Specific heat capacity

One of these assumptions is that particle motion is random.

State what is meant by random particle motion.

State two other assumptions required for this derivation.

When the temperature of a fixed mass of an ideal gas increases at constant pressure, the gas must expand.

Explain this expansion using the kinetic theory model.

Helium behaves as an ideal gas. The temperature of a sample of helium gas is increased from 27 °C to 77 °C.

Calculate the change in the average kinetic energy of a helium atom in the sample.

When an ideal gas is heated at constant volume, all of the energy supplied by heating is used to increase the average kinetic energy of the particles.

When water vapour is heated at constant volume, its specific heat capacity

Deduce whether water vapour behaves as an ideal gas when heated at constant volume.

mass of a water molecule = 3.0 × 10⁻²⁶ kg

A radioisotope thermoelectric generator (RTG) uses radioactive decay to generate electrical energy.

A vehicle that is used to examine the surface of Mars includes an RTG that initially contains 3.23 × 10₂₄ atoms of plutonium-238 238Pu 94 .

Determine the initial mass of plutonium-238 in the RTG.

The plutonium-238 undergoes alpha decay. The energy released by this decay is converted into electrical energy with an efficiency of 4.58%.

The power provided by the RTG decreases over time as the plutonium-238 decays.

The vehicle requires a minimum electrical power of 29.6 W to operate.

Calculate the initial activity of the plutonium-238. Go on to determine, in Earth years, the time for which the RTG can provide sufficient power for the vehicle to operate.

decay constant of plutonium-238 = 2.51 × 10⁻¹⁰ s⁻¹ energy of alpha particle emitted by plutonium-238 = 8.94 × 10⁻¹³ J

The vehicle has a battery-powered laser that is used to vaporise small samples of ice. On Mars, ice changes directly from solid to vapour at a temperature of 0 °C. The ice does not form a liquid.

The energy required to vaporise 1.0 g of solid ice directly into water vapour is the same as the total energy required to melt 1.0 g of ice and then to vaporise the melted ice.

The laser produces an output power of 1.0 kW for a maximum time of 3.0 s before its battery needs to be recharged.

The laser heats the ice from an initial temperature of −25 °C to a temperature of 0.0 °C. The laser then transfers energy to vaporise the ice at 0.0 °C.

Determine whether the laser is able to vaporise 1.0 g of the ice that is initially at −25 °C before the battery needs to be recharged.

specific heat capacity of ice = 2.1 kJ kg⁻¹ K⁻¹ specific latent heat of fusion of ice = 334 kJ kg⁻¹ specific latent heat of vaporisation = 2500 kJ kg⁻¹

A room contains dry air at a temperature of 20.0 °C and a pressure of 105 kPa.

Show that the amount of air in each cubic metre is about 40 mol.

The density of the dry air is 1.25 kg m⁻³.

Calculate c_rms for the air molecules. Give your answer to an appropriate number of significant figures.

Calculate, in K, the change of temperature that will double c_rms for the air molecules.

A room contains moist air at a temperature of 20 °C. A dehumidifier cools and then condenses water vapour from the moist air. The final temperature of the liquid water that collects in the dehumidifier is 10 °C. Drier air leaves the dehumidifier at a temperature of 20 °C.

Table 1 compares the air flowing into and out from the dehumidifier.

In one hour, a volume of 960 m³ of air flows through the dehumidifier. Assume that the density of the air remains constant at 1.25 kg m⁻³.

Determine how much heat energy is removed in one hour from the water vapour by the dehumidifier.

specific heat capacity of water vapour = 1860 J kg⁻¹ K⁻¹ specific latent heat of vaporisation of water = 2.3 × 10⁶ J kg⁻¹

A 1000 W heater is 75% efficient. The heater is used to increase the temperature of some water from 10 °C to 85 °C in 7 hours.

What mass of water is heated?

specific heat capacity of water = 4200 J kg⁻¹ K⁻¹

Figure 1 shows an electric steam iron.

Water from a reservoir drips onto an electrically-heated metal plate. The water boils and steam escapes through holes in the metal plate.

The electrical power of the heater inside the iron is 2.1 kW.

Assume that all the energy from the heater is transferred to the metal plate.

The metal plate has a mass of 1.2 kg and is initially at a temperature of 20 °C. The heater is switched on. After a time t the metal plate reaches its working temperature of 125 °C.

Calculate t.

specific heat capacity of the metal = 450 J kg⁻¹ K⁻¹

The metal plate is maintained at its working temperature. Water at 20 °C drips continuously onto the metal plate. Steam at 100 °C emerges continuously from the iron.

