Chapter 7: Review

The chapter problem, solved

A pressure cooker holds 1.50 kg of water at 20 °C. The water boils at 121 °C. The specific heat capacity of water is 4200 J kg⁻¹ K⁻¹ and the specific latent heat of vaporisation is $2.26 \times 10^{6}\,\text{J}\,\text{kg}^{-1}$ . The steam above the water is at $2.00 \times 10^{5}\,\text{Pa}$ and 121 °C, in a space of volume $2.00 \times 10^{-3}\,\text{m}^{3}$ . The cooker boils for 12 minutes, and 0.050 kg of water turns into steam. The constants are $R = 8.31\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1}$ , $k = 1.38 \times 10^{-23}\,\text{J}\,\text{K}^{-1}$ and $N_A = 6.02 \times 10^{23}\,\text{mol}^{-1}$ .

1. The temperature change. The water starts at 20 °C and boils at 121 °C.

\Delta\theta = 121 - 20 = 101\,\text{K}

2. The energy to heat the water.

E = mc\Delta\theta = 1.50 \times 4200 \times 101 = 6.36 \times 10^{5}\,\text{J}

3. The energy to turn some water into steam.

E = mL = 0.050 \times 2.26 \times 10^{6} = 1.13 \times 10^{5}\,\text{J}

4. The total energy.

E_{\text{total}} = 6.36 \times 10^{5} + 1.13 \times 10^{5} = 7.49 \times 10^{5}\,\text{J}

5. The average rate of energy transfer while boiling. The boiling lasts $12 \times 60 = 720\,\text{s}$ .

P = \frac{E}{t} = \frac{1.13 \times 10^{5}}{720} = 157\,\text{W}

6. The number of moles of steam. The temperature is $121 + 273 = 394\,\text{K}$ .

\begin{aligned} n &= \frac{pV}{RT} \\ &= \frac{2.00 \times 10^{5} \times 2.00 \times 10^{-3}}{8.31 \times 394} \\ &= 0.122\,\text{mol} \end{aligned}

7. The number of molecules.

N = nN_A = 0.122 \times 6.02 \times 10^{23} = 7.36 \times 10^{22}

8. The average kinetic energy of a steam molecule.

\tfrac{3}{2}kT = 1.5 \times 1.38 \times 10^{-23} \times 394 = 8.16 \times 10^{-21}\,\text{J}

9. The rms speed of a steam molecule. The molar mass of water is $0.018\,\text{kg}\,\text{mol}^{-1}$ .

c_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3 \times 8.31 \times 394}{0.018}} = 739\,\text{m}\,\text{s}^{-1}

10. The judgement. The cooker reaches 121 °C because the lid is clamped shut and the valve holds the pressure at $2.00 \times 10^{5}\,\text{Pa}$ . Water boils when its vapour pressure matches the pressure above it, so a higher pressure raises the boiling temperature above 100 °C. The hotter water cooks the food faster. The steam cannot escape, so it pushes outwards on the lid, and the clamp must be strong enough to hold it.

11. The first prediction. As the water is heated with the lid clamped shut, the pressure rises. The volume of the space is fixed, and the temperature rises, so $pV = NkT$ means the pressure must rise. The water can get hotter than 100 °C before it boils, because the higher pressure raises the boiling point.

Mixed practice — choose the method first

For each question, say which rule or equation you will use before you calculate.

The pressure cooker holds 1.50 kg of water at 20 C and the water boils at 121 C. Find the energy needed to heat the water to boiling. c = 4200 J/kg/K.
Find the energy needed to turn 0.050 kg of boiling water into steam. L = 2.26e6 J/kg.
Find the total energy supplied to heat the water and to turn 0.050 kg of it into steam.
The cooker boils for 12 minutes, and 0.050 kg of water turns to steam in that time. Find the average rate at which energy is used for boiling.
The steam above the water is at 2.00e5 Pa and 121 C, in a volume of 2.00e-3 m3. Treat it as an ideal gas and find the number of moles. R = 8.31 J/K/mol.
Find the number of molecules in the steam. Use n = 0.1222 mol and N_A = 6.02e23 /mol.
Find the average random kinetic energy of a steam molecule at 121 C. k = 1.38e-23 J/K.
Find the rms speed of a steam molecule at 121 C. The molar mass of water is 0.018 kg/mol and R = 8.31 J/K/mol.
Explain, in terms of the steam molecules, why the steam exerts a pressure on the inside of the cooker.
At the start, the lid is clamped shut and the water is heated. Explain why the pressure rises and why the water can reach 121 C before it boils.
A 2.0 kg metal block is heated through 25.0 K. The mass has a percentage uncertainty of 0.5% and the temperature change 0.8%. The specific heat capacity is taken as exact. Find the energy supplied and its absolute uncertainty.
An experiment gives an energy of 210000 J. Write this value to two significant figures.
A molecule of mass 4.8e-26 kg moves at 500 m/s. Find its kinetic energy.
A molecule of mass 4.8e-26 kg hits a wall at 500 m/s and rebounds at 500 m/s. Find the change in its momentum.
Choose the equation to use for each situation, and give a reason. (a) Heating water in a kettle with no boiling. (b) Melting ice at 0 C. (c) Finding the number of moles of steam in a sealed space.

