Thermodynamics

Prerequisites: Energy, Entropy. Full path (8 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

Intuitive

Thermodynamics describes systems with an enormous number of particles, like the gas in a cylinder or the water in a kettle, using only a handful of quantities we can measure from the outside: temperature, pressure, volume, energy. It was developed in the nineteenth century to understand steam engines, before it was known for certain that matter is made of atoms, and it still works without any reference to them.

The theory rests on a few laws. Two bodies in contact end up at the same temperature, and temperature is the quantity that decides in which direction heat flows. Energy is conserved, and heat is one form in which energy is transferred. The second law says that there is a quantity, entropy, that never decreases in an isolated system. Heat flows by itself from hot to cold and never the other way. A cup of coffee cools down in a room. It never spontaneously heats up while the room gets colder, although energy conservation alone would allow it.

The second law has practical consequences. No engine that takes heat from a hot source can turn all of it into work. Some heat must always be dumped into a colder environment, and the best possible efficiency depends only on the two temperatures. This limit holds for every heat engine, whatever it is built from.

Statistical mechanics later explained why these laws hold: entropy counts the number of microscopic arrangements of the particles that look the same from the outside, and a system drifts towards macrostates that have overwhelmingly many such arrangements.

• Feynman’s Ratchet and the perpetual motion gambling scheme | Gravity and Levity

Concrete

A simple system, for example a gas, in equilibrium is described by state variables such as internal energy $U$, volume $V$, particle number $N$, temperature $T$, pressure $p$ and entropy $S$. Only a few of them are independent; the relations between them are equations of state, such as $pV = Nk_BT$ for the ideal gas.

The laws.

• Zeroth law: thermal equilibrium is transitive, which allows the definition of temperature.
• First law: $dU = \delta Q - \delta W$. The change in internal energy equals the heat added to the system minus the work done by the system. $U$ is a function of state; $\delta Q$ and $\delta W$ are not exact differentials, since they depend on the process.
• Second law: there is a state function $S$ such that $dS \geq \delta Q/T$ for any process, with equality for reversible processes. For an isolated system $\delta Q = 0$ and $S$ cannot decrease.
• Third law: the entropy of a system approaches a constant as $T \to 0$ (zero for a non-degenerate ground state).

For reversible changes of a simple system with fixed $N$, $\delta Q = T\,dS$ and $\delta W = p\,dV$, which gives the fundamental relation

$$dU = T\,dS - p\,dV, \qquad T = \left(\frac{\partial U}{\partial S}\right)_V, \quad p = -\left(\frac{\partial U}{\partial V}\right)_S .$$

Example: the Carnot limit. An engine works in a cycle between a hot reservoir at $T_h$ and a cold one at $T_c$. In one cycle it takes heat $Q_h$ from the hot reservoir, gives heat $Q_c$ to the cold one and does work $W = Q_h - Q_c$, by the first law, because $U$ returns to its starting value. The engine's own entropy also returns to its starting value, so the total entropy change is that of the reservoirs, $\Delta S = -Q_h/T_h + Q_c/T_c$. The second law requires $\Delta S \geq 0$, so $Q_c \ge Q_h T_c/T_h$ and the efficiency obeys

$$\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h} \leq 1 - \frac{T_c}{T_h}.$$

Equality holds for a reversible engine. A power station with steam at 800 K and cooling water at 300 K cannot exceed $\eta = 1 - 300/800 \approx 0.63$.

Potentials. Different experimental conditions are handled by Legendre transforms of $U$: the Helmholtz free energy $F = U - TS$ with $dF = -S\,dT - p\,dV$, the enthalpy $H = U + pV$ and the Gibbs free energy $G = U - TS + pV$. At fixed $T$ and $V$ a system in contact with a heat bath minimises $F$; at fixed $T$ and $p$ it minimises $G$. Equality of mixed second derivatives gives Maxwell relations such as $\left(\partial S/\partial V\right)_T = \left(\partial p/\partial T\right)_V$, which relate quantities that are hard to measure to ones that are easy to measure.

Recommended Lecture Notes:

• IB classical thermodynamics (Lent 2003-2004)

Recommended Papers:

• Gibbs vs Boltzmann Entropies by E. T. Jaynes; especially the section ""Intuitive meaning of the second law".

Abstract

Fundamental relation and convexity. In the Gibbs formulation an equilibrium system is described by a fundamental relation $S = S(U, V, N)$ (or equivalently $U = U(S,V,N)$). $S$ is positively homogeneous of degree one (extensivity), increasing in $U$, and concave. Homogeneity gives the Euler relation $U = TS - pV + \mu N$, and together with the fundamental relation the Gibbs-Duhem relation $S\,dT - V\,dp + N\,d\mu = 0$. Concavity of $S$, equivalently convexity of $U(S,V,N)$, is the condition of thermodynamic stability. It implies positive heat capacity and compressibility, and it guarantees that the Legendre transforms to $F$, $H$ and $G$ are well defined. Where $S$ fails to be strictly concave, the system separates into coexisting phases.

Contact geometry. The thermodynamic phase space of a simple system with fixed $N$ is $\mathbb{R}^5$ with coordinates $(U, S, V, T, p)$ and the contact form

$$\theta = dU - T\,dS + p\,dV .$$

$\theta \wedge (d\theta)^2 \neq 0$, so $\theta$ defines a contact structure. The equilibrium states of a given substance form a Legendrian submanifold $L$: a 2-dimensional submanifold on which $\theta|_L = 0$. If $L$ is a graph over the $(S,V)$ plane it is generated by $U(S,V)$ through $T = \partial U/\partial S$ and $p = -\partial U/\partial V$. Legendre transforms to other potentials are changes of the generating function, and they correspond to describing the same $L$ over different coordinate planes. Reversible processes are curves in $L$, and the first law in the form $dU = T\,dS - p\,dV$ is the statement $\theta|_L = 0$.

Axiomatic foundations. Carathéodory reformulated the second law as a statement about accessibility: arbitrarily close to every equilibrium state there are states that cannot be reached by adiabatic processes. By Carathéodory's theorem on Pfaffian forms, this implies that the heat form $\delta Q$ for quasi-static processes has an integrating factor, $\delta Q = T\,dS$, which gives the existence of entropy and absolute temperature. Lieb and Yngvason gave a later axiomatisation in which entropy is derived from the preorder of adiabatic accessibility on state space, together with axioms for scaling and combining systems. In the Lieb-Yngvason approach entropy is unique up to an affine change of scale for each system.

Recommended Papers:

• On Geometry of Phenomenological Thermodynamics by Jerzy Kocik

Why is it interesting?

“….[classical thermodynamics] is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.“

Einstein

“The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations — then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation — well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.“

Eddington

“If physical theories were people, thermodynamics would be the village witch,” the physicist Lídia del Rio and co-authors wrote last year in Journal of Physics A. “The other theories find her somewhat odd, somehow different in nature from the rest, yet everyone comes to her for advice, and no one dares to contradict her.”

https://www.quantamagazine.org/the-quantum-thermodynamics-revolution-20170502/

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Newton's Second Law
4. Statistics
5. Vector Calculus
6. Entropy
7. Newtonian Mechanics
8. Energy

Leads to: Free Energy, Maxwell Relations.