B.1 Thermal energy transfers

A block of metal has a mass of 0.500 kg and a volume of 2.5 × 10⁻⁴ m³.

What is its density?

A sample of water is at 300 K. Its temperature then rises by 20 °C.

What is its initial temperature in degrees Celsius, and what is the rise in kelvin?

An electric heater stands on the floor of a closed room. A convection current is set up in the air of the room.

Which statement explains why the air just above the heater rises?

Which statement about the internal energy of a substance is correct?

Two stars have the same luminosity, but one of them is three times as far from the Earth as the other.

What is the ratio (apparent brightness of the more distant star) / (apparent brightness of the nearer star)?

Energy reaches the Earth from the Sun across empty space.

By which mechanism is it transferred?

The surface of the Sun behaves as a black body at a temperature of about 5800 K. Wien's displacement law constant is 2.9 × 10⁻³ m K.

At what wavelength does the Sun's emission spectrum peak?

A substance can exist as a solid, as a liquid and as a gas.

Which statement describes the molecules of the substance when it is a liquid?

Which statement about a black body is correct?

A small steel rivet at 450 °C is dropped into a tank containing 200 kg of water at 15 °C. The internal energy of the water in the tank is far greater than the internal energy of the rivet.

Which statement about the thermal energy transfer once the rivet is in the water is correct?

The front glass wall of a tropical aquarium is 0.60 m wide, 0.40 m high and 8.0 mm thick. The thermal conductivity of the glass is 0.96 W m⁻¹ K⁻¹. The inner surface of the glass is at 24.5 °C and the outer surface is at 23.9 °C.

What is the rate of thermal energy transfer through the front glass?

A titanium hip implant has been inside a patient’s body for several months. The implant and the bone in contact with it are both at 37 °C.

Which statement describes the particles where the implant touches the bone?

A sports physiotherapist places an ice pack on an athlete's sprained ankle. The ice in the pack melts at a constant 0 °C. Which describes the changes in the molecules of the ice while it melts?

A laptop processor is cooled by a copper heat sink rather than a plastic block of the same shape. What is the main reason copper conducts thermal energy much better than plastic?

A communications satellite orbits far above the atmosphere. Which mechanism(s) can transfer thermal energy from the Sun to the satellite?

A walk-in freezer has an insulating wall panel. The panel is replaced by one made of the same material with double the thickness and double the area. The temperature difference across the panel is unchanged. How does the rate of thermal energy transfer through the new panel compare with that through the original panel?

A small steel rivet at 400 °C is dropped into a large tank of water at 20 °C during a shipyard quenching process. The water in the tank has far more internal energy than the rivet. Which statement about the thermal energy transfer is correct?

A biological sample is stored in liquid nitrogen at 77 K. It is removed and left in a laboratory until it reaches 20 °C. What is the change in temperature of the sample?

A nurse places a digital thermometer under a patient's tongue and waits before reading it. Why is the wait necessary?

A hospital patient's body temperature rises from 37.0 °C to 39.5 °C during a fever. What is the change in the patient's absolute (thermodynamic) temperature?

A ski-boot liner is redesigned. The new liner has double the surface area and double the thickness of the old one. On a milder day, the temperature difference across it is half that of the original test. What is the new rate of thermal energy transfer by conduction compared with the original?

The air in a sealed laboratory chamber starts at 27 °C. It is heated until its absolute temperature has doubled. What is its new temperature in degrees Celsius?

A frozen tissue sample is stored at −196 °C. It is moved to a transport container and warms to −171 °C. What is the change in temperature of the sample in kelvin?

A deep-sea diving support vessel stores helium gas and xenon gas in separate cylinders kept at the same temperature. Which quantity is the same for the atoms in both cylinders?

The infrared detectors on a space telescope are cooled to a temperature close to absolute zero. Which statement describes absolute zero?

The heating element of an electric grill glows when it is switched on. At its working temperature T (in kelvin), the radiation it emits has its peak at wavelength λ. When the grill is turned down, the peak wavelength becomes 1.5λ. Assume the element behaves as a black body. What is the new temperature of the element?

An LED-replacement project tests an old-style lamp filament. The filament is replaced by one with half the surface area that operates at twice the absolute temperature. The emissivity is unchanged. What is the ratio of the new radiated power to the original radiated power?

