TheIBAxolotl
The particle model and temperature
Matter is made of particles in constant random motion. In solids they vibrate about fixed positions, in liquids they move past one another while staying close, and in gases they are far apart and move freely. Density \(\rho = \frac{m}{V}\) is high for solids and liquids and far lower for gases because of this spacing.
Temperature measures the average random kinetic energy of the particles. The Kelvin (absolute) scale starts at absolute zero, where this kinetic energy is at its minimum:
\[T(\text{K}) = \theta(^\circ\text{C}) + 273\]
A temperature difference is the same size in K and in °C. For a simple model of a gas, the average kinetic energy per particle is \(\bar{E}_k = \frac{3}{2}k_B T\).
Internal energy
The internal energy of a substance is the total of the random kinetic energies and the intermolecular potential energies of all its particles. Heating can raise either part:
- Raising the temperature increases the random kinetic energy.
- Changing phase (melting, boiling) at constant temperature increases the potential energy as particles are pulled further apart against attractive forces.
Specific heat capacity and latent heat
The energy needed to change the temperature of a mass \(m\) by \(\Delta T\) is
\[Q = mc\Delta T\]
where \(c\) is the specific heat capacity. Water has an unusually large value, about \(4200\ \text{J kg}^{-1}\ \text{K}^{-1}\).
During a phase change the temperature stays constant, and
\[Q = mL\]
where \(L\) is the specific latent heat (fusion for melting, vaporization for boiling). A heating curve shows sloping parts where \(Q = mc\Delta T\) applies and flat parts where \(Q = mL\) applies. In mixing problems, energy lost by the hot body equals energy gained by the cold body, provided no energy escapes.
Conduction, convection and radiation
- Conduction: energy passes through a material by particle collisions and, in metals, by free electrons. The rate through a slab is \(\frac{\Delta Q}{\Delta t} = kA\frac{\Delta T}{\Delta x}\), where \(k\) is thermal conductivity.
- Convection: warmer fluid expands, becomes less dense and rises while cooler fluid sinks, setting up a current. It needs a fluid.
- Radiation: all bodies emit electromagnetic waves (mainly infrared at everyday temperatures). It needs no medium, so it is how the Sun's energy crosses space.
Black-body radiation
A black body absorbs all radiation falling on it and is the most efficient possible emitter. Its emission spectrum depends only on temperature:
- Stefan–Boltzmann law: total power emitted \(L = \sigma A T^4\). Doubling \(T\) multiplies power by 16.
- Wien's displacement law: peak wavelength \(\lambda_{\max} T = 2.9 \times 10^{-3}\ \text{m K}\). Hotter bodies peak at shorter wavelengths, so hot stars look blue and cooler stars red.
Luminosity and apparent brightness
A star's luminosity \(L\) is the total power it radiates. For a spherical star, \(L = \sigma 4\pi R^2 T^4\). Far away, this power spreads over a sphere of radius \(d\), so the apparent brightness received per square metre is
\[b = \frac{L}{4\pi d^2}\]
Two stars of equal luminosity can look very different in brightness if their distances differ.
Worked example 1: heating water
How much energy is needed to warm \(0.50\ \text{kg}\) of water from \(20\ ^\circ\text{C}\) to \(80\ ^\circ\text{C}\)? Take \(c = 4200\ \text{J kg}^{-1}\ \text{K}^{-1}\).
- \(\Delta T = 60\ \text{K}\)
- \(Q = mc\Delta T = 0.50 \times 4200 \times 60 = 1.3 \times 10^5\ \text{J}\) (2 s.f.)
Worked example 2: a star's temperature
A star's spectrum peaks at \(480\ \text{nm}\). Estimate its surface temperature.
- \(T = \frac{2.9 \times 10^{-3}}{480 \times 10^{-9}} = 6.0 \times 10^3\ \text{K}\) (2 s.f.)
Common mistakes
- Using °C instead of K in the Stefan–Boltzmann law or Wien's law.
- Saying temperature rises during melting; it stays constant while potential energy rises.
- Confusing luminosity (total power, W) with apparent brightness (power per area, W m\(^{-2}\)).
- Forgetting to convert nm to m in Wien's law.
- In mixing problems, forgetting that melted ice must also warm up after it melts.
Exam tips
- On heating curves, label which stage uses \(mc\Delta T\) and which uses \(mL\).
- Ratio questions (\(T\) doubles, \(R\) halves) are fastest with proportionality, not full calculation.
- In "explain" answers on phase change, mention both kinetic and potential energy.
- State the assumption that no energy is lost to the surroundings in mixing problems.
- Keep powers of ten tidy: stellar radii and distances make it easy to slip by \(10^3\).