B.1 Thermal energy transfers

A black-body sphere at 500 K is placed in surroundings at 300 K. σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

What is the net power lost per unit area of the sphere?

The temperature of a sample of gas is raised from 18 °C to 63 °C.

Which row gives the change in temperature of the gas and the ratio (final average kinetic energy of the particles) / (initial average kinetic energy of the particles)?

A steel block of mass 4.0 kg at 30 °C is placed in contact with a steel nail of mass 0.050 kg at 90 °C. The two objects are isolated from their surroundings.

Which row identifies the object with the greater internal energy before contact, and the direction of the net transfer of thermal energy?

One end of a long ceramic rod is held in a flame. After some time the other end of the rod becomes warm.

Which statement best explains how thermal energy is transferred along the rod?

In a cold store, the cooling unit is mounted just below the ceiling. Air that passes over the cold pipes of the unit is cooled.

Which statement best explains how the air throughout the store, including the air near the floor, is cooled?

A liquid can change into a gas by evaporation or by boiling.

Which statements about evaporation are correct?

I. It takes place only at the surface of the liquid.
II. It can take place at temperatures below the boiling point of the liquid.
III. When no energy is supplied to the liquid, it increases the mean kinetic energy of the molecules that remain in the liquid.

A biological sample is taken out of liquid nitrogen at −196 °C and allowed to warm to a laboratory temperature of 20 °C. What is the change in temperature of the sample?

A block of pure ice at 0 °C is heated steadily and melts. Which statement describes the water molecules during melting?

On board an orbiting space station, a candle flame is roughly spherical and hot gases do not rise away from it. What is the best explanation?

A block of gallium is placed on a warm hand and melts at a constant temperature of 30 °C. During melting, what happens to the mean random kinetic energy and the intermolecular potential energy of the gallium atoms?

The argon in a sealed light bulb is heated from 27 °C to 327 °C. By what factor does the mean random kinetic energy of the argon atoms increase?

Which statement correctly distinguishes evaporation of a liquid from boiling?

A hospital autoclave sterilises instruments using water at a pressure greater than atmospheric pressure. The water in the autoclave boils at a temperature above 100 °C. What is the reason for this?

Air passes through an F1 brake duct at constant pressure and its temperature rises from 300 K to 450 K. What is the value of (final density) ÷ (initial density) of the air? Treat the air as an ideal gas.

A fixed mass of an ideal gas in a flexible bag on a space probe is heated at constant pressure until its thermodynamic temperature doubles. The initial density of the gas is ρ. What is the final density of the gas?

Air near the heater of a coffee roaster rises through the roasting chamber and cooler air flows in to replace it. Which statement best explains why the heated air rises?

A drum contains 0.25 kg of coffee beans with a specific heat capacity of 1600 J kg⁻¹ K⁻¹. The beans are heated from 20 °C to 220 °C. What is the energy transferred to the beans?

Two black bodies are spheres. Sphere X has radius r and temperature T. Sphere Y has radius 2r and temperature T/2.

What is the ratio (power radiated by X) / (power radiated by Y)?

A star has a luminosity of 25L_⊙, where L_⊙ is the luminosity of the Sun. The apparent brightness of the star at the Earth is 1.0 × 10⁻¹⁰ times the apparent brightness of the Sun at the Earth.

What is the distance of the star from the Earth?

Water in an electric kettle is kept boiling by a 2.20 kW heating element. 90 % of the energy supplied by the element is transferred to the water; the rest is lost to the surroundings. The graph shows how the mass m of water in the kettle varies with time t once it is boiling.

What is the specific latent heat of vaporization of water given by these data?

A metal block of mass m and specific heat capacity c, at a temperature θ in degrees Celsius, is placed in a mass 2m of a liquid at 0 °C. The specific heat capacity of the liquid is 4c. No energy is transferred to the surroundings.

What is the final temperature, in degrees Celsius?

