# Worksheet: Thermal Equilibrium

In this worksheet, we will practice describing the exchange of energy between objects that heat each other and observing how their temperatures change as a result.

**Q1: **

Two perfectly insulated gas canisters contain air with a specific heat capacity of and a density of 1.35 kg/m. One cylinder contains 25 L of air at a temperature of and the other contains 15 L of air at a temperature of . The canisters are connected and the gas in both is allowed to come to an equilibrium temperature. Find the equilibrium temperature to three significant figures.

**Q2: **

Some hot water that has a mass of 750 g and a temperature of is poured over 1,500 g of an unknown substance that has a temperature of . The temperatures of the water and the substance are both monitored. At an instant , the water temperature is and the substance temperature is . Find the specific heat capacity of the substance. Use a value of for the water’s specific heat capacity. Assume that the water does not heat its surroundings.

- A
- B
- C
- D
- E

**Q3: **

A 2.5 L volume of water at a temperature of is added to a perfectly insulated container that holds a 45 kg block of lead. After the water and the lead have equalized each other’s net heating, the temperature of both substances is . Find the initial temperature of the lead block. Use a value of for the specific heat capacity of water and a value of for the specific heat capacity of lead. Answer to three significant figures.

**Q4: **

A brick has a mass of 4.5 kg, a temperature of , and a specific heat capacity of . The brick is dropped into a perfectly insulated water tank containing 15 kg of water at a temperature of . At an instant after the brick has been dropped into the water, the temperature of the water is . Find the temperature of the brick at . Answer to the nearest degree. Use a value of for the specific heat capacity of water.

**Q5: **

The graph shows the changes in temperature with time of different objects. Each object has a different temperature after eight minutes of cooling, and all the objects’ initial temperatures are different from each other. Which of the following statements most correctly expresses what is shown by the curves drawn in the graph?

- AAll the objects cool at the same average rate.
- BThe magnitude of the instantaneous rate of cooling of all the objects at a given time is the same.
- CThe object with the lowest temperature after eight minutes has the greatest rate of cooling.
- DThe object with the highest initial temperature has the greatest rate of cooling.

**Q6: **

The graph shows the changes in temperature with time of different objects. Each object has a different temperature after eight minutes of cooling, but all the objects’ temperatures are initially the same. At any given time, one object is cooling at a greater rate than any of the other objects at that same time. Find the temperature of this object after eight minutes. Answer to one significant figure.

**Q7: **

The graph shows the changes in temperature with time of different objects. Each object has a different initial temperature, but all the objects’ temperatures become more similar to each other the longer the objects cool.

What temperature would all the objects eventually have, given unlimited cooling time?

At any given time, one object is cooling at a greater rate than any of the other objects at that same time. What is the initial temperature of this object?

**Q8: **

An open-topped cylinder contains water and a metal cube, as shown in the diagram. The water and the cube are both at the same temperature. The metal cube heats the water in the cylinder at a rate of 36 W and the water heats the cube at a rate of 12 W. The air directly above the cylinder is heated at a rate of 14 W. At what rate is the cylinder heated?

**Q9: **

The objects A, B, and C in the diagram have the temperatures , , and respectively. The temperature of object A is the same as the temperature of object C. The temperature of object C is the same as the temperature of object B. How does compare to ?

- A
- B
- C

**Q10: **

A high-temperature reservoir consists of
100 kg
of water that is at an initial temperature of
.
A low-temperature reservoir consists of 100 kg
of water that is initially at a temperature of
.
The high-temperature reservoir heats the low-temperature reservoir until both reservoirs reach an equilibrium temperature,
taking 7,550 seconds.
Assume that both reservoirs are perfectly insulated except for the contact channel between them.
The contact channel between the reservoirs is a metal block with an area of
0.012 m^{2}
and a length of 8.5 cm.
The metal has thermal conductivity of
.

Find the average rate of net heating of the low-temperature reservoir. Use a value of for the specific heat capacity of water and answer to three significant figures.

What is the initial rate at which the high-temperature reservoir heats the low-temperature reservoir? Answer to three significant figures.

**Q11: **

Two equal masses of water heat each other and reach an equilibrium temperature, as shown by the temperature changes with time of the masses of water in the diagram. Which of the following diagrams correctly shows the changes of temperature with time if the initially lower temperature water has a greater mass than the initially higher temperature water?

- A
- B
- C
- D

**Q12: **

The graph shows the temperature changes over time of two objects. The horizontal axis values represent time, and the vertical axis values represent temperature. The temperature values for both curves are the same for the time values and .

Are the average cooling rates of both objects over the time interval shown in the graph equal?

- ANo
- BYes

Are the instantaneous cooling rates of both objects over the time interval shown in the graph equal for all times between and ?

- AYes
- BNo

**Q13: **

A bowl of soup is heated in a microwave oven. The heated bowl is removed and placed on a table where the bowl and the soup both decrease in temperature by in a time of 420 s. The bowl has a mass of 0.75 kg and a specific heat capacity of . The soup has a mass of 0.95 kg and a specific capacity of . Find the average rate of heating of the table and the air around it. Give your answer to the nearest watt.

**Q14: **

Two objects, object 1 and object 2, heat the air around them at different rates while air flows between them, as shown in the diagram. The air around the objects heats them at different rates. The direction of the flow of air between the objects depends only on the difference between the net heating of the air near each object. The difference in the net heating of the air around each object produces heat convection in the air between them. Assume that the heating rates shown in the diagram remain constant over a short time interval during which heat convection occurs.

Which object is the heat convection directed toward during the time interval ?

- AObject 1
- BObject 2

For the object that the heat convection is directed toward during the time interval , what is the ratio of the rate at which the object heats the air to the rate at which heat convection occurs?