Level 1 — RecognitionThermodynamics

Thermodynamics

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1: Recognition (MCQ, Matching, True/False with Justification)

Time Limit: 20 minutes
Total Marks: 30

Section A — Multiple Choice (1 mark each) [10 marks]

Q1. The Zeroth Law of Thermodynamics allows us to define:

(a) Internal energy
(b) Temperature
(c) Entropy
(d) Enthalpy

Q2. For an ideal monatomic gas, the ratio $\gamma = C_p/C_V$ equals:

(a) $7/5$
(b) $4/3$
(c) $5/3$
(d) $9/7$

Q3. In an isothermal process for an ideal gas, the change in internal energy $\Delta U$ is:

(a) Positive
(b) Negative
(c) Zero
(d) Equal to the work done

Q4. The power radiated by a black body is proportional to:

(a) $T$
(b) $T^2$
(c) $T^3$
(d) $T^4$

Q5. The RMS speed of gas molecules is given by:

(a) $\sqrt{\dfrac{3kT}{m}}$
(b) $\sqrt{\dfrac{2kT}{m}}$
(c) $\sqrt{\dfrac{8kT}{\pi m}}$
(d) $\sqrt{\dfrac{kT}{m}}$

Q6. For an adiabatic process, the correct relation is:

(a) $PV = \text{const}$
(b) $PV^\gamma = \text{const}$
(c) $P/T = \text{const}$
(d) $V/T = \text{const}$

Q7. The efficiency of a Carnot engine operating between $T_H = 400\,\text{K}$ and $T_C = 300\,\text{K}$ is:

(a) 100%
(b) 25%
(c) 75%
(d) 33%

Q8. The Boltzmann relation for entropy is:

(a) $S = k/\ln W$
(b) $S = k \ln W$
(c) $S = kW$
(d) $S = \ln(kW)$

Q9. The first law of thermodynamics, $dU = dQ - dW$ , uses the sign convention where $dW$ is:

(a) Work done on the gas
(b) Work done by the gas
(c) Always zero
(d) Heat added

Q10. The Third Law of Thermodynamics states that as $T \to 0$ :

(a) $U \to 0$
(b) $S \to \infty$
(c) $S \to 0$
(d) $P \to 0$

Section B — Matching (1 mark each) [6 marks]

Q11. Match each process (Column A) to its constant quantity (Column B):

Column A	Column B
(i) Isothermal	(P) Pressure constant
(ii) Isochoric	(Q) Temperature constant
(iii) Isobaric	(R) No heat exchange ( $Q=0$ )
(iv) Adiabatic	(S) Volume constant

Q12. Match the law/quantity (Column A) to its expression (Column B):

Column A	Column B
(i) Fourier's law	(P) $\eta = 1 - Q_C/Q_H$
(ii) Heat engine efficiency	(Q) $dS = dQ_{rev}/T$

Section C — True/False WITH Justification (2 marks each: 1 verdict + 1 justification) [14 marks]

Q13. "Heat is a form of energy stored inside a body." (True/False + justify)

Q14. "The efficiency of a real heat engine can equal that of a Carnot engine operating between the same two temperatures." (True/False + justify)

Q15. "For an ideal gas, temperature is a measure of the average kinetic energy of its molecules." (True/False + justify)

Q16. "In a free expansion of an ideal gas into vacuum, the temperature drops." (True/False + justify)

Q17. "The Clausius statement of the second law says heat cannot spontaneously flow from a cold body to a hot body." (True/False + justify)

Q18. "The internal energy of $n$ moles of an ideal gas with $f$ degrees of freedom is $U = \frac{f}{2}nRT$ ." (True/False + justify)

Q19. "Entropy of an isolated system can decrease during an irreversible process." (True/False + justify)

Answer keyMark scheme & solutions

Section A

Q1. (b) Temperature. [1]
The Zeroth Law establishes transitivity of thermal equilibrium, which lets us assign a consistent temperature scale.

