6.4 Thermal Energy Transfer

A metal ball of mass m is released from rest and falls through a distance of h, reaching speed v. The specific heat capacity of the metal is c. 

Show that the increase in temperature, ΔT, of the ball is given by: 

            ΔT =

A meteorite of pure iron falls to Earth. It begins to accelerate uniformly from the edge of the atmosphere. It initially accelerates, then reaches a constant velocity and continues to fall. 

The meteorite falls into a circular paddling pool of water at a temperature of 15 °C, from the edges of the atmosphere at a height, = 200 km. The mass of the meteorite remains at 4.1 kg throughout, and it enters the pool with a velocity of 120 m s^–1, as shown in Figure 1.

Figure 1

The specific heat capacity of iron is 450 J kg­­^–1­°C^–1­, and the meteorite was at a temperature of -265 °C before it started to fall. 

The paddling pool has a circular base of radius 1.5 m, and is full to a height of 40 cm. 

Determine the increase in temperature of the water, assuming that the meteorite and the water reach thermal equilibrium and no thermal energy is lost to the surroundings. 

            The specific heat capacity of water is 4200 J kg­­^–1­ K^–1­

            The density of water is 1000 kg m^–3

A car of mass 850 kg is travelling on a flat road with a constant speed of 32 m s^–1. When an obstacle appears in the road, the brakes are applied and the car comes to a stop. 

The car has four brake disks with a mass of 1.1 kg each, and the specific heat capacity of the brake disk material is 460 J kg­­^–1­ K^–1­. 

Calculate the increase in temperature of the disks.

When brakes are applied in a car, incompressible brake fluid forces the brake pads into place, as shown in Figure 1. Brake fluid heats up due to being in contact with the brake pads, but must not boil, otherwise the fluid will compress and the brakes will not work. 

Figure 1

A brand of brake fluid uses a material called glycol. It is suggested that mixing some water with the glycol may make the brake fluid safer from overheating. 

                     Table 1

Use the data in Table 1, and your answer to part (a) to evaluate whether adding water to glycol will make the brake fluid safer from overheating.

A car manufacturer wishes to create brakes that bring a car to a stop over the same distance, whether going up or down hill. Figure 2 shows a section of road in which cars A and B are approaching each other and need to stop. The road is at an angle θ

from horizontal, and the distance between the cars is x. 

Figure 2

A and B have an identical mass, M, velocity, v, and four identical brake pads of mass . 

Determine an expression for the difference in increase in temperature of the brake pads of car A and car B when they come to a stop after both braking over distance .

An electrical immersion heater with a power of 4 kW is used to heat water flowing past it in a cylinder. The water flows through the heating cylinder at a rate of 0.11 kg s^–1 as shown in Figure 1.Valves at the beginning and end of the cylinder prevent the water from flowing backwards. 

Figure 1

The specific heat capacity of water is 4200 J kg^–1 K^–1

Calculate the rise in temperature of the water as it flows through the heater, assuming all the energy is transferred to the water.

A fault in the pump that pushes water through the heater causes the water to stop flowing. The values at each end of the heating cylinder close, as shown in Figure 2 and the water inside continues to heat. The closed cylinder has a length of 24 cm and a radius of 5 cm. 

Figure 2

The water was at a temperature of 23 °C when the valves shut. 

Water has a density of 1000 kg m^–3. 

Calculate the time taken for the water to boil (at 100 °C) if the immersion heater continues supplying energy at the same rate.

In order to measure the specific heat capacity of liquids, different methods can be used. Figure 3 shows a diagram of two of these methods. 

Figure 3

Method A: Immersion Heater involves submerging an immersion heater in the liquid to be tested. 

Method B: Continuous Flow involves flowing the liquid to be testing past a heater. 

Discuss the two different methods for measuring the specific heat capacity of a liquid. In your answer: 

• Explain how a value for the specific heat capacity is obtained
• Explain any systematic problems with the methods, and how they will affect the final result
• Explain how a continuous flow method can compensate for energy lost as thermal radiation during the experiment 

The quality of your written communication will be assessed in your answer.