Two stars X and Y have black-body spectra with the same peak wavelength. The radius of X is three times the radius of Y, and X is twice as far from the Earth as Y.

The apparent brightness of Y is b. What is the apparent brightness of X?

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 (mean separation of the molecules in the steam) / (mean separation of the molecules in the liquid water)?

A solid sample is heated at constant power with negligible energy losses. The graph shows how its temperature varies with time as it warms, melts and then warms as a liquid.

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

The heating element of an electric grill behaves as a grey body of constant emissivity and surface area. Its temperature is raised from 327 °C to 627 °C.

Which row gives the ratio (peak wavelength of the emitted radiation at 627 °C) / (peak wavelength at 327 °C) and the ratio (power emitted at 627 °C) / (power emitted at 327 °C)?

Two solid spheres P and Q are made of the same metal and have the same black surface. The radius of Q is twice the radius of P. Both spheres are at the same temperature, suspended in an evacuated enclosure whose walls are so cold that the radiation absorbed by the spheres is negligible. The temperature is uniform throughout each sphere.

What is (initial rate of fall of temperature of P)/(initial rate of fall of temperature of Q)?

Star X has twice the surface temperature of star Y and half the radius of star Y. Both stars may be treated as black bodies. What is \(\frac{L_X}{L_Y}\), the ratio of the luminosity of X to the luminosity of Y?

Star P has a luminosity of 9 L_☉ and is at a distance d from Earth. Star Q has a luminosity of 1 L_☉ and is at a distance d/3 from Earth. What is \(\frac{b_P}{b_Q}\), the ratio of the apparent brightness of P to the apparent brightness of Q?

A medical thermal-imaging camera is used to detect radiation from a patient's skin. The skin is at 34 °C. Wien's displacement constant is 2.9 × 10⁻³ m K. What is the approximate wavelength at which the skin's emission peaks?

A main-sequence star has a luminosity of 100 L_☉ and a surface temperature twice that of the Sun. What is the radius of the star in terms of the solar radius R_☉?

The wall of a laboratory freezer is made of two flat layers, X and Y, of equal thickness and equal area, in good thermal contact. The thermal conductivity of X is three times that of Y. Energy is conducted through the wall at a steady rate.
What is the ratio \(\frac{\text{temperature difference across X}}{\text{temperature difference across Y}}\)?

Equal masses of water at 20 °C and an engine coolant at 80 °C are mixed in an insulated container. The specific heat capacity of the coolant is half that of water. No energy is transferred to the surroundings or the container.
What is the final temperature of the mixture?

The density of liquid water is 1000 kg m⁻³. The density of steam at 100 °C and atmospheric pressure is 0.60 kg m⁻³.
What is the best estimate of \(\frac{\text{mean molecular spacing in steam}}{\text{mean molecular spacing in liquid water}}\)?

Two stars, X and Y, have the same luminosity. The peak wavelength in the spectrum of X is twice that of Y. Both stars behave as black bodies.
What is \(\frac{\text{radius of X}}{\text{radius of Y}}\)?

A shallow dish of water in a beehive loses water by evaporation and the remaining water becomes cooler. Which explanation is correct?

A solid of mass m is heated by a heater of constant power. Its temperature rises from 20 °C to 60 °C in 50 s. It then melts at 60 °C, which takes a further 100 s. After it has melted, the liquid rises from 60 °C to 100 °C in 150 s. Assume all the energy from the heater goes into the sample. What is the value of (specific heat capacity of the liquid) ÷ (specific heat capacity of the solid)?

A beekeeper dries a thin film of nectar in a warm hive. Which single change would decrease the rate at which water evaporates from the film?

The surface of a space probe's radiator behaves as a black body. Its thermodynamic temperature is increased to three times its original value. The original peak wavelength was λ and the original total power emitted was P. Which row gives the new peak wavelength and the new total power emitted?