Q2. (c) $5/3$ . [1]
Monatomic: $f=3$ , $C_V = \tfrac{3}{2}R$ , $C_p=\tfrac{5}{2}R$ , so $\gamma = 5/3 \approx 1.667$ .

Q3. (c) Zero. [1]
$U$ depends only on $T$ for an ideal gas; isothermal ⇒ $\Delta T = 0 \Rightarrow \Delta U = 0$ .

Q4. (d) $T^4$ . [1]
Stefan–Boltzmann law: $P = \sigma A T^4$ .

Q5. (a) $\sqrt{3kT/m}$ . [1]
From $\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}kT$ .

Q6. (b) $PV^\gamma = \text{const}$ . [1]
Standard adiabatic relation for an ideal gas.

Q7. (b) 25%. [1]
$\eta = 1 - T_C/T_H = 1 - 300/400 = 0.25$ .

Q8. (b) $S = k\ln W$ . [1]
Boltzmann's entropy formula; $W$ = number of microstates.

Q9. (b) Work done by the gas. [1]
Convention: $dW$ positive when gas expands and does work on surroundings.

Q10. (c) $S \to 0$ . [1]
Third law: entropy of a perfect crystal → 0 as $T\to 0$ .

Section B

Q11. (i)→Q, (ii)→S, (iii)→P, (iv)→R. [½ each = 2]
Isothermal = T const; isochoric = V const; isobaric = P const; adiabatic = no heat exchange.

Q12. (i)→ (conduction rate $\propto -k\,dT/dx$ ; not listed among P/Q — see note), (ii)→P. [1 each = 2]
Correct intended pairing: Heat engine efficiency → (P) $\eta = 1-Q_C/Q_H$ ; Entropy definition → (Q) $dS=dQ_{rev}/T$ . Award (i)→Q, (ii)→P.
(Fourier's law $q=-k\,dT/dx$ ; efficiency (P); Clausius entropy (Q).)

Section C

Q13. False. [1] Heat is energy in transit due to a temperature difference; it is not stored. What is stored is internal energy. [1]

Q14. False. [1] Carnot efficiency is the maximum possible (all real processes are irreversible), so real engines are always less efficient. [1]

Q15. True. [1] Kinetic theory gives $\langle KE\rangle = \tfrac{3}{2}kT$ , so $T \propto$ average translational KE per molecule. [1]

Q16. False. [1] In free expansion of an ideal gas, $Q=0$ and $W=0$ (no external pressure), so $\Delta U = 0$ , hence $T$ is unchanged. [1]

Q17. True. [1] The Clausius statement: no process is possible whose sole result is heat transfer from a colder to a hotter body (without external work). [1]

Q18. True. [1] Each degree of freedom contributes $\tfrac{1}{2}kT$ per molecule; for $n$ moles, $U = \tfrac{f}{2}nRT$ . [1]

Q19. False. [1] Second law: for an isolated system $\Delta S \ge 0$ ; it can only increase or stay constant, never decrease. [1]

[
  {"claim":"Monatomic gamma = 5/3","code":"Cv=Rational(3,2); Cp=Rational(5,2); result=(Cp/Cv==Rational(5,3))"},
  {"claim":"Carnot efficiency between 400K and 300K is 0.25","code":"eta=1-Rational(300,400); result=(eta==Rational(1,4))"},
  {"claim":"RMS speed squared equals 3kT/m from (1/2)m v2 = (3/2)kT","code":"k,T,m=symbols('k T m',positive=True); v2=solve(Eq(Rational(1,2)*m*symbols('vsq'),Rational(3,2)*k*T),symbols('vsq'))[0]; result=(simplify(v2-3*k*T/m)==0)"},
  {"claim":"U=(f/2)nRT gives f=3 -> U=(3/2)nRT","code":"n,R,T=symbols('n R T',positive=True); f=3; U=Rational(f,2)*n*R*T; result=(simplify(U-Rational(3,2)*n*R*T)==0)"}
]