Thermodynamics Crib Sheet
2 Zeroth Law.
3 First Law.
3.1 Ideal Gas Law.
4 Second Law.
4.1 Heat Engines.
5.1 Irreversible Processes.
5.2 Isothermal Process.
5.3 Adiabatic Process.
5.4 Carnot Engine.
5.5 Engine Efficiency.
5.7 Restatement of Second Law.
7 Gibbs Free Energy.
8 Third Law.
9 Kinetic Theory.
9.1 Energy Distribution.
9.2.1 Velocity Distribution.
9.3 Bose-Einstein Distribution.
10 Black Holes.
Thermodynamic systems are described by a number of physical variables such as volume and pressure, spin, electromagnetic field strength and direction, etc. All the possible states the system can be in form a state space, a subset of RN. Systems in thermodymanic equilibrium are (i) in physical contact, (ii) "heat" can flow freely between them and (iii) the state of the system does not change over time (hence "equilibrium").
The requirement for physical contact could probably be relaxed, but the more limited definition works fine for the purpose of defining temperature. (See below)
What is temperature? It is not obvious that temperature is a well-formed concept. Other concets, such as pressure, are intuitive but turn out to be more subtle than expected. For example, the forces in a moving fluid are often better described by a stress tensor than a single "pressure" scalar.
If two systems are in thermal equilibrium with a third system, then they are in equilibrium with each other.
The Zeroth Law allows temperatures to be consistently assigned to equilibrium states. Systems in thermal equilibrium are said to be "at the same temperature".
Lemma: The relation © = "is in equlibrium with" between thermodynamic systems is a equivalence relation.
The Zeroth Law is a statement that the relation is transitive. I.e. if A © B and
B © C then
A © C.
A temperature function can be defined by assigning the same temperature to each state in the equivalence class. The temperature so defined may indeed not look like the Celsius temperature scale, or even be continuous, but it is a temperature function. In some systems, states of constant temperature may form smooth surfaces in the state space in which case the normal provides a natural ordering of nearby surfaces. It is then simple to construct a global temperature function that provides an continuous ordering of states. For example, if two systems of ideal gas are in equilibrium, then P1V1/N1 = P2V2/N2 where Pi is the pressure in the ith system, Vi is the volume, and Ni is the 'amount' of gas. The surface PV / N = const defines surfaces of equal temperature, and the obvious (but not only) way to label them is to define T so that PV / N = RT where R is some constant. Such a system can be now be used as a thermometer to calibrate other systems
Is temperature defined for non-equlibrium systems? For example, how does the definition of temperature defined from the Zeroth Law extend to turbulent fluid flow? The answer is to (i) apply equilibrium thermodynamics to "finite elements", and/or (ii) associate temperature with energy. (See below)
Definition: An Isothermal is a surface of constant temperature in the state space.
Energy is conserved. Heat is a form of energy.
For any system, DU = Q - W where U = internal energy, Q = heat added to the system, W = work done by the system.
The relationship PV = NRT can be established experimentally.
(There is some circularity here since the equation is partial motivation for the choice of temperature function)
Consider cube of side L filled with N molecules of an ideal gas. Assume the molecule moves from one wall, to the opposite wall and back.
Average impulse on wall per molecule
For N molecules,
A heat engine is a device that absorbs an amount of heat Qh from a hot reservoir, exhausts an amount of heat Qc to a cold reservoir and does work W.
Kelvin-Plank Statement of the Second Law: It is impossible to extract an amount of heat Qh from a hot reservoir and use it all to do work W. Some amount of heat Qc > 0 must be exhausted to a cold reservoir.
The second law states that it is impossible to build a perpetual motion machine. Any real-world motion generates heat through friction, and the 2nd law states that not all of the generated heat can be turned back into work.
An irreversible process goes only in one direction. I.e. the system and the universe cannot be put back into the initial state. Assertion: A process is reversible if and only if it only passes through equilibrium states that are “quasi”-static. Any non-equilibrium state renders the process irreversible. The proof? Good question.
There is no such thing as a “quasi-static” state. It is a useful construct. There is no such thing as a truly reversible process in nature.
In general dQ = dU + dW is not an exact differential. I.e. so the heat absorbed / released depends on path taken between states.