The maker claims that the iron can generate steam at a rate of 60 g min⁻¹.

Determine whether this claim is true.

specific latent heat of vaporisation of water = 2.3 10⁶ J kg⁻¹ specific heat capacity of water = 4200 J kg⁻¹ K⁻¹

A solar panel transfers energy at a rate of 1.2 kW to liquid passing through it. The liquid has a specific heat capacity of 4.0 kJ kg⁻¹ K⁻¹.

When the liquid flows through the solar panel, its temperature increases by 3.0 K.

The flow rate of the liquid is

A perfectly insulated flask contains a sample of metal M at a temperature of –10 °C.

Figure 1 shows how the temperature of the sample changes when energy is transferred to it at a constant rate of 35 W.

State the melting temperature of M.

Explain how the energy transferred to the sample changes the arrangement of the atoms during the time interval t_A to t_B.

State what happens to the potential energy of the atoms and to the kinetic energy of the atoms during the time interval t_A to t_B.

Describe how the motion of the atoms changes during the time interval t_B to t_C.

The sample has a mass of 0.25 kg.

Determine the specific heat capacity of M when in the liquid state. State an appropriate SI unit for your answer.

Table 1 shows the specific latent heats of fusion l for elements that are liquid at similar temperatures to M.

M is known to be one of the elements in Table 1.

Identify M.

Figure 1 shows a perfectly insulated cylinder containing 0.050 kg of liquid nitrogen at a temperature of 70 K. A heater transfers energy at a constant rate of 12 W to the nitrogen. A piston maintains the pressure at 1.0 × 10⁵ Pa during the heating process.

The nitrogen is heated from 70 K and is completely turned into a gas after 890 s.

Calculate the specific heat capacity of liquid nitrogen. Give an appropriate unit for your answer.

specific latent heat of vaporisation of nitrogen = 2.0 × 10⁵ J kg⁻¹

boiling point of nitrogen = 77 K

The work done by the nitrogen in the cylinder when expanding due to a change of state is X. The energy required to change the state of the nitrogen from a liquid to a gas is Y.

Deduce which is greater, X or Y.

density of liquid nitrogen at its boiling temperature = 810 kg m⁻³

density of nitrogen gas at its boiling temperature = 3.8 kg m⁻³

State what is meant by the internal energy of a gas.

Absolute zero of temperature can be interpreted in terms of the ideal gas laws or the kinetic energy of particles in an ideal gas.

Describe these two interpretations of absolute zero of temperature.

A mixture of argon atoms and helium atoms is in a cylinder enclosed with a piston. The mixture is at a temperature of 310 K.

Calculate the root mean square speed (c_rms) of the argon atoms in the mixture.

molar mass of argon = 4.0 × 10⁻² kg mol⁻¹

Compare the mean kinetic energy of the argon atoms and the helium atoms in the mixture.

Explain, in terms of the kinetic theory model, why a pressure is exerted by the gas on the piston.

The mixture of gases in the cylinder stays the same.

Explain, using the kinetic theory model, two changes that can be made independently to reduce the pressure exerted by the gas.

Explain what is meant by specific latent heat of fusion.

Figure 1 shows how the temperature of the water is maintained in a hot tub.

The hot tub system has a volume of 4.5 m³ and is filled with water at a temperature of 28 °C

The heater transfers thermal energy to the water at a rate of 2.7 kW while a pump circulates the water.

Assume that no heat is transferred to the surroundings.

Calculate the rise in water temperature that the heater could produce in 1.0 hour.

density of water = 1000 kg m^–3 specific heat capacity of water = 4200 J kg^–1 K^–1

The pump can circulate the water at different speeds. When working at higher speeds the rise in temperature is greater.

Explain why. Again assume that no heat is transferred to the surroundings.

A number of assumptions are made when explaining the behaviour of a gas using the molecular kinetic theory model.

State one assumption about the size of molecules.

Figure 1 shows how the pressure changes with volume for a fixed mass of an ideal gas.

At A the temperature of the gas is 27^oC. The gas then undergoes two changes, one from A to B and then one from B to C.

Calculate the number of gas molecules trapped in the cylinder using information from the initial situation at A.

Calculate, in K, the change in temperature of the gas during the compression that occurs between A and B.

Deduce whether the temperature of the gas changes during the compression from B to C.

Compare the work done on the gas during the change from A to B with that from B to C on Figure 1.

What is the temperature difference of the water between the top and bottom of the waterfall?

The specific heat capacity of water is 4200 J kg⁻¹ K⁻¹.

What is the average rate at which thermal energy is gained by the water?