Heat and Thermodynamics

Heat and thermodynamics carry about 4 questions (our estimate). Calorimetry (mixing hot and cold substances, ice), thermal expansion, radiation (T^4), the first law on a gas process, kinetic-theory averages and the efficiency of a Carnot or Otto engine are the standard shapes. Numerical questions are mostly one formula with careful unit handling (kelvin, not Celsius, in T^4 and in engine efficiency).

2.1 Temperature and quantity of heat

Thermal equilibrium is the content of the zeroth law: bodies in equilibrium with a third are in equilibrium with each other, which defines temperature. Scales: \frac{C}{5}=\frac{F-32}{9}=\frac{K-273}{5} (the same reading -40 on Celsius and Fahrenheit). Triple point of water: 273.16 K, 0.01 °C, 611.7 Pa (the fixed point of the Kelvin scale).

Heat is energy in transit. Q=mc\Delta T; heat capacity C=mc; Q=mL (L_f=336 kJ kg⁻¹ for ice, L_v=2256 kJ kg⁻¹ for steam; c_{water}=4200 J kg⁻¹ K⁻¹, c_{ice}\approx2100). Method of mixtures: heat lost = heat gained; check first whether all the ice melts. Measuring c by mixtures; L by the electrical method VIt=mL (Joule's constant J=4.18 J cal⁻¹).

Newton's law of cooling: rate of cooling \propto(T-T_s) for a small excess; T=T_s+(T_0-T_s)e^{-kt}; for the average values \frac{T_1-T_2}{t}=K\Big(\frac{T_1+T_2}{2}-T_s\Big).

2.2 Thermal expansion

Linear \Delta L=L\alpha\Delta T; area \beta=2\alpha; volume \gamma=3\alpha (\alpha:\beta:\gamma=1:2:3). Liquids have real expansion \gamma_r=\gamma_a+\gamma_{vessel}. Water contracts from 0 to 4 °C, so its density is maximum at 4 °C (ice floats; lakes freeze from the top). If an expansion is prevented, the thermal stress is Y\alpha\Delta T. A pendulum clock runs slow in summer (l increases). The Bimetallic strip bends towards the metal of lower \alpha. Gas expansion: PV=nRT; Charles (V\propto T at constant P), Boyle (PV const at constant T), Gay-Lussac (P\propto T).

2.3 Transfer of heat

Conduction: \dfrac{Q}{t}=\dfrac{kA(T_1-T_2)}{L} (k in W m⁻¹ K⁻¹). Rods of different material in series: R=\sum\frac{L}{kA} (thermal resistance); in parallel the conductances add. Interface temperature of two slabs in series: T=\frac{k_1T_1/L_1+k_2T_2/L_2}{k_1/L_1+k_2/L_2}. Convection moves heat with a fluid (sea breeze, boiling). Radiation needs no medium.

A black body absorbs all radiation. Stefan–Boltzmann: E=\sigma T^4, \sigma=5.67\times10^{-8} W m⁻² K⁻⁴; net power from a body at T in surroundings T_s: P=e\sigma A(T^4-T_s^4). Doubling T multiplies the radiated power by 16. Wien: \lambda_mT=2.9\times10^{-3} m K. Kirchhoff: a good absorber is a good emitter. Thermos flask: silvering reduces radiation, vacuum stops conduction and convection.

2.4 Thermal properties of matter and kinetic theory

Ideal gas: PV=nRT=NkT, R=8.314 J mol⁻¹ K⁻¹, k=1.38\times10^{-23} J K⁻¹, N_A=6.02\times10^{23}. Kinetic theory: P=\frac{1}{3}\rho\overline{v^2}; average translational KE of a molecule \frac{3}{2}kT (independent of the gas); v_{rms}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3P}{\rho}}, mean v_{av}=\sqrt{\frac{8RT}{\pi M}}, most probable \sqrt{\frac{2RT}{M}} (v_{mp}:v_{av}:v_{rms}=\sqrt{2}:\sqrt{8/\pi}:\sqrt{3}). v_{rms}\propto\sqrt{T/M}: hydrogen is 4 times faster than oxygen at the same temperature. Mean free path \lambda=\frac{1}{\sqrt{2}\pi d^2n}.

Degrees of freedom f (3 monatomic, 5 diatomic at ordinary temperature, 6 non-linear polyatomic); energy per molecule \frac{f}{2}kT, so U=\frac{f}{2}nRT, C_v=\frac{f}{2}R, C_p=C_v+R (Mayer), \gamma=\frac{C_p}{C_v}=1+\frac{2}{f} (5/3, 7/5, 4/3). Solids (Dulong–Petit): molar heat capacity \approx3R=25 J mol⁻¹ K⁻¹. Vapour pressure, humidity, dew point, boiling at lower pressure on mountains (water boils below 100 °C at the summit of Everest).

