Heat and thermodynamics: Class 11 Physics

Key points

Temperature measures how hot or cold a body is. It decides the direction of heat flow. Heat is the energy that moves from a hotter to a colder body because of the temperature difference. It is measured in joules. Temperature is measured in kelvin (K) in the SI system, and T(K) = t(°C) + 273. The zeroth law of thermodynamics says that if two bodies are each in thermal equilibrium with a third, they are in equilibrium with each other. This is the basis of the thermometer.

The heat needed to change the temperature of a mass m by ΔT is Q = mcΔT, where c is the specific heat capacity in J kg⁻¹ K⁻¹. For water c = 4200 J kg⁻¹ K⁻¹ (about), which is high, so water is a good coolant and sea breezes form. The heat capacity of a body is C = mc, in J/K. When hot and cold bodies are mixed, with no heat loss, heat lost by the hot body equals heat gained by the cold body.

During a change of state, the temperature stays constant while heat is supplied. The heat needed is Q = mL, where L is the latent heat. The latent heat of fusion (melting) of ice is 3.34 × 10⁵ J/kg and the latent heat of vaporisation of water is about 2.26 × 10⁶ J/kg. The energy goes into breaking the bonds between molecules, not into raising the temperature of the molecules.

Thermal expansion: most solids expand when heated. A rod of length L₀ expands by ΔL = αL₀ΔT, where α is the coefficient of linear expansion, in K⁻¹. For volume expansion, ΔV = βV₀ΔT, and for solids β is about 3α. Water behaves unusually: it contracts from 0°C to 4°C and then expands, so its density is greatest at 4°C. This keeps the lower layers of lakes liquid in winter, which protects fish.

The ideal gas and kinetic theory:

• The equation of state is PV = nRT, where n is the number of moles, T is the temperature in kelvin and R = 8.31 J mol⁻¹ K⁻¹. It combines Boyle's law (PV constant at constant T), Charles' law (V/T constant at constant P) and the pressure law (P/T constant at constant V).
• An ideal gas has molecules that are point-like, do not attract each other, and collide elastically.
• The kinetic theory gives the pressure as P = ⅓ρ⟨v²⟩, where ⟨v²⟩ is the mean square speed.
• The average translational kinetic energy per molecule is ½m⟨v²⟩ = (3/2)kT, where k = 1.38 × 10⁻²³ J/K is the Boltzmann constant. So the temperature is a measure of the average kinetic energy.
• The internal energy of an ideal gas depends only on its temperature.

The first law of thermodynamics is the law of conservation of energy applied to heat. The heat Q supplied to a system is used partly to increase its internal energy ΔU and partly to do external work W: Q = ΔU + W. Here W is positive when the gas expands and does work on its surroundings. For a gas at constant pressure, W = PΔV. The work done is the area under the pressure volume (PV) graph.

Thermodynamic processes of an ideal gas:

• Isothermal: temperature is constant, so ΔU = 0 and Q = W. It is slow, and PV is constant.
• Adiabatic: no heat enters or leaves, Q = 0, so W = −ΔU. An expanding gas cools. PVᵞ is constant, where γ = Cp/Cv.
• Isobaric: pressure is constant, W = PΔV.
• Isochoric (isovolumetric): volume is constant, W = 0 and Q = ΔU.

There are two molar specific heats for a gas: Cp (at constant pressure) and Cv (at constant volume). Cp is greater than Cv, because at constant pressure the heat must also supply the work of expansion. They are related by Cp − Cv = R (Mayer's relation).

The second law of thermodynamics has two common statements. Kelvin and Planck: it is impossible to take heat from a single reservoir and change all of it into work in a cyclic process. Clausius: heat cannot flow from a colder to a hotter body by itself, without work being done. These statements show that natural processes have a direction, and no heat engine can have 100% efficiency.

A heat engine takes heat Q₁ from a hot reservoir at T₁, converts part of it into work W, and rejects the heat Q₂ to a cold reservoir at T₂. Its thermal efficiency is η = W/Q₁ = (Q₁ − Q₂)/Q₁ = 1 − Q₂/Q₁. The Carnot engine is an ideal reversible engine working between two temperatures, with two isothermal and two adiabatic steps. It has the greatest possible efficiency, η = 1 − T₂/T₁, where the temperatures are in kelvin. To raise the efficiency, the hot reservoir should be hotter or the cold reservoir cooler. A refrigerator is a heat engine working in reverse: it takes heat from a cold space and releases heat to the room, using work from a motor.

Entropy is a measure of the disorder of a system. In an isolated system the total entropy never decreases. It increases in all natural (irreversible) processes, such as heat flowing from hot to cold, gases mixing and ice melting in a warm room. This gives another form of the second law, and explains why time seems to move in one direction.

Study tip: Write the sign convention for the first law before you start. For problems with a change of state, split the heating into steps (warming, melting, warming again) and add the heats. Always give temperatures in kelvin in the gas laws and efficiency formulas.

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Common mistakes

• Using Celsius temperatures in the gas equation or the Carnot efficiency. Always convert to kelvin.
• Mixing up the sign of W in the first law. Work done by the gas is positive in Q = ΔU + W.
• Thinking that heat and temperature are the same. Heat is energy, temperature is a measure of hotness.
• Forgetting the latent heat when ice at 0°C is changed to water, and including only mcΔT.
• Writing the efficiency of an engine as more than 100%, or using Q₂/Q₁ instead of 1 − Q₂/Q₁.

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