Thermodynamics and Kinetic Theory - heat, gases and the first law
Thermodynamics and Kinetic Theory
This domain connects calorimetry (heating, melting, boiling), the gas laws, the kinetic model of a gas, and the first law of thermodynamics. Questions typically ask you to calculate heat quantities, use gas-law ratios, interpret molecular speeds, or apply ΔU = Q − W to an isothermal, adiabatic, isobaric or isochoric process.
Core rules
- Expansion of a solid: ΔL = L₀αΔθ; area expansivity is 2α and volume expansivity 3α. Liquids need the apparent expansion of the containing vessel.
- Q = mcΔθ for a temperature change, Q = mL at a change of state. In mixtures, heat lost by the hot body = heat gained by the cold body, assuming no loss to the surroundings.
- Gas laws, always with T in kelvin: Boyle pV = constant; Charles V/T = constant at constant p; pressure law p/T = constant at constant V.
- Ideal gas: pV = nRT with R = 8.31 J mol⁻¹ K⁻¹, or pV = NkT with k = 1.38 × 10⁻²³ J K⁻¹.
- Kinetic theory: p = ⅓ρc̄², and mean translational kinetic energy per molecule = (3/2)kT, so rms speed ∝ √T.
- First law: ΔU = Q − W, where W is work done by the gas. W = pΔV at constant pressure; isothermal ΔU = 0; adiabatic Q = 0; isochoric W = 0.
Worked example
0.50 kg of ice at 0 °C is melted and the water warmed to 20 °C. Take L = 3.34 × 10⁵ J kg⁻¹ and c = 4.2 × 10³ J kg⁻¹ K⁻¹.
Melting: Q₁ = 0.50 × 3.34 × 10⁵ = 1.67 × 10⁵ J. Warming: Q₂ = 0.50 × 4.2 × 10³ × 20 = 4.2 × 10⁴ J. Total = 2.09 × 10⁵ J.
Traps
Putting °C into a gas-law ratio; omitting the latent-heat stage; confusing specific heat capacity with molar heat capacity; using pV = nRT with mass instead of moles; reversing the sign of W in the first law because the gas is compressed rather than expanded; treating rms speed as an arithmetic mean.