Question

Two solid objects, $A$ and $B$, are placed in boiling water and allowed to come to temperature there. Each is then lifted out and placed in separate beakers containing $1000\mathrm{g}$ water at ${10.0}^{\circ }\mathrm{C}$. Object 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 A has a larger heat capacity as it transfers more heat ($14630\mathrm{J}$) to the water in the beaker compared to object B ($10868\mathrm{J}$). We can't determine the specific heats of A and B without more information, but since A has a larger heat capacity, its specific heat is likely to be higher than the specific heat of B.

Step-by-step solution

Define heat capacity and specific heat

Heat capacity is the amount of heat required to change the temperature of an object by 1 degree Celsius (or other temperature unit). Specific heat is the heat capacity per unit mass, i.e., the amount of heat required to change the temperature of 1 gram of an object by 1 degree Celsius.

Calculate the heat gained by water in each beaker

We are given that object A increased the water temperature by ${3.50}^{\circ }\mathrm{C}$ and object B increased the water temperature by ${2.60}^{\circ }\mathrm{C}$. The heat gained by the water in each beaker can be calculated using the formula: Heat gained = $mc\mathrm{\Delta }T$ where m is the mass of water (1000 g), c is the specific heat of water $(4.18\mathrm{J}/{\mathrm{g}}^{\circ }\mathrm{C})$, and ΔT is the change in temperature. Let's calculate the heat gained by the water for both beakers. For object A: Heat gained = $(1000)(4.18)(3.50)=14630\mathrm{J}$ For object B: Heat gained = $(1000)(4.18)(2.60)=10868\mathrm{J}$

Determine the larger heat capacity

Since the two objects were initially at the same temperature (boiling water) and ended up transferring different amounts of heat to the water in the beakers, we can conclude that the object with larger heat capacity absorbed more heat from the boiling water and, therefore, transferred more heat to the water in the beaker. In our case, object A had more heat transferred to the water ($14630\mathrm{J}$) than object B ($10868\mathrm{J}$). Hence, object A has the larger heat capacity.

Discuss the specific heats of A and B

We can't determine the specific heats of A and B without more information. However, we can make a general statement based on heat capacity differences. Since object A has a larger heat capacity than object B, for the same mass and temperature change, object A will require more heat than object B. Therefore, we can say that the specific heat of A (per unit mass) is likely to be higher than the specific heat of B.

Key concepts

Specific Heat

Specific heat is a fundamental concept that tells us how a material reacts to heat energy. It is defined as the amount of heat required to change the temperature of 1 gram of a substance by 1 degree Celsius. This value helps us understand how different materials react to the same amount of heat.

• Materials with a high specific heat require more energy to change their temperature.
• This property is intrinsic to the material—it doesn't change based on how much of the material you have.
• Knowing the specific heat of a substance helps in applications such as designing heating or cooling systems.

In the exercise, the specific heat can't be directly determined because we don't have the mass of objects A and B. But we can infer certain things. For example, if an object with higher heat capacity requires more heat for the same temperature change, it might have a higher specific heat.

Thermal Energy Transfer

Thermal energy transfer is the process of heat being exchanged between substances. This is central to thermodynamics and helps explain how heat moves.

• Energy is transferred from the hotter object to the cooler one until they reach thermal equilibrium—this is when both objects are at the same temperature.
• This process depends on the specific heat and mass of the materials involved.
• In a practical scenario, thermal energy transfer is involved in processes like heating buildings, cooking, and even biological systems.

In our example, when objects A and B were placed into the cold water, they transferred heat to the water until they cooled down to the water's initial temperature. A greater change in water temperature indicates more heat was transferred.

Thermodynamic Properties

Thermodynamic properties are characteristics that define the thermal and energy state of a system. They include variables like temperature, pressure, volume, and heat capacity.

• Understanding these properties helps us predict how systems will behave when subjected to different conditions.
• Heat capacity, an important thermodynamic property, tells us how much heat energy is needed to change the temperature of an object, without changing its physical state.
• Each material and object will have its unique set of thermodynamic properties.

In the given exercise, comparing the heat capacities of objects A and B allowed predictions about their thermal behavior without needing to know their specific masses or the exact material properties.