Thermal physics

Revision notes

Temperature is linked to the microscopic energy distribution of particles. In an ideal gas, particles are treated as point-like, moving randomly with negligible intermolecular forces except during elastic collisions. For a fixed amount of ideal gas, pV = nRT. Gas pressure arises from momentum changes when particles collide with container walls.

Temperature, internal energy and thermal equilibrium

Temperature is associated with the distribution of microscopic particle energies, while internal energy is the total microscopic kinetic and potential energy of a system. Two objects in thermal equilibrium have the same temperature and no net thermal-energy transfer between them.

Absolute temperature is measured in kelvin. Temperature differences in kelvin and degrees Celsius have the same numerical size, but equations involving ratios or ideal gases require absolute temperature.

Specific heat capacity

For heating without a change of state, Q = mcΔT. Specific heat capacity is the energy required per kilogram per kelvin of temperature rise. In calorimetry, energy losses to the surroundings or the apparatus can cause systematic differences between the ideal model and measured value.

The kinetic model

An ideal gas is modelled as a large number of particles moving randomly with negligible volume and negligible intermolecular forces except during elastic collisions. Pressure arises from momentum changes when particles collide with container walls.

Increasing temperature increases mean kinetic energy. At fixed volume this increases the rate and magnitude of momentum transfer to the walls, increasing pressure.

Ideal-gas equation

For n moles of ideal gas, pV = nRT. For a fixed mass of gas, relationships between pressure, volume and absolute temperature follow from this equation. Always convert temperature to kelvin and volume to cubic metres when using SI units.

The kinetic theory equation pV = ⅓Nm(\(c_{\text{rms}}\))² connects the two models. Comparing it with pV = NkT gives the mean kinetic energy of a molecule, ½m(\(c_{\text{rms}}\))² = \(\dfrac{3}{2}kT\).

Internal energy of gases

For an ideal monatomic gas, internal energy depends only on temperature in the school-level model because intermolecular potential energy is neglected. Work done on or by a gas and heating can therefore change its internal energy.

Exam technique

Use kelvin in gas-law calculations. Distinguish temperature from internal energy. For state changes, explain where transferred energy goes when temperature is constant. In practical calorimetry, identify heat-loss mechanisms and their effect on the calculated quantity.

Kinetic theory and the Boltzmann constant

The ideal gas equation can be written for molecules as pV = NkT, where N is the number of molecules and k = \(\dfrac{R}{N_{A}}\) = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant. One mole contains \(N_{A}\) = 6.02 × 10²³ particles, and the number of moles is n = mass/molar mass.

Kinetic theory models pressure as the result of molecular collisions and gives pV = ⅓Nm(\(c_{\text{rms}}\))², where \(c_{\text{rms}}\) is the root-mean-square speed. Combining this with pV = NkT gives the mean kinetic energy of one molecule: ½m(\(c_{\text{rms}}\))² = \(\dfrac{3}{2}kT\). Mean molecular kinetic energy is therefore proportional to absolute temperature.

Gas laws and absolute zero

For a fixed mass of gas: at constant temperature pV is constant (Boyle's law); at constant pressure V/T is constant (Charles's law); at constant volume \(\dfrac{p}{T}\) is constant. Extrapolating a graph of pressure or volume against temperature to zero gives absolute zero, 0 K = −273.15 °C.

The first law of thermodynamics

The first law is energy conservation for a gas: ΔU = Q − W, where ΔU is the increase in internal energy, Q the energy supplied by heating and W the work done by the gas. When a gas expands at constant pressure p by ΔV, it does work W = pΔV.

Compressing a gas quickly does work on it with little time for heat transfer (Q ≈ 0), so its internal energy and temperature rise. For an ideal monatomic gas, U = \(\dfrac{3}{2}nRT\), so a change in internal energy shows up as a change in temperature.