Thermal Physics

Internal energy, gas laws, and kinetic theory of ideal gases.

Spec Points Covered
  • I can explain the concept of thermal equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹..The state in which two objects in thermal contact have no net heat transfer between them because they are at the same temperature. and how it relates to temperature.
  • I can convert between Celsius and Kelvin temperature scales and explain the significance of absolute zeroThe lowest possible temperature (0 K = -273.15 °C), at which particles have minimum possible kinetic energyThe capacity to do work. Measured in joules (J)...
  • I can define internal energyThe capacity to do work. Measured in joules (J).The sum of the randomly distributed kinetic and potential energies of all the molecules in a system. as the sum of the random kinetic and potential energies of molecules.
  • I can use the equation Q = $mc\Delta\theta$ to calculate energyThe capacity to do work. Measured in joules (J). transferred during temperature changes.
  • I can use the equation $Q = mL$ to calculate energy transferred during changes of state.
  • I can state the assumptions of the ideal gasA theoretical gas composed of many randomly moving point particles that do not interact except during brief elastic collisions. model.
  • I can apply the gas laws (Boyle's, Charles's, and pressure lawFor a fixed mass of gas at constant volume, pressureForce per unit area. Measured in pascals (Pa), where 1 Pa = 1 N m⁻². is directly proportional to absolute temperature.) to solve problems.
  • I can use $pV = nRT$ and $pV = NkT$ to solve ideal gasA theoretical gas composed of many randomly moving point particles that do not interact except during brief elastic collisions. problems.
  • I can derive and use the relationship between molecular kinetic energyThe energy an object possesses due to its motion. and temperature: $\frac{1}{2}$m\langle \(c^{2}\) \rangle = $\frac{3}{2}$kT.
  • I can calculate root-mean-square speeds of gas molecules and explain why different gases at the same temperature have different rms speeds.
  • I can relate the Boltzmann constantThe gas constant per molecule. Relates the average kinetic energyThe energy an object possesses due to its motion. of gas molecules to temperature. k = 1.38 x 10⁻²³ J K⁻¹. to the gas constant and Avogadro's number: $k = R/N_A$.
  • I can use pV = $\frac{1}{3}$Nm\langle \(c^{2}\) \rangle to link macroscopic gas behaviour to molecular motion.
Notes
01 Thermal equilibrium Thermal equilibrium → 02 Internal energy Internal energy → 03 Specific heat capacity (c) Specific heat capacity (c) → 04 Ideal gas Ideal gas → 05 Boyle's law Boyle's law → 06 The two forms are related $pV = nRT$ → 07 The mean square speed \langle \(c^{2}\) \rangle is th... $pV = \frac{1}{3}Nm\langle c^2 \rangle$ → 08 From $\frac{1}{2}$m\langle \(c^{2}\) \rangle = $\frac... → 09 The Boltzmann constant k is the gas constant per molecule, just as R is the gas constant per mole $k = \frac{R}{N_A}$ →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Energy for temperature change
$$Q = mc\Delta\theta$$
  • Where:
    • $Q$ = J
    • $m$ = kg
    • $c$ = J kg^{-1} \(K^{-1}\)
    • $\Delta\theta$ = K or \degree C
Use when a substance changes temperature without changing state.
Energy for change of state
$$Q = mL$$
  • Where:
    • $Q$ = J
    • $m$ = kg
    • $L$ = J kg^{-1}
Use when a substance changes state at constant temperature.
Ideal gas equation (moles)
$$pV = nRT$$
  • Where:
    • $p$ = Pa
    • $V$ = \(m^{3}\)
    • $n$ = mol
    • $R$ = 8.31 J mol^{-1} \(K^{-1}\)
    • $T$ = K
Use when working with moles of gas. T must be in Kelvin.
Ideal gas equation (molecules)
$$pV = NkT$$
  • Where:
    • $p$ = Pa
    • $V$ = \(m^{3}\)
    • $N$ = number of molecules
    • $k$ = 1.38 \times \(10^{-23}\) J \(K^{-1}\)
    • $T$ = K
Use when working with number of molecules. T must be in Kelvin.
Mean kinetic energy of a molecule
$$\frac{1}{2}m\langle c^2 \rangle = \frac{3}{2}kT$$
  • Where:
    • $m$ = kg (mass of one molecule)
    • $\langle c^2 \rangle$ = \(m^{2}\) \(s^{-2}\)
    • $k$ = J \(K^{-1}\)
    • $T$ = K
Links molecular KE to temperature. Average KE is the same for all ideal gas molecules at the same temperature regardless of mass.
Kinetic theory equation
$$pV = \frac{1}{3}Nm\langle c^2 \rangle$$
  • Where:
    • $p$ = Pa
    • $V$ = \(m^{3}\)
    • $N$ = number of molecules
    • $m$ = kg
    • $\langle c^2 \rangle$ = \(m^{2}\) \(s^{-2}\)
Derived from kinetic theory. Links macroscopic gas properties (p, V) to molecular motion. The 1/3 comes from averaging over 3 dimensions.
Kelvin-Celsius conversion
$$T(K) = T(\degree C) + 273$$
  • Where:
    • $T$ = K or \degree C
Must memorise. Always convert to Kelvin before using gas law or kinetic theory equations.
Boltzmann constant
$$k = \frac{R}{N_A}$$
  • Where:
    • $k$ = J \(K^{-1}\)
    • $R$ = J mol^{-1} \(K^{-1}\)
    • $N_A$ = mol^{-1}
Must memorise. Links the molar gas constant to the molecular Boltzmann constant.