A sample of 0.30 kg of ice at 0 °C is taken from the deck of a bridge. A heater of power 500 W, with all of its energy going to the ice, melts the sample completely. The specific latent heat of fusion of ice is 3.3 × 10⁵ J kg⁻¹. What is the time taken?

The graph of intensity against wavelength for a hot black-body surface has a single peak. The surface temperature is raised. Which statement about the new graph is correct?

Hot coffee of mass 0.30 kg at 80 °C is mixed in an insulated cup with cold milk of mass 0.20 kg at 20 °C. Assume the coffee and the milk have the same specific heat capacity and that no energy is transferred to the surroundings. What is the final temperature of the mixture?

Two rods of the same cross-sectional area are joined end to end. Rod X has length L and thermal conductivity 2k; rod Y has length 2L and thermal conductivity k. The free end of X is at 100 °C and the free end of Y at 0 °C, and the sides are insulated.

What is the steady temperature of the junction?

A solid is heated by a heater of constant power, and no energy is lost to the surroundings. While the substance is solid, its temperature rises at 2.0 K s⁻¹. When it reaches its melting point, it takes 120 s to melt completely. What is the ratio \(\frac{L_f}{c_s}\) of the specific latent heat of fusion to the specific heat capacity of the solid?

A small spherical space probe behaves as a black body and has no internal power source. At distance d from the Sun its equilibrium surface temperature is T. The probe then moves to a distance 4d from the Sun. What is its new equilibrium temperature?

A small black sphere at a temperature of 400 K sits inside a large evacuated enclosure whose walls are at 200 K. The net power radiated by the sphere is P.
The sphere is heated to 800 K. The walls stay at 200 K. What is the new net power radiated by the sphere?

A hospital warmer contains an 80 W heater that is switched on for 5.0 minutes. Only 60% of the energy it transfers reaches a block of ice already at 0 °C, which melts. The specific latent heat of fusion of ice is 3.34 × 10⁵ J kg⁻¹. What is the mass of ice that melts?

An insulated container holds 0.50 kg of water at 30 °C. A block of ice of mass 0.10 kg at 0 °C is added and all of it melts. The specific heat capacity of water is 4.18 × 10³ J kg⁻¹ K⁻¹ and the specific latent heat of fusion of ice is 3.34 × 10⁵ J kg⁻¹. Energy transfer to the container and surroundings is negligible. What is the final temperature?

Two spherical radiators on a space probe emit as black bodies. Radiator X has radius r and thermodynamic temperature T. Radiator Y has radius 2r and thermodynamic temperature T/2. What is the value of (power emitted by Y) ÷ (power emitted by X)?

A student tests the Stefan–Boltzmann law using the tungsten filament of a small lamp in an evacuated bulb. The kelvin temperature T of the filament is found from its resistance, and the electrical power P supplied is assumed to be radiated. The filament is modelled as a grey body:

P = eσAT⁴

where e is the emissivity and A = 1.0 × 10⁻⁵ m² is the surface area of the filament. The table and the graph show log₁₀ P against log₁₀ T with the line of best fit. The power is uncertain by ±2 %, the temperature by ±1.5 % and the area by ±4 %.

A cardboard tube, closed at both ends, contains 0.80 kg of lead shot. When the tube is quickly inverted, the shot falls through a mean height of 1.10 m and stops at the bottom. The student inverts the tube N times, then pours the shot into a polystyrene cup and measures the rise in temperature ΔT with a digital thermometer of resolution 0.1 K. The shot is cooled back to room temperature before each run. The table and the graph show the results with the line of best fit.

A student determines the density of air. A rigid glass flask with a tap has an internal volume of 1.25 × 10⁻³ m³ (±1 %). The flask is connected to a vacuum pump and a pressure gauge. The student reduces the pressure p of the air in the flask in steps; at each step the tap is closed and the mass m of the flask and the air inside it is measured on a balance. The temperature of the air in the flask is 20 °C throughout.

The table and the graph show m against p with the line of best fit.

