Question

Two solid objects, \(A\) and B, are placed in boiling water and allowed to come to the temperature of the water. Each is then lifted out and placed in separate beakers containing \(1000 \mathrm{~g}\) water at \(10.0^{\circ} \mathrm{C}\). Object \(\mathrm{A}\) increases the water temperature by \(3.50^{\circ} \mathrm{C} ;\) B increases the water temperature by \(2.60^{\circ} \mathrm{C}\). (a) Which object has the larger heat capacity? (b) What can you say about the specific heats of \(\mathrm{A}\) and \(\mathrm{B}\) ?

Short answer

The object with the larger heat capacity is object A, as it transfers more heat to the water, increasing its temperature by a greater amount. We cannot directly determine the specific heats of objects A and B from the given information, but we can infer that object A must have a larger specific heat and/or mass than object B, as it is able to transfer more heat to the water in the beaker.

Step-by-step solution

Calculate Heat Transfer for Objects and Water in the beakers

First, let's calculate how much heat each object transferred to the water in their beakers using the final temperature of the water. For object A: \(Q_A = m_{w}c_{w}\Delta T_A\) For object B: \(Q_B = m_{w}c_{w}\Delta T_B\) Where: - \(m_{w}\) is the mass of water (1000g) - \(c_{w}\) is the specific heat capacity of water (4.18 J/g°C) - \(\Delta T_A\) and \(\Delta T_B\) are the temperature changes in the water after adding object A and object B, respectively (3.50°C for A and 2.60°C for B) Now, calculate the heat transfers: \(Q_A = 1000 \times 4.18 \times 3.50 = 14630 \) J \(Q_B = 1000 \times 4.18 \times 2.60 = 10868 \) J

Compare Heat Capacities

To determine which object has a larger heat capacity, we can compare the heat transfers of objects A and B. The object with the larger heat transfer has the larger heat capacity. Since \(Q_A > Q_B\), object A has a larger heat capacity.

Discuss Specific Heats

We cannot determine the specific heats of object A and object B directly from the given information, as we do not know their masses or the exact heat capacities. However, we can infer some information about their specific heats. Since object A has a larger heat capacity and both objects reach the same initial temperature in boiling water, object A must have a larger specific heat and/or mass than object B, as it is able to transfer more heat to the water in the beaker.

Key concepts

Heat Capacity

Heat capacity is an essential concept in understanding heat transfer. It refers to the amount of heat energy required to change the temperature of an object by one degree Celsius (or Kelvin). In this exercise, we compared two objects, A and B, to identify which had the larger heat capacity. When these objects were placed in water, they released energy, causing the water's temperature to rise.

In simpler terms, heat capacity can be thought of as a thermal reservoir – the bigger the capacity, the more heat that can be stored or released. From our experiment, object A, which caused a larger temperature change in the water, demonstrated a larger heat capacity than object B.

To find the heat transferred by these objects, we used the formula:

• For object A: \(Q_A = m_{w}c_{w}\Delta T_A\)
• For object B: \(Q_B = m_{w}c_{w}\Delta T_B\)

The object with the greater value of \(Q\) – in our case, A – has a larger heat capacity because it transfers more heat to the water, resulting in a larger change in temperature.

Specific Heat

Specific heat is a material property that tells us how much heat energy is needed to change the temperature of one gram of a substance by one degree Celsius. It reflects how a material absorbs and retains heat. In the exercise, we explored the relationship between specific heat and heat capacity.

While we didn't calculate the exact specific heats for objects A and B, we inferred insights based on their heat capacity. Specific heat contributes to an object's overall heat capacity, alongside its mass. Thus, a material could have a high specific heat, allowing it to absorb more heat without significantly changing temperature, or the object could simply be large or heavy.

For object A, we determined from its greater heat capacity that it could have a higher specific heat than object B. This would be one reason why A transferred more heat to the water, elevating the temperature more than object B did. When considering specific heat, think about how much energy a given substance can hold without drastic temperature shifts.

Temperature Change

Temperature change is a vital measurement in heat transfer scenarios, indicating the amount of heat energy exchanged. In the exercise, this was measured by observing the change in water temperature after introducing the hot objects.

Whenever heat moves from one body to another, it changes the temperature of at least one of them. The water's temperature change, due to heat gained from objects A and B, was critical in determining which object had a larger heat capacity. More energy resulted in a more significant temperature shift for the water, highlighting greater heat movement.

In practical terms, analyzing temperature change helps us understand energy transformation and storage in materials. For instance, since the temperature of the water increased differently with objects A and B, the corresponding energy transfers were a product of both objects' unique thermal characteristics.

By calculating the temperature changes, we could determine that object A caused a 3.50°C increase. In contrast, object B caused a 2.60°C rise, which confirmed our findings about their relative heat capacities.