The temperature remains constant in an Isothermal process. For an ideal gas,̃
dQ = 0 for Adiabatic processes, so dU = -dW
Þ Put g = 5/3, then T1V1 g-1 = T2V2 g-2. Substituting T = PV/NR gives P1V1g = P2V2g.
Þ where l = P1V1g
The most efficient heat engine possible. (If a more efficient engine could be built, the more efficient engine could be used to drive the Carnot Engine as a refrigerator and produce perpetual motion) The engine cycle consists of 4 stages shown in the P-V diagram below.
The four stages are
Total work done in 1 cycle = WAB+ WBC+ WCD+ WDA = WIsothermal + WAdiabatic
WIsothermal = , WAdiabatic = -DUTotal = 0 since the cycle is a closed path which saves doing a great deal of algebra.
Efficiency E is defined to be . For a Carnot Engine, .
The following relationship holds for a Carnot engine: Qh /Th = Qc/Tc Þ = 0.
The area inside any closed path can be approximated by Carnot “tiles” as shown in the diagram below. . I.e. there exists a function S, called Entropy, that for any path from A to B.
The relationship between entropy, heat and temperature is often stated in the simplified form as DS = Q/T where S = entropy, Q = heat flow, T = temperature.
The second law implies that for a non-Carnot engine ³ 0. I.e. DS ³ 0.
Query: If any cycle can be approximated by a mesh of Carnot engines, and DS = 0 for Carnot engines, then why doesn’t DS = 0 for all engines? Answer: The Carnot engine assumes an Ideal gas operating in a perfect environment. Real-world engines are more wasteful with their heat flows so DS ³ 0.
In any cyclic process the entropy will either increase or remain the same.
For systems characterised by pressure and volumne, Enthalpy H = U + PV where U = internal energy, V = volume of system, P = pressure.
At constant pressure, dW = PdV, so dQ = dU + PdV = dH
Gibbs Free Energy = G = H - T.S where H = enthalpy, T = temperature and S = entropy.
The sign of
It is impossible to cool a body to absolute zero by any finite process. What is absolute zero?
Nernst’s Heat Theorem: If it is possible to reach absolute zero, all systems at that temperature would have the same entropy (defined to be zero). In particular, as T ® 0, S ® 0.
Corollary: A body at absolute zero is associated with a definite energy called the zero-point energy.
f (E) = probability distribution for particles with given energy. Three versions:
Let dN = F(vx ,vy ,vz) dvxdvydvz where F is the velocity distribution function, dN is the number of particles with velocities in the x,y,z direction in the intervals vx ® vx+dvx, vy ® vy+dvy, vz ® vz+dvz
Assume x,y,z are orthogonal and vx,vy,vz are independent, then F(vx ,vy ,vz) = f(vx)g(vy)h(vz)
Assume isotropy, then F(vx ,vy ,vz) = f(vx)f(vy)f(vz)
Isotropy implies also F(vx ,vy ,vz) = Y(v) for some Y where v = (vx2 + vy2 + vz2)½
Hence F = Y(v(vx ,vy ,vz)) = f(vx)f(vy)f(vz)
Historically this was a VERY HARD problem - understanding black body radiation. Planck came up with a solution by assuming E = nhw where h = planks constant, w = frequency, and began the quantum revolution.
Black holes are completely described by their mass, charge and angular momentum Þ T2 violated.
Common sense suggest that the entropy of a system (e.g. computer chip) is related to volume (more chips = more entropy). Volume of system proportional system diameter3. Boundary (area) of system proportional system diameter2. Expect volume (and hence entropy) to outpace boundary area and so exceed holographic bound. Actually system collapse into black hole before bound can be exceeded!
Many alternative bounds have been proposed. Universal bound more restrictive than Holographic bound.
Holographic bound restricts the number of degrees of freedom inside a surface, however QFT and Superstrings allow infinite degrees of freedom Þ QFT and Superstrings cannot be the final story.
Please feel free to send suggestions on how this page could be improved to the author Shaun O'kane. I will no longer respond to comments challenging the relationship between the Zeroth Law and temperature. Please note that articles on Wikipedia are sometimes spectacularly inaccurate.
 "Information in the Holographic Universe" by Jacob D. Bekenstein. Scientific American, August 2003.
Copyright © Shaun O’Kane, 2000, 2001, 2003