Level 2 — RecallThermodynamics

Thermodynamics

40 marksprintable — key stays hidden on paper

Level 2 (Recall & Standard Problems)

Time: 30 minutes Total marks: 40

Use $R = 8.314\ \mathrm{J\,mol^{-1}K^{-1}}$ , $\sigma = 5.67\times10^{-8}\ \mathrm{W\,m^{-2}K^{-4}}$ , $k_B = 1.38\times10^{-23}\ \mathrm{J/K}$ where needed.

Q1. State the Zeroth law of thermodynamics and explain briefly why it justifies the use of a thermometer. (3 marks)

Q2. State the first law of thermodynamics, defining each symbol and its sign convention using the form $dU = dQ - dW$ . (3 marks)

Q3. A copper block of mass $0.50\ \mathrm{kg}$ is heated from $20^\circ\mathrm{C}$ to $70^\circ\mathrm{C}$ . Given specific heat capacity $c = 385\ \mathrm{J\,kg^{-1}K^{-1}}$ , calculate the heat supplied. (3 marks)

Q4. How much heat is required to convert $200\ \mathrm{g}$ of ice at $0^\circ\mathrm{C}$ into water at $0^\circ\mathrm{C}$ ? (Latent heat of fusion $L_f = 3.34\times10^5\ \mathrm{J\,kg^{-1}}$ .) Then how much additional heat raises this water to $30^\circ\mathrm{C}$ (take $c_{water}=4186\ \mathrm{J\,kg^{-1}K^{-1}}$ )? (4 marks)

Q5. A metal rod of length $1.00\ \mathrm{m}$ has linear expansion coefficient $\alpha = 1.2\times10^{-5}\ \mathrm{K^{-1}}$ . Find its increase in length when heated by $80\ \mathrm{K}$ . State the volumetric expansion coefficient in terms of $\alpha$ . (4 marks)

Q6. (a) Write the ideal gas law $PV = nRT$ . (b) Calculate the pressure of $2.0\ \mathrm{mol}$ of an ideal gas in a $0.050\ \mathrm{m^3}$ container at $300\ \mathrm{K}$ . (4 marks)

Q7. For one mole of an ideal gas undergoing an isothermal expansion from volume $V_1$ to $V_2$ at temperature $T$ , derive the expression for the work done by the gas. (4 marks)

Q8. (a) Write the relation for the RMS speed of gas molecules in terms of temperature and molar mass. (b) Compute $v_{rms}$ for oxygen ( $M = 0.032\ \mathrm{kg/mol}$ ) at $300\ \mathrm{K}$ . (4 marks)

Q9. A Carnot engine operates between a hot reservoir at $500\ \mathrm{K}$ and a cold reservoir at $300\ \mathrm{K}$ . (a) Find its efficiency. (b) State how efficiency depends on the reservoir temperatures. (4 marks)

Q10. State the Clausius definition of entropy change and compute the entropy change when $500\ \mathrm{J}$ of heat is transferred reversibly to a reservoir at $250\ \mathrm{K}$ . (3 marks)

End of paper

Answer keyMark scheme & solutions

Q1. (3 marks)

Statement: If two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other. (2)
Justification: this transitivity means a thermometer (the "third system") gives a consistent temperature reading; two bodies at the same thermometer reading are in equilibrium. (1)

Q2. (3 marks)

$dU = dQ - dW$ . (1)
$dU$ = change in internal energy; $dQ$ = heat added to the system (positive when absorbed); $dW$ = work done by the system (positive when gas expands). (2)

Q3. (3 marks)

$Q = mc\Delta T$ (1)
$\Delta T = 50\ \mathrm{K}$ ; $Q = 0.50 \times 385 \times 50$ (1)
$Q = 9625\ \mathrm{J} \approx 9.6\ \mathrm{kJ}$ (1)

Q4. (4 marks)

Melting: $Q_1 = mL_f = 0.200 \times 3.34\times10^5 = 6.68\times10^4\ \mathrm{J}$ (2)
Heating: $Q_2 = mc\Delta T = 0.200 \times 4186 \times 30 = 25116\ \mathrm{J} \approx 2.51\times10^4\ \mathrm{J}$ (2)

Q5. (4 marks)

$\Delta L = \alpha L_0 \Delta T = 1.2\times10^{-5}\times1.00\times80$ (1)
$\Delta L = 9.6\times10^{-4}\ \mathrm{m} = 0.96\ \mathrm{mm}$ (2)
Volumetric coefficient $\beta = 3\alpha = 3.6\times10^{-5}\ \mathrm{K^{-1}}$ (1)

Q6. (4 marks)

(a) $PV = nRT$ (1)
(b) $P = \dfrac{nRT}{V} = \dfrac{2.0\times8.314\times300}{0.050}$ (2)
$P = 99768\ \mathrm{Pa} \approx 9.98\times10^4\ \mathrm{Pa}$ (1)

Q7. (4 marks)

$W = \int_{V_1}^{V_2} P\,dV$ (1)
Isothermal: $P = \dfrac{nRT}{V}$ (from $PV=nRT$ , $T$ const) (1)
$W = nRT\int_{V_1}^{V_2}\dfrac{dV}{V}$ (1)
$W = nRT\ln\!\left(\dfrac{V_2}{V_1}\right)$ (with $n=1$ : $W = RT\ln(V_2/V_1)$ ) (1)

Q8. (4 marks)

(a) $v_{rms} = \sqrt{\dfrac{3RT}{M}}$ (1)
(b) $v_{rms} = \sqrt{\dfrac{3\times8.314\times300}{0.032}}$ (2)
$= \sqrt{233831} \approx 483\ \mathrm{m/s}$ (1)

Q9. (4 marks)

(a) $\eta = 1 - \dfrac{T_C}{T_H} = 1 - \dfrac{300}{500} = 0.40$ (40%) (2+1)
(b) Efficiency increases as $T_H$ rises or $T_C$ falls; maximum when $T_C\to0$ . (1)

Q10. (3 marks)

$dS = \dfrac{dQ_{rev}}{T}$ (1)
$\Delta S = \dfrac{500}{250}$ (1)
$\Delta S = 2.0\ \mathrm{J/K}$ (1)

[
{"claim":"Q3 heat for copper block = 9625 J","code":"result = (0.50*385*50 == 9625)"},
{"claim":"Q4 total heat = 66800 + 25116 J","code":"Q1=0.200*3.34e5; Q2=0.200*4186*30; result = (abs(Q1-66800)<1e-6) and (abs(Q2-25116)<1e-6)"},
{"claim":"Q6 pressure approx 99768 Pa","code":"P=2.0*8.314*300/0.050; result = abs(P-99768)<1"},
{"claim":"Q8 v_rms oxygen approx 483 m/s","code":"v=sqrt(3*8.314*300/0.032); result = abs(float(v)-483)<1"},
{"claim":"Q9 Carnot efficiency = 0.40","code":"result = (1 - Rational(300,500) == Rational(2,5))"},
{"claim":"Q10 entropy change = 2.0 J/K","code":"result = (Rational(500,250) == 2)"}
]