A test tube containing 0.025 kg of molten lauric acid is clamped in still air in a room at 20 °C. A temperature sensor records the temperature θ of the lauric acid every minute as it cools and freezes. The graph shows the data points and a smooth curve through them. The specific heat capacity of liquid lauric acid is 2.3 × 10³ J kg⁻¹ K⁻¹. The rate of energy loss to the room depends only on the temperature difference between the tube and the room.

A student investigates how the apparent brightness b of the radiation from a small filament lamp, supplied with 12 W of electrical power, varies with distance. A radiation sensor is placed at distance d from the front of the glass bulb. The filament is an unknown distance d₀ behind the front of the bulb, so the student models the lamp as a point source of luminosity L:

b = L / 4π(d + d₀)²

The graph shows b^−1/2 against d with the line of best fit.

A student measures the thermal conductivity k of expanded polystyrene. A small heater inside a closed polystyrene box keeps the air in the box 20.0 K above room temperature. When conditions are steady, the electrical power P supplied to the heater is recorded. The experiment is repeated with boxes that have the same inner surface area A = 0.240 m² but walls of different thickness x.

The student models the walls as flat slabs, so that P = kAΔT/x + P₀, where ΔT = 20.0 K and P₀ is the power lost through the gap around the lid, which does not depend on x. The uncertainty in each value of P is ±0.3 W. The graph shows P against 1/x with error bars, the line of best fit (solid) and the steepest and shallowest lines consistent with the error bars (dashed).

A student tests Wien's displacement law using an electrically heated ceramic element. An infrared spectrometer measures the wavelength λ_max at which the emitted radiation is most intense, with an uncertainty of ±0.03 μm. The temperature θ of the element is measured with a thermocouple embedded in the middle of the element.

The table gives the results. The graph shows 1/λ_max against the thermocouple temperature T in kelvin, with the line of best fit.

A physiotherapist takes an ice pack from a freezer and places it on an athlete's injured ankle. The pack contains 0.25 kg of ice at −10 °C. The ice warms to 0 °C and then melts completely.
Specific heat capacity of ice = 2100 J kg⁻¹ K⁻¹
Specific latent heat of fusion of ice = 3.34 × 10⁵ J kg⁻¹

A hospital ward kettle contains 1.5 kg of water at 20 °C. Its heater has a power of 2.0 kW and all the energy it transfers goes into the water. Specific heat capacity of water = 4.18 × 10³ J kg⁻¹ K⁻¹. Specific latent heat of vaporisation of water = 2.26 × 10⁶ J kg⁻¹.

A 0.50 kg block of a solid material for storing heat in a Formula 1 team garage is heated by an immersed 250 W heater. Its temperature is 20 °C at t = 0 and 60 °C at t = 300 s. The temperature then stays at 60 °C until t = 800 s, when all the solid has melted. Assume all the energy from the heater goes into the material.

Air in a coffee roaster's chimney is at a thermodynamic temperature of 353 K, while the air outside is at 293 K. Both are at a pressure of 1.01 × 10⁵ Pa. The molar mass of air is 0.0290 kg mol⁻¹. Use the molar gas constant R = 8.31 J mol⁻¹ K⁻¹.

A student determines the specific heat capacity of an aluminium block of mass 0.80 kg that will be used in a probe's thermal test. A heater of power 60 W is switched on for 5.0 minutes and the temperature of the block rises by 21.0 K.

A mountaineer melts snow in a pan on a small gas stove. The pan contains 0.80 kg of snow, initially at −24 °C, and the stove transfers thermal energy to the contents of the pan at a constant rate P. Neglect the energy absorbed by the pan and the energy lost to the surroundings.

The graph shows how the temperature θ of the contents varies with time t until the water reaches 25 °C. Specific heat capacity of ice = 2.1 × 10³ J kg⁻¹ K⁻¹; specific latent heat of fusion of ice = 3.34 × 10⁵ J kg⁻¹.