An engineer chooses a coating for the radiator fins of a motorcycle engine. Which statement correctly defines the emissivity of a surface?

An electric heater of power P warms a liquid of mass m for a time t. The temperature of the liquid rises by Δθ. Only 80 % of the energy supplied by the heater is transferred to the liquid.

What is the specific heat capacity of the liquid?

A block of ice at 0 °C is heated at a constant rate until all of the water formed from it has boiled away. No energy is lost to the surroundings.

The specific latent heat of vaporization of water is about seven times its specific latent heat of fusion.

Which sketch graph shows the variation with time t of the temperature θ of the sample?

A hot metal block is placed in cold water in an insulated container. Neither the block nor the water changes phase. After some time thermal equilibrium is reached.

Which statement about the block and the water at thermal equilibrium is correct?

The temperature of a sample of an ideal gas is increased from 27 °C to 327 °C.

What is the ratio (average kinetic energy of the molecules at 327 °C) / (average kinetic energy of the molecules at 27 °C)?

One end of a long glass rod is held in a flame. Some time later the other end of the rod has become warm.

Which statement describes how energy is conducted along the rod?

In each graph the solid line shows the variation with wavelength λ of the intensity I emitted by a black body at temperature T.

The body is then heated to a higher temperature. Which graph shows, as a dashed line, the emission of the body at the higher temperature?

When an energy Q is supplied to block X, its temperature rises by ΔT. Block Y has twice the mass of X, and the specific heat capacity of Y is 1.5 times that of X.

The same energy Q is supplied to Y. No energy is lost. What is the rise in temperature of Y?

The water in an electric kettle is boiling at 100 °C. The heating element supplies energy at a rate of 2.0 kW, and all of this energy is transferred to the water.

Specific latent heat of fusion of water = 3.34 × 10⁵ J kg⁻¹. Specific latent heat of vaporization of water = 2.26 × 10⁶ J kg⁻¹.

Which row gives the mass of water turned into steam in 60 s, and the change in the intermolecular potential energy of the water that boils?

Two rods of the same length carry the same temperature difference between their ends. Rod X has four times the thermal conductivity of rod Y and twice the cross-sectional area.

The rate of thermal energy transfer along X is how many times that along Y?

A metal block of mass M is heated to a temperature θ_h and then placed in an insulated cup containing a mass m of water at temperature θ_c. The final temperature of the block and the water is θ_f. The specific heat capacity of water is c_w. Energy transferred to the cup and to the surroundings is negligible.

What is the specific heat capacity of the metal?

The thermal radiation emitted by the outer surface of a mug of hot tea at 57 °C has its peak at a wavelength of 8.79 μm. The radiation emitted by the heating element of an oven has its peak at 2.93 μm. Both surfaces behave as black bodies.

What is the temperature of the heating element?

Liquid X has density ρ and liquid Y has density 3ρ. Equal masses of X and Y are mixed. The liquids do not react, and the volume of the mixture is equal to the total volume of X and Y before mixing.

What is the density of the mixture?

A sealed lamp contains a mixture of helium atoms and xenon atoms that has been at a steady temperature for a long time. The mass of a xenon atom is about 33 times the mass of a helium atom.

Which row compares the xenon atoms with the helium atoms?

Water is stored in a porous clay pot that stands in dry, moving air. Water seeps through the clay and evaporates from the outer surface of the pot. The water inside cools to a temperature below that of the air.

Which statement explains why evaporation lowers the temperature of the water that remains?

A small electrically heated plate behaves as a black body. Energy is lost from the plate only by radiation, and the radiation it absorbs from the surroundings is negligible. The electrical power supplied to the plate is doubled.

When the temperature of the plate is steady again, what is (new kelvin temperature) / (original kelvin temperature)?

Star Y is 2.0 × 10⁵ times as far from the Earth as the Sun is. The apparent brightness of Y at the Earth is 1.0 × 10⁻¹⁰ times the apparent brightness of the Sun.

What is the luminosity of Y? L_☉ is the luminosity of the Sun.

A sample of an ideal gas is heated from 0 °C to 300 °C.

Which graph shows how the average kinetic energy E_k of the molecules of the gas varies with the temperature θ of the gas in degrees Celsius?