2.5 Laws of thermodynamics

First law: \Delta U=Q-W (heat added to the system, work done by it); W=\int P\,dV (area under the P–V curve).

Adiabatic: PV^\gamma= const, TV^{\gamma-1}= const, T^\gamma P^{1-\gamma}= const. Compressing a gas adiabatically 8-fold (monatomic) raises P by 8^{5/3}=32. A cyclic process returns \Delta U=0; the work is the enclosed area.

Second law (Kelvin–Planck: no engine converts heat wholly into work; Clausius: heat does not flow spontaneously from cold to hot). Heat engine: \eta=\frac{W}{Q_1}=1-\frac{Q_2}{Q_1}. Carnot cycle (two isotherms, two adiabats): \eta=1-\frac{T_2}{T_1} (kelvin), the maximum possible between two temperatures, reversible. Otto (petrol): \eta=1-\frac{1}{r^{\gamma-1}} (r=8, \gamma=1.4: 56.5 %). Diesel: higher compression ratio, injection at constant pressure. Refrigerator: COP =\frac{Q_2}{W}=\frac{T_2}{T_1-T_2}. Entropy \Delta S=\frac{Q}{T}; the entropy of an isolated system never decreases; melting at T: \Delta S=\frac{mL}{T}.

Formula and fact sheet

Practice questions

Set A — Recall (10 questions)

1. The temperature of the triple point of water isa273.15 Kb273.16 Kc0 Kd373.15 K
2. The heat required to raise 2 kg of water by 10 °C (c=4200 J kg⁻¹ K⁻¹) isa8400 Jb42000 Jc84000 Jd840 J
3. If the absolute temperature of a black body is doubled, the radiated power becomesa2 timesb4 timesc8 timesd16 times
4. For a solid, the coefficients of linear and volume expansion are related bya\gamma=3\alphab\gamma=2\alphac\gamma=\alpha/3d\gamma=\alpha
5. In an isothermal process of an ideal gasaQ=0bW=0c\Delta U=0d\Delta P=0
6. For an ideal gas C_p-C_v equalsaRbR/2c2Rd0
7. The efficiency of a Carnot engine working between 600 K and 300 K isa25 %b50 %c75 %d100 %
8. The mean translational kinetic energy of a gas molecule at temperature T isa\frac{1}{2}kTbkTc\frac{3}{2}kTd3kT
9. In a spontaneous process the entropy of the universeaincreasesbdecreasescis zerodstays constant
10. The SI unit of thermal conductivity isaW m⁻²bJ K⁻¹cW KdW m⁻¹ K⁻¹

Set B — Understanding (7 questions)

1. 100 g of ice at 0 °C is added to 0.5 kg of water at 40 °C (L_f=336 kJ kg⁻¹, c=4.2 kJ kg⁻¹ K⁻¹). The final temperature isa10 °Cb25 °Cc20 °Cd30 °C
2. A 1 m rod expands by 1.2 mm when heated through 100 °C. Its coefficient of linear expansion isa1.2\times10^{-6} K⁻¹b1.2\times10^{-5} K⁻¹c1.2\times10^{-4} K⁻¹d1.2\times10^{-3} K⁻¹
3. A slab of k=0.8 W m⁻¹ K⁻¹, area 2 m² and thickness 0.1 m has 25 K across it. The heat conducted per second isa200 Wb800 Wc400 Wd40 W
4. A monatomic ideal gas (\gamma=\frac{5}{3}) is compressed adiabatically to \frac{1}{8} of its volume. Its pressure rises by a factora8b32c16d40
5. One mole of an ideal gas at 300 K expands isothermally to twice its volume. The work done is nearly (R=8.314)a1.73 kJb2.49 kJc0.86 kJd3.46 kJ
6. A refrigerator keeps its inside at 270 K in a room at 300 K. Its maximum coefficient of performance isa9b0.9c10d1/9
7. The ratio of v_{rms} of hydrogen to oxygen at the same temperature isa16b2c8d4

Set C — Application (5 questions)

1. A Carnot engine between 500 K and 300 K absorbs 1000 J per cycle. The work done per cycle isa600 Jb400 Jc200 Jd500 J
2. A 0.5 kg metal block at 100 °C is dropped into 1 kg of water at 20 °C and the final temperature is 25 °C (c_w=4200). The specific heat of the metal isa560 J kg⁻¹ K⁻¹b420c840d280
3. An Otto engine has compression ratio 8 and \gamma=1.4. Its efficiency is abouta43.5 %b65 %c87.5 %d56.5 %
4. A steel rail fixed at both ends is heated by 20 °C (Y=2\times10^{11} Pa, \alpha=12\times10^{-6} K⁻¹). The thermal stress isa4.8\times10^7 Pab2.4\times10^7 Pac4.8\times10^6 Pad9.6\times10^7 Pa
5. The internal energy of 2 mol of a monatomic ideal gas at 300 K is nearlya5.0 kJb12.5 kJc7.5 kJd3.7 kJ