A thermal imaging camera detects the infrared radiation emitted by people at night. A person has a surface area of 1.8 m² at a mean surface temperature of 33 °C. The emissivity of the surface is 0.98.

A runner of mass 65 kg runs at a constant speed of 4.0 m s⁻¹ on level ground against a total resistive force of 15 N. Her muscles convert chemical energy into mechanical work with an efficiency of 20 %; the rest of this energy is transferred as thermal energy to her body. Specific latent heat of vaporization of sweat at skin temperature = 2.4 × 10⁶ J kg⁻¹; mean specific heat capacity of the body = 3.5 × 10³ J kg⁻¹ K⁻¹.

In laser spot welding, a single pulse of infrared light melts a small cylinder of steel of diameter 0.40 mm and depth 0.20 mm. The steel is initially at 20 °C. Each pulse has an energy of 0.50 J and the wavelength of the light is 1.06 μm.

Data for steel: density = 7.9 × 10³ kg m⁻³; specific heat capacity = 490 J kg⁻¹ K⁻¹; melting point = 1800 K; specific latent heat of fusion = 2.7 × 10⁵ J kg⁻¹.

A reusable hand warmer is a sealed plastic pouch containing 0.090 kg of liquid sodium acetate trihydrate at room temperature, 20 °C. The liquid is below its normal freezing temperature. When a small metal disc in the pouch is clicked, the liquid starts to crystallise (solidify). Its temperature rises rapidly to the melting point, 58 °C, and then stays at 58 °C for a long time while crystallisation continues.

Specific heat capacity of the liquid = 3.0 × 10³ J kg⁻¹ K⁻¹; specific heat capacity of the solid = 2.0 × 10³ J kg⁻¹ K⁻¹; specific latent heat of fusion = 2.64 × 10⁵ J kg⁻¹.

During a hot-weather marathon, a runner's body must lose excess thermal energy at a rate of 600 W to keep a constant core temperature. Assume that all of this energy is removed by the evaporation of sweat.
Specific latent heat of vaporization of sweat at skin temperature = 2.4 × 10⁶ J kg⁻¹
Mass of runner = 65 kg
Average specific heat capacity of the human body = 3500 J kg⁻¹ K⁻¹

An engineer tests an aluminium alloy block of mass 0.80 kg for use as a heat sink. A 50 W electrical heater is embedded in the block. The temperature of the block is recorded every 60 s.

Time / s: 0, 60, 120, 180, 240
Temperature / °C: 20.0, 24.0, 28.1, 32.0, 36.0

A 2.2 kW electric kettle contains 1.2 kg of water at 15 °C. It takes 215 s to bring the water to its boiling point of 100 °C. The kettle's switch is faulty, so the kettle stays on after the water boils.
Specific heat capacity of water = 4180 J kg⁻¹ K⁻¹
Specific latent heat of vaporization of water = 2.26 × 10⁶ J kg⁻¹

A cyclist adds 0.050 kg of crushed ice at 0 °C to a bottle containing 0.40 kg of sports drink at 28 °C. The bottle is well insulated and has negligible heat capacity. All the ice melts.
Specific heat capacity of the drink and of water = 4180 J kg⁻¹ K⁻¹
Specific latent heat of fusion of ice = 3.34 × 10⁵ J kg⁻¹

A student finds the specific heat capacity of steel. Steel ball bearings are heated in boiling water and then quickly transferred into water in an insulated polystyrene cup. The data are shown below.
Quantity — Value
mass of steel balls — 0.250 kg
initial temperature of balls — 100.0 °C
mass of water in cup — 0.180 kg
initial temperature of water — 18.0 °C
final temperature of mixture — 28.5 °C
Specific heat capacity of water = 4180 J kg⁻¹ K⁻¹. Assume the cup absorbs no energy.