A marathon runner is cooled as sweat evaporates from their skin at a temperature well below 100 °C. Which statement best explains why evaporation cools the remaining sweat and the skin?

The superconducting magnet of a hospital MRI scanner is kept in liquid helium at its boiling point. Thermal energy leaks into the helium at a steady rate of 0.20 W. The specific latent heat of vaporisation of helium is 2.0 × 10⁴ J kg⁻¹. What mass of helium boils off in 5.0 × 10⁴ s?

Air-conditioning units that cool a room are usually mounted high on a wall, close to the ceiling. Which statement explains why this position cools the whole room effectively?

On a hot day, the air in a car tyre warms from 20 °C to 40 °C. By what factor does the average random kinetic energy of the air molecules increase?

Two brake discs, A and B, absorb the same amount of energy during an emergency stop. Disc B has twice the mass of disc A. The specific heat capacity of the material of B is one third of that of A. What is \(\frac{\Delta T_B}{\Delta T_A}\), the ratio of the temperature rise of B to that of A?

Two modules are docked in space and come into thermal contact. They are isolated from their surroundings. Module X has a heat capacity of 300 J K⁻¹ and is at 80 °C. Module Y has a heat capacity of 100 J K⁻¹ and is at 0 °C. What is their final common temperature?

A sample of engine coolant of mass m is heated by an electric heater of constant power P. A data logger shows that the temperature rises at a constant rate R (in K s⁻¹). Energy losses are negligible. What is the specific heat capacity of the coolant?

The absolute temperature of an ideal gas in a spacecraft propellant tank rises from 150 K to 300 K. What happens to the average random kinetic energy of the gas molecules?

A processor dissipates 50 W into a copper heat sink of mass 0.25 kg. The specific heat capacity of copper is 400 J kg⁻¹ K⁻¹. Assume no energy leaves the heat sink. At what rate does the temperature of the heat sink rise?

A space probe travels from Earth's orbit (1.0 AU from the Sun) to Jupiter's orbit (5.2 AU from the Sun). Approximately what fraction of the apparent brightness of the Sun at Earth does the probe receive at Jupiter?

A red giant star has a radius three times that of star S and a surface temperature half that of star S. What is the ratio of the luminosity of the red giant to the luminosity of star S?

An outside wall is made from a layer of brick (thermal conductivity 0.80 W m⁻¹ K⁻¹) and a layer of insulating foam (thermal conductivity 0.040 W m⁻¹ K⁻¹) of equal thickness and equal area, in contact with each other. Thermal energy is conducted through the wall at a steady rate. Which statement is correct?

Two stars, X and Y, have the same apparent brightness when observed from Earth. The luminosity of X is 16 times the luminosity of Y. What is the ratio \(\frac{d_X}{d_Y}\) of their distances from Earth?

An astronomer wants to determine the radius of a distant star, treating it as a black body. Which set of measurements is sufficient?

Thermal conductivity is measured in W m⁻¹ K⁻¹. What is this unit expressed in SI base units?

Gas flowing through a heat exchanger is heated from 27 °C to 327 °C. What is the ratio \(\frac{\text{final average kinetic energy of a molecule}}{\text{initial average kinetic energy of a molecule}}\)?

A 3D-printed lattice bracket for an aircraft is made from a titanium alloy of density 4.5 g cm⁻³. The bracket has a mass of 144 g and occupies an external volume of 80 cm³. What fraction of the external volume is empty space?

Crushed ice at 0 °C in a sports injury pack melts while it absorbs energy from an athlete's ankle. The temperature of the pack stays at 0 °C. What happens to the average kinetic energy and the average potential energy of the water molecules during melting?

Two copper heat sinks in a computer have masses of 0.20 kg and 0.40 kg. Both are at 60 °C. Which statement is correct?

The tip of a soldering iron is heated from 27 °C to 327 °C. The tip can be treated as a black body, and its area does not change. By what factor does the power radiated by the tip increase?

Hot water at 80 °C is poured into a ceramic mug at 20 °C. The mass of the water equals the mass of the mug. The specific heat capacity of the ceramic is one quarter of that of water. Assume no energy is transferred to the surroundings. What is the final common temperature?

Two stars X and Y have the same radius. The peak wavelength in the spectrum of X is half that of Y. What is the ratio of luminosity of X to luminosity of Y?

