -Q_B\) and \(C_A \Delta T_A = -C_B \Delta T_B\), where \(\Delta T\) is the change in temperature.

Apply the Principle to Find the Relationship

The negative sign indicates that one object gains heat while the other loses it, which means if object A loses a certain amount of heat, object B gains that exact amount of heat. The relationship between temperatures will depend on their specific heat capacities and mass. If the two objects have equal heat capacity and mass, they will have equal but opposite temperature changes.

Key concepts

Law of Conservation of Energy

In any given scenario where energy interactions occur, such as when two objects at different temperatures come into contact, the Law of Conservation of Energy becomes a fundamental principle to consider. This law simply states that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred from one object to another.

Imagine you have a hot cup of coffee and a cold steel spoon. According to this law, when you place the spoon into the coffee, the heat energy from the hot coffee is transferred to the colder spoon, and there is no loss of energy during this process. The coffee cools down slightly while the spoon warms up. In an isolated system where no heat is lost to the environment, the heat lost by the coffee will exactly equal the heat gained by the spoon.

So in educational terms, if object A (coffee) loses a certain quantity of heat, object B (spoon) gains the same amount of heat, assuming a perfect isolated system. This reinforces the concept of energy balance and is a pivotal lesson in understanding how energy behaves in different contexts.

Thermal Equilibrium

Thermal equilibrium is a key term in the study of heat transfer. It occurs when two objects that are in physical contact no longer exchange heat, meaning they have reached the same temperature. To visualize this, think of two connected rooms of different temperatures. If you open the door between them, heat will naturally flow from the warmer room to the cooler one until both rooms have the same temperature. They have reached thermal equilibrium.

In the case of our previous coffee and spoon example, the hot coffee and the cold spoon will exchange heat (the coffee gives away heat while the spoon absorbs it) until they are both at the same temperature. This state where the net heat exchange between the objects is zero is what we call reaching thermal equilibrium. Teaching this concept helps students understand that achieving thermal equilibrium is the driving force behind heat transfer.

Specific Heat Capacity

Specific heat capacity is a property of a material that indicates how much heat energy is required to change the temperature of a certain mass of the substance by a given amount. It is usually denoted by the symbol C and can vary widely from one material to another.

Using water and metal as examples, we can explain that water has a high specific heat capacity, meaning it takes a lot of energy to raise its temperature. Metal, on the other hand, typically has a lower specific heat capacity, so it heats up and cools down more quickly.

When we look at our objects A and B, with their specific heat capacities likely being different, it gives insight into how the temperature of each object will change as heat is transferred. The formula \( C_A \Delta T_A = -C_B \Delta T_B \) informs us that the temperature change of an object is directly proportional to the heat transferred and inversely proportional to its mass and specific heat capacity. This equation simplifies the process of understanding and calculating the temperature changes that occur during heat transfer, ensuring students can predict and quantify changes in a hands-on and applicable way.