An unopened fizzy drink can is cooled using an electric chiller that is powered through a usb connection with a laptop. The chiller is advertised as using 39 W of power, and cools drinks to 15 °C from whatever the room temperature is, and then maintains that temperature.

A can of soda has a mass of 15 g when empty, and contains 320 g of soda. The can is made of a metal alloy with a specific heat capacity of 900 J kg^–1 K^–1 and the soda has a specific heat capacity of 3800 J kg^–1 K^–1. 

Calculate the time it will take to cool the can to 15 °C, after starting cooling it from a room temperature of 21 °C.

The first law of thermodynamics is defined by the equation:

            U = Q – W 

Explain how each of each of the terms in the following equation relates to the situation in part (a).

An ice cube maker is an alternative way of cooling drinks. A particular model advertises that it can produce 10 kg of ice in 24 hours and requires 220 W when working, and produces ice cubes of a temperature of – 6 °C.

The specific latent heat of fusion of ice is 0.336 MJ kg⁻¹

The specific heat capacity of ice is 2100 J kg^–1 K^–1

The specific heat capacity of water is 4200 J kg^–1 K^–1 

Determine whether the ice cube maker, or the electric chiller in question (a), is the most energy efficient method for cooling a drink from 21 °C to 15 °C. Assume that only the required amount of ice is produced.

Explain the meaning of the statement the specific heat capacity of water is 4200 J kg^–1 K^–1.

An engineer is designing an ice-making machine. Water will enter the device at 21°C and the ice cubes are to be cooled to –4°C before release. 

Show that about 0.6 MJ of energy must be removed from 1.5 kg of water at 21°C to change it into ice at –4°C.        

Set out the stages in your answer clearly. 

Specific heat capacity of water = 4.2 × 10³ J kg^–1 K^–1
Specific heat capacity of ice = 2.1 × 10³ J kg^–1 K^–1
Specific latent heat of fusion of ice = 3.3 × 10⁵ J kg^–1

The design brief requires that 4.5 kg of water is frozen in 400 s. 

Calculate the rate at which energy must be removed from the machine.

The specific latent heat of vaporisation of water is 2.3 × 10⁶ J kg^–1. 

Suggest why the specific latent heat of vaporisation of water is much greater than the specific latent heat of fusion of water.

Explain what is meant by the internal energy of a body.

When a substance changes state, there is no change in temperature. However, its internal energy still changes. 

Explain why this happens.

The first law of thermodynamics is defined by the equation: 

            ΔU = Q – W 

State qualitatively what is meant by the first law of thermodynamics and define the variables in the equation.

When a gas expands, it does 190 J of work and loses 95 J of internal energy. 

Calculate the heat transfer to or from the gas.

Explain what is meant by specific latent heat of vaporisation.

Explain why energy is needed to melt ice at 0 ºC to water at 0 ºC and why this energy is lower than what is needed to boil water at 100 ºC to steam at 100 ºC. 

A cup contains 0.75 kg of water at a temperature of 12 °C. The water is heated by passing steam at 100 °C into it. 

Specific heat capacity of water = 4200 J kg^–1 K^–1

Specific latent heat of vaporisation of water = 2.3 × 10⁶ J kg^–1

Boiling point of water = 100 °C. 

Use the above data to calculate the minimum mass of water that is in the cup when the temperature of the water reaches its boiling point.

Explain why there is likely to be a greater mass of water in the cup than you have calculated in part (c).

A student immerses a 3.0 kW electric heater in an insulated beaker of water. The heater is switched on and after 140 s the water reaches boiling point. 

Table 1 gives data collected during the experiment.

                     Table 1

Calculate the specific heat capacity of water if the thermal capacity of the beaker is negligible. State an appropriate unit.

A sample of solid material, which has a mass of 0.17 kg, is supplied with energy at a constant rate. The specific heat capacity of the material is 1500 J kg^–1 K^–1 when in the solid state. During heating, its temperature is recorded at various times and the following graph is plotted as shown in Figure 1. 

Figure 1

Assume there is no heat exchange with the surroundings. 

Show that energy is supplied to the material at a rate of 34 W.

Calculate the specific latent heat of fusion of the material.

Calculate the specific heat capacity of the material when in the liquid state.