A patient's exposed skin has area A, emissivity e and absolute temperature T_s. The hospital room walls are at absolute temperature T_r, where T_r < T_s. Which expression gives the net rate of energy loss by radiation from the skin?

A student measures the specific heat capacity of a metal block. She heats the block in boiling water, records the temperature of the boiling water, then carries the block to a calorimeter of cold water. The block cools noticeably in the air during the transfer. How does this affect the calculated specific heat capacity of the metal?

A star of radius R behaves as a black body with surface temperature T. It is at a distance d from the Earth. σ is the Stefan–Boltzmann constant.

What is the apparent brightness of the star at the Earth?

At 100 °C and atmospheric pressure the density of liquid water is 958 kg m⁻³ and the density of steam is 0.598 kg m⁻³.

What is the best estimate of the ratio (mean separation of the molecules in steam) / (mean separation of the molecules in liquid water)?

Star P has twice the radius of star Q, but only half its surface temperature. Both behave as black bodies.

The luminosity of P is how many times that of Q?

A lagged composite bar is made of a copper section and a stainless-steel section of equal length and equal cross-sectional area, joined end to end. The thermal conductivity of copper is about 25 times that of stainless steel. The free end of the copper is held at 100 °C, the free end of the steel at 0 °C, and the temperatures are steady.

Which graph shows how the temperature θ varies with the distance x along the bar from its hot end?

A liquid in a container loses energy to its surroundings at a constant rate. The graph shows how its temperature θ varies with time. The liquid cools by Δθ in a time t₁; its temperature is then constant for a time t₂ while it freezes.

What is (specific latent heat of fusion) / (specific heat capacity of the liquid)?

Ice at 0 °C is added to 0.300 kg of fruit juice at 25.0 °C in an insulated jug. When all of the ice has melted, the temperature of the mixture is 5.0 °C. No energy is exchanged with the jug or the surroundings.

Specific heat capacity of juice and of water = 4.18 × 10³ J kg⁻¹ K⁻¹. Specific latent heat of fusion of ice = 3.34 × 10⁵ J kg⁻¹.

What mass of ice was added?

A closed cool box has walls of expanded polystyrene. A second cool box is made of the same material, and every dimension of the second box, including the thickness of its walls, is twice the corresponding dimension of the first box.

The temperatures inside and outside the two boxes are the same. Energy enters each box only by conduction through its walls, and the walls are thin compared with the size of the box.

What is (rate of thermal energy transfer into the second box) / (rate of thermal energy transfer into the first box)?

On the Earth a candle flame is tall and pointed. On board an orbiting space station, where the station and everything in it are in free fall, a candle flame in still air is almost spherical and much dimmer.

Which statement explains this?

A solar thermal store contains a solid wax. A heater supplies energy to the wax at a constant rate, and all of it is absorbed. The temperature of the solid rises by 20 K in 2.0 minutes. The wax then reaches its melting point and takes a further 10 minutes to melt completely. What is the value of \(\frac{L}{c}\), where L is the specific latent heat of fusion and c is the specific heat capacity of the solid wax?

A cool box at a sports event contains 0.20 kg of water at 40 °C. Ice at 0 °C is added, and the system is insulated. What is the minimum mass of ice needed to bring all the water to 0 °C?
Specific heat capacity of water = 4.2 × 10³ J kg⁻¹ K⁻¹
Specific latent heat of fusion of ice = 3.36 × 10⁵ J kg⁻¹

An engineer measures the specific heat capacity of a battery cell for an electric vehicle. An electric heater supplies a measured amount of energy, and the engineer uses c = E/(mΔT). The cell loses a significant amount of energy to the surrounding air during heating. How does the calculated value compare with the true value?

Two blocks, P and Q, are placed in thermal contact inside an insulated container. By the time they reach equilibrium, the temperature of P has fallen by 30 K and the temperature of Q has risen by 10 K. The masses of the blocks are not known. Which statement must be correct?

A ceramic heating element in a toaster warms up from 27 °C to 327 °C. Its emissivity and surface area stay constant. By what factor does the power it radiates increase?