Liquid nitrogen is used in medicine to freeze and destroy skin lesions.
Density of liquid nitrogen = 808 kg m⁻³.
Density of nitrogen gas at 20 °C and atmospheric pressure = 1.16 kg m⁻³.
Molar mass of nitrogen = 0.028 kg mol⁻¹. N_A = 6.02×10²³ mol⁻¹.

A home energy store uses 12 kg of paraffin wax as a phase-change material. The wax is heated from a solid at 20 °C to a liquid at 70 °C. The melting point of the wax is 58 °C.
Specific heat capacity of solid wax = 2100 J kg⁻¹ K⁻¹.
Specific heat capacity of liquid wax = 2400 J kg⁻¹ K⁻¹.
Specific latent heat of fusion of wax = 2.0×10⁵ J kg⁻¹.
Specific heat capacity of water = 4180 J kg⁻¹ K⁻¹.

The envelope of a hot-air balloon used for aerial filming has a volume of 2800 m³. The outside air is at 15 °C and has a density of 1.22 kg m⁻³. The air inside the envelope is heated to 100 °C. The air inside and outside is at the same pressure. At constant pressure, the density of air is inversely proportional to its kelvin temperature.

Air in a beehive is at a temperature of 35 °C (308 K) and a pressure of 1.01 × 10⁵ Pa. The molar mass of air is 0.0290 kg mol⁻¹ and the molar gas constant is R = 8.31 J mol⁻¹ K⁻¹. Treat air as an ideal gas.

During a 90-minute race a Formula 1 driver loses 2.1 kg of sweat, of which 70% evaporates from the skin. Specific latent heat of vaporisation of sweat = 2.4 × 10⁶ J kg⁻¹. The driver's mass is 72 kg and the specific heat capacity of the body is 3500 J kg⁻¹ K⁻¹. The driver's body generates thermal energy at an average rate of 920 W.

A 0.40 kg block of beeswax is initially a solid at 25 °C. It is heated in a water bath by a 600 W heater, of which 80% of the energy transferred goes into the wax. The wax melts at a constant temperature of 62 °C and is then heated to 80 °C. Specific heat capacity of solid wax = 2500 J kg⁻¹ K⁻¹. Specific heat capacity of liquid wax = 2900 J kg⁻¹ K⁻¹. Specific latent heat of fusion of wax = 1.8 × 10⁵ J kg⁻¹.

A hospital sterilises 0.80 kg of steel instruments, initially at 20 °C, in a steam chamber. Steam at 100 °C condenses on the instruments until they reach 100 °C. Assume that all the energy released by the condensing steam goes to the instruments and that the condensed water stays at 100 °C. Specific heat capacity of steel = 480 J kg⁻¹ K⁻¹. Specific latent heat of vaporisation of water = 2.26 × 10⁶ J kg⁻¹. Specific heat capacity of water = 4.18 × 10³ J kg⁻¹ K⁻¹.

Honeybees keep a hive cool in hot weather by fanning their wings over water droplets that they have placed on the comb.

A re-entry probe has a nose cone that can be modelled as a black body with a total surface area of 8.0 m². Its surface is at a thermodynamic temperature of 1400 K. Use the Stefan–Boltzmann constant σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴ and Wien's constant 2.90 × 10⁻³ m K.

A coffee roaster heats 12.0 kg of green beans, of which 1.2 kg is water. Assume the following.
• The 10.8 kg of dry bean material, specific heat capacity 1600 J kg⁻¹ K⁻¹, is heated from 22 °C to 200 °C.
• The 1.2 kg of water, specific heat capacity 4200 J kg⁻¹ K⁻¹, is heated from 22 °C to 100 °C and then completely vaporised. The specific latent heat of vaporisation of water is 2.26 × 10⁶ J kg⁻¹.
• Energy needed to heat the drum is ignored.

A probe's thermal camera records the radiation emitted by two hot components, X and Y, which can be modelled as black bodies. The peak wavelength of the radiation from X is 1.45 µm and that from Y is 4.35 µm. Wien's constant is 2.90 × 10⁻³ m K.