Unit 13: Thermal Physics — Long Questions

Consider N molecules, each of mass m, inside a cubical box of side L (volume V = L³). Consider one molecule moving with velocity component v_x perpendicular to a wall.
1. Collision with the wall: the molecule rebounds elastically, so its momentum reverses from mv_x to −mv_x. Change in momentum, Δp = −mv_x − (mv_x) = −2mv_x (magnitude 2mv_x).
2. Time between collisions: the molecule must travel a distance 2L (to the opposite wall and back) before striking the same wall again, so the time between successive collisions is Δt = 2L/v_x.
3. Force from one molecule: by Newton's second law, F = Δp/Δt = 2mv_x/(2L/v_x) = mv_x²/L.
4. Total force from N molecules: summing the contributions of all molecules, F = (m/L)(v_x1² + v_x2² + ... + v_xN²) = (m/L)N<v_x²>.
5. Pressure: P = F/Area = F/L² = (Nm/L³)<v_x²> = (Nm/V)<v_x²>.
6. Since molecular motion is random in three dimensions, <v_x²> = <v_y²> = <v_z²> = (1/3)<v²>, where <v²> is the mean-square speed.
7. Substituting: P = (1/3)(Nm/V)<v²>, or equivalently PV = (1/3)Nm<v²>.
Using density ρ = Nm/V (total mass/volume), this can also be written as P = (1/3)ρ<v²>.

For a large number of molecules, statistical physics (using probability theory) is used instead of tracking individual particles. Atoms and molecules of a system can exist in different discrete energy states/levels E1, E2, E3, ... because of their different speeds. Boltzmann showed that if N1 and N2 are the number of molecules (population) in a lower energy state E1 and a higher energy state E2 respectively, their populations are related by the Boltzmann factor: N2/N1 = e^(−ΔE/kBT), where ΔE = E2 − E1 is the energy difference between the states. This is known as the Boltzmann distribution law. It states that the probability of finding molecules in a particular energy state decreases exponentially as that energy increases (relative to kBT), so more molecules are found in lower energy states than in higher ones. Since number density is directly related to pressure, this law is important in describing the statistical mechanics of systems with a large number of molecules, and it underlies the Maxwell–Boltzmann distribution of molecular speeds.

Neutron stars are formed when a massive star's iron core collapses under gravity during a supernova, compressing matter to extraordinarily high densities — among the densest known objects in the universe after black holes (a teaspoon of neutron-star material can have a mass over 5.5 × 10¹² kg). At such densities, the matter inside behaves like a degenerate gas, and it is the outward degeneracy pressure of this extremely dense matter that balances the enormous inward gravitational force, in accordance with the hydrostatic equilibrium condition (ΔP/Δr = −GM_rρ_r/r²). As long as this internal pressure can support the star against its own gravity, the neutron star remains stable; if the star's mass is too large for the degenerate pressure to support it, the gravitational collapse can continue further, potentially forming a black hole. The collapse that forms the neutron star also compresses and intensifies its magnetic field, since the field lines are squeezed closer together — this gives rise to pulsars, rapidly rotating neutron stars that emit beams of radiation.

From kinetic theory, PV = (1/3)Nm<v²>, where N is the number of molecules and m is the mass of one molecule.
The ideal gas law gives PV = nRT = nNAkBT, and since n = N/NA, this is PV = NkBT.
Equating the two expressions for PV: (1/3)Nm<v²> = NkBT, so m<v²> = 3kBT, or (1/2)m<v²> = (3/2)kBT.
The quantity (1/2)m<v²> (more precisely (1/2)mv(rms)²) is the average translational kinetic energy of a molecule, denoted <K.E.>. Therefore:
<K.E.> = (3/2)kBT.
Since this equation shows the average translational kinetic energy of the gas molecules is a simple constant ((3/2)kB) multiplied by the absolute temperature T, the temperature of a gas is a direct measure of the average translational kinetic energy of its molecules — it depends on nothing else (not on the mass of the molecules, the pressure, or the volume).

Formation all stars form from contracting clouds of gas and dust (nebulae). As the cloud contracts under gravity, its density and temperature rise (since K.E. ∝ T), forming a proto-star.
Main sequence (stable star): a stable star is reached when the inward gravitational force is exactly balanced by the outward internal (thermal) pressure, described by the hydrostatic equilibrium equation ΔP/Δr = −GM_rρ_r/r². As core temperature rises enough, nuclear fusion begins (hydrogen fusing into helium), releasing energy that sustains this equilibrium for most of the star's life.
Imbalance and evolution: (i) if gravitational force exceeds internal pressure, the star contracts, its core heats further, and nuclear reactions can progress to heavier elements; (ii) as fusion products and rising temperature change the pressure balance, the star can expand significantly, e.g. becoming a red giant, where stars with at least about half the Sun's mass begin helium fusion to form carbon and oxygen; in more massive stars fusion continues to successively heavier elements up to iron.
Final fate (depends on mass): once nuclear fuel is exhausted, the star can no longer support itself against gravity.

• Low and intermediate mass stars (such as the Sun): the core contracts into a dense white dwarf while outer layers are expelled as a planetary nebula.
• Very massive stars (roughly ten times the Sun's mass): the inert iron core collapses catastrophically in a supernova explosion, forming a neutron star (supported by degenerate neutron pressure) or, if the remnant mass is too great for that pressure to support it, a black hole. Red dwarf stars evolve far more slowly, due to low mass and efficient fuel use; none have yet completed their life cycle since the universe is not old enough.