Thermal & Statistical Physics I — formula sheet

The Ideal Gas Law from Kinetic Theory

The pressure a gas exerts is the accumulated recoil of countless molecular impacts: PV equals two-thirds of the total translational kinetic energy of the gas. Temperature is not a substance but a measure of that random kinetic energy — heating a gas literally means speeding up its molecules. The Boltzmann constant k_B = 1.381×10⁻²³ J/K is the conversion factor between the kelvin scale and joules of molecular energy. Writing N = nN_A recovers the chemist's form PV = nRT with R = N_Ak_B.

Carnot's Theorem and Maximum Engine Efficiency

Equality holds if and only if the cycle is reversible. Nothing about the working substance appears — steam, air, or a single atom, the bound is identical. It also shows that η = 1 requires T_C = 0, which is the Kelvin statement of the third law's practical consequence.

The Clausius Inequality and Entropy as a State Function

Weighting each heat exchange by the reciprocal of the temperature at which it happens produces a quantity that returns to zero over any reversible cycle. That closure is what promotes S to a state variable: its change between two states is fixed regardless of path, even a wildly irreversible one, because you may compute it along any convenient reversible substitute. For real processes the inequality records the shortfall — entropy is generated, never destroyed.

Thermodynamic Potentials via Legendre Transforms

Each free energy is the internal energy with a "stored" contribution removed so that an intensive variable you can clamp becomes independent. Subtracting TS removes the reversible heat reservoir bookkeeping and hands control to temperature; adding PV removes the atmospheric flow work and hands control to pressure. The sign of each differential coefficient tells you the conjugate variable: e.g. −(∂G/∂T)_P = S and (∂G/∂P)_T = V.

Equilibrium as Free-Energy Minimization

The one master inequality dA ≤ 0 contains every equilibrium rule. Whichever pair of variables the environment clamps, the availability collapses onto the matching potential — S (max), U, F, H, or G — and that potential is driven to a minimum (or S to a maximum). Equilibrium is not a special force balance; it is the exhaustion of the system's capacity to lower its free energy.

The Maxwell Relations

Stability, Convexity, and Positive Response Functions

Stability against fluctuations is two independent statements. Thermal stability (C_V > 0): if a subregion spontaneously absorbs a little heat its temperature rises above its surroundings, and heat then flows back out — the fluctuation is self-correcting. Mechanical stability (κ_T > 0): if a subregion spontaneously compresses, its pressure rises above its surroundings, and it re-expands. A negative C_V or κ_T would make the fluctuation grow without bound — the homogeneous state is then not equilibrium at all. Geometrically these are the convexity conditions on the potentials.

The Clausius-Clapeyron Relation

The slope of a first-order phase boundary in the P–T plane equals the latent heat divided by the absolute temperature times the volume change on crossing. A large latent heat steepens the line; a large volume change flattens it. The sign of the slope is fixed by the sign of Δv (since L > 0 and T > 0 for absorption of heat by the higher-entropy phase): if the higher-entropy phase is also the more voluminous, the boundary slopes up; if it is denser (as liquid water is, relative to ice), the boundary slopes down.

Boltzmann Entropy and Statistical Temperature

Entropy is k times the logarithm of the number of microstates consistent with the macrostate. Two bodies are in equilibrium when the logarithmic slope of their microstate counts with energy is equal; that common slope is 1/(k T). A system whose state count rises steeply with added energy is "cold" (large β, small T) and greedily absorbs heat; one whose count is already flattening is "hot" and readily gives energy up. Heat therefore flows from hot to cold because that raises the total log-count.

The Boltzmann Distribution

States of higher energy are exponentially less likely, with k_BT setting the scale over which "less likely" becomes "essentially never". Every thermodynamic quantity follows from Z: U = −∂ln Z/∂β, F = −k_BT ln Z, and so on.

The Partition Function and F = -kT ln Z

Everything thermodynamic is a derivative of ln Z. Compute the single scalar Z from the spectrum; its logarithm is (up to −kT) the free energy. From F(T,V,N) the standard thermodynamic derivatives deliver the rest: S=−(∂F/∂T)_V,N, pressure P=−(∂F/∂V)_T,N, chemical potential μ=(∂F/∂N)_T,V, and U=F+TS. The differentiation with respect to β pulls down factors of E_i, which is why log-derivatives of Z generate energy moments (mean, variance, …).

Energy Fluctuations and Heat Capacity

The mean-square spread of a system's instantaneous energy about its average is set entirely by how much its average energy changes with temperature. A body that soaks up heat readily (large C_V) is also a body whose energy visibly trembles; a body that barely responds to temperature is one whose energy is nearly frozen. Since ⟨(ΔE)²⟩ ≥ 0 always, the relation also proves C_V ≥ 0: a stable system cannot have negative heat capacity.

Equivalence of Ensembles in the Thermodynamic Limit

The canonical free energy is the Legendre transform of the microcanonical entropy: pass from the variable E to its conjugate 1/T = S′(E), and F = E* − TS emerges automatically. The neglected term is a single logarithm — smaller than F by a factor (ln N)/N. Physically, a system held at fixed T has an energy that is not fixed but is pinned to E*(T) with fractional scatter only 1/√N; on macroscopic scales it is indistinguishable from a system held at fixed energy E*. That is the equivalence of ensembles.

The Maxwell-Boltzmann Speed Distribution

A competition between two factors: v² rising and the exponential falling. The peak sits where they balance. The long high-speed tail — not the mean — governs reaction rates, atmospheric escape and evaporation.

Equipartition and Its Failure

Monatomic gas: three translational terms give U = 3⁄2 Nk_BT, C_V = 3⁄2 Nk_B. A diatomic gas adds two rotational terms → 5⁄2 Nk_B; adding the vibrational mode (kinetic and potential, two more terms) would give 7⁄2 Nk_B. A solid with three vibrational modes per atom, each counted twice, gives C = 3Nk_B — the Dulong–Petit law.

Sackur-Tetrode Entropy and the Gibbs Paradox

Absolute entropy — not a difference — for a classical ideal gas, and it contains h. Classical physics has no way to produce h, so a purely classical entropy is undefined up to an additive constant. Nature is not.

The Grand Canonical Ensemble and Chemical Potential

Opening the walls to particle flow replaces the Boltzmann weight e^−βE with e^{−β(E−μN)}: each particle the system holds is “rebated” by μ against its energy cost, because holding it denies the reservoir that particle. The reservoir sets the price μ; the system decides how many particles to buy. Averages follow by differentiating ln Ξ: N̅ = kT(∂ ln Ξ/∂μ)_T,V = −(∂Φ/∂μ)_T,V and U̅ − μN̅ = −(∂ ln Ξ/∂β)_βμ.

The First Law of Thermodynamics and Work in Ideal-Gas Processes

The first law partitions every joule of heat into internal-energy change and \(p\,dV\) work. For an ideal gas the internal energy is a pure function of temperature, so the entire thermodynamic state is pinned down by \(T\) (or equivalently \(p\) and \(V\)). The four standard processes are the four ways of holding one variable fixed while the other two adjust, and each gives a closed-form expression for \(W\), \(Q\), and \(\Delta U\).

Heat Conduction: Fourier's Law and the Heat Equation

Fourier's law states that heat flux is proportional to the negative temperature gradient. The heat equation describes how temperature evolves in time due to spatial variations in temperature. In steady state, the temperature profile is linear in a slab, and the heat current is proportional to the temperature difference and inversely proportional to the thickness.