Question

A cold object at \(5^{\circ} \mathrm{C}\) is placed in an insulated cup of water at \(25^{\circ} \mathrm{C}\). Determine whether or not each of the following will occur. (a) The increase in temperature of the object will be equal to the decrease in temperature of the water. (b) Only the object will change temperature. (c) Only the water will change temperature. (d) The heat gain by the object will equal the heat lost by the water. (c) The final temperature will be somewhere between \(5^{\circ} \mathrm{C}\) and \(25^{\circ} \mathrm{C}\).

Short answer

Based on the analysis and solutions steps, only scenarios (d) and (e) are true. The remaining scenarios (a), (b), and (c) cannot be confirmed due to lack of information or are against the basic principles of thermodynamics.

Step-by-step solution

Analyze the statement for Scenario (a)

The statement in scenario (a) suggests that the temperature change in the object is equal to the water's temperature change. This statement could be true if the masses and specific heats of the water and the object were the same. Since there is no relevant information about the latter, this statement can't be confirmed.

Analyze the statement for Scenario (b)

The statement for scenario (b) alleges that only the object will change its temperature. This is against the law of thermodynamics where heat energy is transferred from the hot body to the cold body until they reach thermal equilibrium. Therefore scenario (b) is false.

Analyze the statement for Scenario (c)

Scenario (c) suggests that only the water's temperature will change. This statement, as for scenario (b), is also false because both the object and water will change their temperature until equilibrium is reached.

Analyze the statement for Scenario (d)

According to scenario (d), the heat gained by the object equals the heat lost by the water. This is absolutely correct because, according to the law of conservation of energy, the total amount of energy inside a closed system will not change. In this case, the heat loss from the water equals the heat gained by the object.

Analyze the statement for Scenario (e)

Finally, scenario (e) states that the final temperature will lie within the initial temperatures of the object and the water. This statement is true because energy transfer stops when both object and water are at the same temperature, which will definitely be somewhere between \(5^{\circ} \mathrm{C}\) and \(25^{\circ} \mathrm{C}\).

Key concepts

Thermal Equilibrium

When two objects of different temperatures come into contact within an insulated system, they both undergo temperature changes. This process continues until they reach a point of thermal equilibrium, meaning they share the same temperature and no further heat flow occurs between them. It's helpful to imagine a cup of hot coffee left on the counter. Over time, the coffee cools because heat is transferred to its cooler surroundings. Eventually, it settles at room temperature – reaching thermal equilibrium with the environment.

Applying this concept to our exercise, when the cold object is placed in the warmer water, heat flows from the water to the object. Both temperatures will adjust. Scenario (b) and (c) can be dismissed as false because in a closed system like the insulated cup, both the object and the water will undergo temperature changes until they reach thermal equilibrium, which contradicts the statements that only one will change temperature.

Law of Conservation of Energy

One of the keystones of physics is the law of conservation of energy, which states that energy cannot be created or destroyed, only converted from one form to another. Within an isolated system, the total energy remains constant.

During heat transfer, the energy lost by the warmer body is gained by the cooler body. This tenet is pivotal in understanding why scenario (d) from our exercise is true. The heat lost by the warmer water is precisely the amount of heat gained by the colder object. It's all about balance; the energy within the insulated cup remains unchanged. The energy manifests as internal energy or heat; as one body loses heat, the other gains it, adhering to the conservation principle.

Specific Heat Capacity

The specific heat capacity is a property of a material that indicates how much energy is needed to change the temperature of one kilogram of the material by one degree Celsius. Each substance has its own specific heat capacity, denoted as c. The relationship is given by the formula: \[ Q = m \times c \times \triangle T \] where Q is the heat energy transferred, m is the mass, c is the specific heat capacity, and \(\triangle T\) is the change in temperature.

In the context of the exercise we're considering, scenario (a) cannot be confirmed without knowing the masses and specific heat capacities of the water and the object. If the object had a higher specific heat capacity than water, it would need to absorb more heat energy for the same temperature change compared to water, and vice versa. This shows why a direct temperature equivalence specified in scenario (a) is not guaranteed without more information.