The atmosphere of a distant exoplanet contains helium (molar mass 4 g mol⁻¹) and methane (molar mass 16 g mol⁻¹) at the same temperature. What is the ratio \(\frac{\text{root mean square speed of helium atoms}}{\text{root mean square speed of methane molecules}}\)?

A jeweller makes an alloy by melting together equal masses of two metals with densities ρ₁ and ρ₂. Assume that the volume of the alloy equals the sum of the volumes of the two metals. What is the density of the alloy?

Two stars A and B have the same apparent brightness when observed from Earth. The surface temperature of A is twice that of B. A is twice as far from Earth as B. What is the ratio of the radius of A to the radius of B?

An athlete drops ice at 0 °C into an equal mass of sports drink at 60 °C in an insulated flask. Treat the drink as water: specific heat capacity 4200 J kg⁻¹ K⁻¹, specific latent heat of fusion of ice 3.3×10⁵ J kg⁻¹. Which describes the final state of the contents?

A beaker of water stands on a top-pan balance. An immersion heater in the water is supplied with a power of 150 W ± 2 %, measured with a joulemeter. When the water is boiling steadily, the balance reading m is recorded every 60 s.

Steel ball bearings are left in boiling water for 10 minutes and then transferred quickly into water in an insulated polystyrene cup. The mixture is stirred and the highest temperature reached is recorded. Masses are measured with a balance of resolution 0.1 g and temperatures with a thermometer read to ±0.5 °C.

The specific heat capacity of water is 4180 J kg⁻¹ K⁻¹. Assume that no energy is transferred to the cup or to the air.

A student measures the specific heat capacity c of sunflower oil. An immersion heater of power P = 48.0 W ± 2 % is placed in a mass m of oil in a glass beaker. The oil is stirred, and the time t for its temperature to rise from 25.0 °C to 35.0 °C is measured. The thermometer reads to 0.1 °C. The experiment is repeated for different masses of oil, each time starting with the beaker and heater at 25.0 °C.

In this experiment energy is also needed to warm the beaker and the heater, so Pt = (mc + C)ΔT, where C is the energy needed to raise the temperature of the beaker and heater by 1 K and ΔT = 10.0 K. Each value of t has an uncertainty of ±2 s, shown by the error bars.

A student determines the density ρ of the steel used to make ball bearings. She measures the diameter d of each of six ball bearings with a micrometer screw gauge and the mass m of each ball with a balance of resolution 0.001 g.

For a sphere m = (πρ/6)d³, so a graph of m against d³ is expected to be a straight line through the origin. The table shows the results. The graph shows five of the six data points.

A student finds a published table of the density ρ of pure water at atmospheric pressure for temperatures θ from 0 °C to 20 °C. The table and the graph show the data.

A student determines the specific heat capacity c of brass. A brass cylinder of mass 0.745 ± 0.001 kg has two holes drilled in it: one holds an electric heater and the other a temperature probe connected to a data logger. The cylinder is wrapped in a layer of insulating foam.

The p.d. across the heater is constant at 11.9 ± 0.1 V and the current is constant at 1.52 ± 0.02 A. The heater is switched on at t = 0 and the temperature θ is recorded every 60 s. The table and the graph show the results.

A small filament lamp is treated as a point source that radiates uniformly in all directions. A radiation sensor measures the intensity b of the radiation at a distance d from the lamp. d is measured with a metre rule from a mark on the front edge of the lamp holder to the face of the sensor. The room is dark and the sensor is zeroed before the lamp is switched on. Each reading of b has an uncertainty of ±2 %.

The student plots b^−½ against d. The graph shows six of the seven points.

A student measures the thermal conductivity k of copper. One end of a copper bar is heated by an electric heater; the other end is cooled by running water. The bar is surrounded by thick lagging. The p.d. across the heater is 12.0 ± 0.1 V and the current is 2.00 ± 0.02 A. The diameter of the bar, measured with vernier calipers, is 25.0 ± 0.1 mm.

When the temperatures are steady, thermocouples read the temperature θ at six distances x from the heated end. Each reading has an uncertainty of ±0.5 °C, shown by the error bars.

A student uses a published database to investigate how the specific heat capacity c of a metal at room temperature depends on the molar mass M of the metal. She suggests that c = KMⁿ, where K and n are constants, and tests the hypothesis that n = −1.

The table shows the data for six metals. The graph shows the points for four of them.