Answers and concise solutions

1. The energy to heat the water

Answer: $E = mc\Delta\theta = 1.50 \times 4200 \times 101 = 6.36 \times 10^{5}\,\text{J}$ .

2. The energy to boil the water

Answer: $E = mL = 0.050 \times 2.26 \times 10^{6} = 1.13 \times 10^{5}\,\text{J}$ .

3. The total energy

Answer: $E = 6.36 \times 10^{5} + 1.13 \times 10^{5} = 7.49 \times 10^{5}\,\text{J}$ .

4. The average rate

Answer: The time is $12 \times 60 = 720\,\text{s}$ . The rate is $\dfrac{1.13 \times 10^{5}}{720} = 157\,\text{W}$ .

5. The number of moles of steam

Answer: $n = \dfrac{pV}{RT} = \dfrac{2.00 \times 10^{5} \times 2.00 \times 10^{-3}}{8.31 \times 394} = 0.122\,\text{mol}$ .

6. The number of molecules

Answer: $N = nN_A = 0.1222 \times 6.02 \times 10^{23} = 7.36 \times 10^{22}$ .

7. The average molecular kinetic energy

Answer: $\tfrac{3}{2}kT = 1.5 \times 1.38 \times 10^{-23} \times 394 = 8.16 \times 10^{-21}\,\text{J}$ .

8. The rms speed

Answer: $c_{\text{rms}} = \sqrt{\dfrac{3RT}{M}} = \sqrt{\dfrac{3 \times 8.31 \times 394}{0.018}} = 739\,\text{m}\,\text{s}^{-1}$ .

9. A model answer

The steam molecules move quickly and at random.
They collide with the inside surface of the cooker.
Each molecule rebounds, so its momentum changes.
The change in momentum means the surface feels a force.
Very many collisions each second give a steady force on each unit of area, which is the pressure.

Answer: The molecules move at random and strike the surface. Each collision changes a molecule's momentum, so the surface feels a force. Very many collisions each second give a steady force per unit area, which is the pressure.

10. A model answer

The temperature of the water and the steam rises as energy is supplied.
The volume is fixed and the temperature rises, so pV = NkT means the pressure rises.
Water boils when its saturated vapour pressure equals the pressure above it, so a higher pressure means a higher boiling temperature.

Answer: At constant volume, a higher temperature means a higher pressure. Water boils when its vapour pressure matches the pressure above it, so a higher pressure raises the boiling temperature to 121 C.

11. The energy and its uncertainty

Answer: The energy is $E = mc\Delta\theta = 2.0 \times 4200 \times 25.0 = 2.1 \times 10^{5}\,\text{J}$ . The percentage uncertainties add: $0.5\% + 0.8\% = 1.3\%$ . The absolute uncertainty is $210000 \times \dfrac{1.3}{100} = 2700\,\text{J}$ .

12. The energy to two significant figures

Answer: $210000\,\text{J} = 2.1 \times 10^{5}\,\text{J}$ to two significant figures.

13. The kinetic energy

Answer: $E_k = \tfrac{1}{2}mv^{2} = \tfrac{1}{2} \times 4.8 \times 10^{-26} \times 500^{2} = 6.0 \times 10^{-21}\,\text{J}$ .

14. The change in momentum

Answer: The velocity reverses, so the change in momentum is $2mv = 2 \times 4.8 \times 10^{-26} \times 500 = 4.8 \times 10^{-23}\,\text{kg}\,\text{m}\,\text{s}^{-1}$ .

15. A model answer

For heating the water, use E = mc(theta). The temperature changes but the water does not change state.
For melting the ice, use E = mL. The state changes at a constant temperature, so the energy goes into breaking bonds.
For the steam, use pV = nRT. It is an ideal gas, and the equation links the pressure, volume, temperature and number of moles.

Answer: (a) E = mc(theta), because the temperature changes. (b) E = mL, because the state changes at constant temperature. (c) pV = nRT, because the steam is an ideal gas.

What you can now do

explain internal energy, and how it differs from temperature;
use $E = mc\Delta\theta$ for a temperature change;
find a specific heat capacity from an electrical heating experiment;
explain what happens to the particles during a change of state;
use $E = mL$ for melting, boiling, freezing and condensing;
read a heating curve and explain its flat parts;
describe the ideal gas model;
use Boyle's law, Charles's law and the pressure law;
convert a temperature between degrees Celsius and kelvin;
explain absolute zero;
use $pV = nRT$ and $pV = NkT$ ;
explain pressure in terms of molecular collisions;
find the average molecular kinetic energy and the rms speed of a gas;
test Boyle's law and read the graph;
explain why a pressure cooker cooks food faster.

Your relevance reflection

Return to the prompt from the chapter opener: think of a time you have seen something heat up or change state, such as ice melting in a drink, a metal spoon warming in soup, or a bicycle pump getting hot. Write one more example now that you have finished the chapter. Has the chapter changed how you think about the energy in a hot drink, a boiling pan, or the air in a warm room?