Some ideas from thermodynamicsThe branch of physics called thermodynamicsdeals with the laws governing the behavior ofphysical systems at finite temperature. These lawsare phenomenological, in the sense that they repre-sent empirical relations between various observedquantities. These quantities include pressure, tem-perature, density and other measurable parametersthat define the state of a given system. Thus, indiscussing magnetic systems, e.g., the variablesmight include temperature, magnetization and ex-ternal magnetic field; in discussing elastic systemswe might want to use temperature, stress and strainas defining variables. Some useful references forthe student who wants to pursue the subject fur-ther are1. F.W. Sears, Thermodynamics, 2nd ed. (Addison-Wesley Publishing Co., Reading, MA, 1963).2. K. Huang, Statistical Mechanics (John Wiley andSons, Inc., New York, 1963).3. R.P. Feynman, Statistical Mechanics (W.A. Ben-jamin, Inc., Reading, MA, 1972).1. The First Law of ThermodynamicsWe imagine a system such as a gas, where tempera-ture, pressure and density are the appropriate sys-tem-defining variables. Then conservation ofenergy states that the following differential rela-tionship must hold:dU = dQ − dW (1)where dQ represents (a small amount of) heatflowing into the system from some heat source;dW the mechanical work done by the system, anddU the change in internal energy.The quantity U depends, by assumption, only onthe present state of the system and not on thehistory of how the system got into that state. Inthat case, dU must be, in the mathematical sense,an exact differential. To understand what thismeans, consider the work done by sliding a heavyblock up an inclined plane. There is a force actingdownhill (gravity) as well as friction. Obviously thework done traversing the wiggly path up the hill isgreater than that along the straight path, sincefriction always acts opposite to the direction ofmotion. That is, the part of the work arising fromovercoming gravity is independent of path (be-cause gravity is a conservative force) but that usedto overcome friction is path-dependent (becausefriction is a dissipative force).In other words, the work output depends on howthe job is done, so dW is not an exact differential.Similarly, the heat input depends on the details ofhow the heat enters the system—for example, itcan depend on the rate at which heat is supplied—so dQ is also not an exact differential. But experi-ence indicates that the internal energy of a systemdoes not depend on how it got there, hence wepostulate that dU is exact.Now suppose this postulate were wrong: then con-clusions we can deduce from it would at some pointdisagree with experience. So far no such cases havearisen; moreover, the microscopic theory of ther-modynamics (that is, statistical mechanics) sug-gests strongly that dU is exact.Physics of the Human Body 53Chapter 6 Some ideas from thermodynamics2. Equation of stateWe can use different sets of variables to define thestate of a thermodynamic system. For example wecan use temperature and pressure, pressure andvolume, or temperature and volume. The reasonwe have only two independent variables is that thethree variables p, V, T are related by an equation ofstate which takes the formf(p, V, T) = 0 . (2)For example one typical equation of state is theperfect gas law,pV = νRT (3)where R is the perfect gas constant (numericalvalue ≈ 8.3 J ⁄ oK ⁄ gm−mol) and ν is the number ofgram-moles present.Another well-known equation of state is the Vander Waals equation, p + an2 1 − nV0 = n kBT (4)where n is the number-density of the molecules, ais a constant representing an average (attractive)inter-molecular interaction energy, and V0 is pro-portional to the volume of a molecule (that is, ifthere are N molecules the available volume isV − NV0 rather than V). Also kB = R ⁄ NA isBoltzmann’s constant.The Van der Waals equation of state represents anattempt to take into account two physical effectsthat cause actual gases to deviate from the behav-ior of an ideal gas: intermolecular forces, and thefinite size of molecules. Although not a preciserepresentation of any gas (except possibly over alimited range of the variables) Eq. (4) is useful inunderstanding qualitatively the behavior of gasesin phase transitions.3. Thermodynamic relationsBecause the internal energy differential dU is (byhypothesis) perfect, we immediately derive certainrelationships. That is, imagine it depends on pres-sure and volume:U ≡ U(p, V)so thatdU = ∂U∂pV dp + ∂U∂Vp dV .(5)Now imagine that we go from internal energyU1 = U(p1, V1) to U2 = U(p2, V2) via two differ-ent paths as shown above. Since the result isindependent of path, it must be true that∫ Γ dU = ∫ Γ f(p, V) dp + g(p, V) dV = 0 ,where the integral is taken around the closed pathΓ. Now, by noting that if the result is true for a largepath it must also be true for the small square pathshown below, we find−g(p + dp)dVg(p)dV−f(V)dp f(V+dV)dp54 Physics of the Human BodyEquation of stateg(p,V) − g(p+dp, V) dV ++ f(p, V+dV) − f(p, V) dp = 0or∂ f∂V − ∂ g∂p dp dV = 0 ,or finally∂∂V ∂U∂pV = ∂∂p ∂U∂Vp .In other words, for dU to be an exact differentialthe order of its second derivatives with respect top and V is immaterial. (This result is just Stokes’sTheorem in two dimensions.)Since the the work output can be writtendW = pdV ,we can express the heat input into a system in thealternate formsdQ = ∂U∂pV dp + p + ∂U∂Vp dV (5a)dQ = p ∂V∂Tp + ∂U∂Tp dT + (5b)+ p ∂V∂pT + ∂U∂pT dpdQ = ∂U∂TV dT + p + ∂U∂VT dV . (5c)The specific heat at constant volume is defined bycV = ∆Q∆TV ≡ ∂U∂TV ,(6)where the last relation follows from letting dV=0above. The specific heat at constant pressure isdefined ascp = ∆Q∆Tp ≡ ∂H∂TV(7)where the enthalpy H is defined asH = U + pV .If a thermally isolated ideal gas is allowed to expandslowly into a vacuum, as shown below,it is found that the temperature does not