Physics 301 15-Sep-2004 3-1PressureLast lecture, we considered two systems with entropy as a function of internal energy,volume and number of particles,σ(U, V, N, U1, V1, N1) = σ1(U1, V1, N1) + σ2(U2, V2, N2) .We allowed them to exchange internal energy (that is, they were placed in thermal con-tact), and by requiring that the entropy be a maximum, we were able to show that thetemperature is1τ=∂σ∂UV,N.Suppose we continue to consider our two systems, and ask what happens if we allowthem to exchange volume as well as energy? (We’re placing them in mechanical as well asthermal contact.) Again, the total entropy must be a maximum with respect to exchangesof energy and exchanges of volume. Working through similar mathematics, we find anexpression for t he change in total ent ropy and insist that it be zero (so the entropy ismaximum) at equilibrium,0 = δσ=∂σ1∂U1δU1+∂σ2∂U2δU2+∂σ1∂V1δV1+∂σ2∂V2δV2=∂σ1∂U1−∂σ2∂U2δU1+∂σ1∂V1−∂σ2∂V2δV1,from which we infer that at equilibrium,∂σ1∂U1=∂σ2∂U2,which we already knew, and∂σ1∂V1=∂σ2∂V2.This last equation is new, and it must have something to do with the pressure. Why?Because, once the temperatures are the same, two systems exchange volume only if onesystem can “push harder” and expand while the other contracts. We define the pressure:p = τ∂σ∂VU,N.We will see later that this definition agrees with the conventional definitio n of pressure asforce per unit area.Copyrightc 2004, Princeton University Physics Department, Edward J. GrothPhysics 301 15-Sep-2004 3-2Chemical PotentialWell, there’s one variable l eft, guess what we’re going to do now! Suppose we allowthe two systems to exchange particles as well as energy and volume. Again, we want tomaximize the entropy with resp ect to changes in al l the independent variables and thisleads to,0 = δσ=∂σ1∂U1δU1+∂σ2∂U2δU2+∂σ1∂V1δV1+∂σ2∂V2δV2+∂σ1∂N1δN1+∂σ2∂N2δN2=∂σ1∂U1−∂σ2∂U2δU1+∂σ1∂V1−∂σ2∂V2δV1+∂σ1∂N1−∂σ2∂N2δN1.So, when the systems can exchange particles as well as energy and volume,∂σ1∂N1=∂σ2∂N2.The fact that these derivatives must be equal in equilibrium allows us to define yet anotherquantity, µ, the chemical potentialµ = −τ∂σ∂NU,V.If two systems are allowed to exchange particles and the chemical potentials are unequal,there will be a net flow of particles until the chemical potentials are equal. Like temperatureand pressure, chemical potential is an intensive quantity. Unlike temperature and pressure,you probably have not come across chemical potential in your elementary thermodynamics.You can think of it very much like a potential energy per particle. Systems with highchemical potential want to send particles to a system wit h low potential energy per particle.Note that we can write a change in the entropy of a system, specified in terms of U , V ,and N asdσ =1τdU +pτdV −µτdN ,or rearranging,dU = τ dσ − p dV + µ dN .Which is the conservation of energy (first law of t hermodynamics) written for a sy stemwhich can absorb energy in the form of heat, which can do mechanical pV work, and whichcan change its energy by changing the number of particles.Copyrightc 2004, Princeton University Physics Department, Edward J. GrothPhysics 301 15-Sep-2004 3-3First Derivatives versus Second DerivativesYou will notice that the quantitites defined by first derivatives are not material specific.For exa mple, whether it’s nitrogen gas or a block of steel, the rate of change of energywith entropy (at constant volume a nd particle number) is the temperature.We’ll eventually define Helmholtz and Gibbs free energies and enthalpy (differentindependent variables) and it will always be the case that the first derivatives of thesequantities produce other quantities that are not ma terial specific.To get to materia l specific quantities, one must go to second derivatives. For example,suppose we have a block of steel and nitrogen gas inside a container that i s t hermallyinsulating, fixed in volume, and impermeable. Then at equilibrium,∂Usteel∂VsteelS,N=∂Unitrogen∂VnitrogenS,N= −p .If we make a change to the volume of the container, we might be interested in∂p∂V= −∂2U∂V2.This quantity is related to the compressibility of the material. Nitrogen gas i s much morecompressible than steel and most of the volume change will be t aken up by the gas, notthe steel. In other words, the material specific quantity (second derivative) is different forthe two materials.Copyrightc 2004, Princeton University Physics Department, Edward J. GrothPhysics 301 15-Sep-2004 3-4ProbabilityHere, we will introduce some basic concepts of probability. To start with, one imaginessome experiment or other process in which several possible outcomes may occur. Thepossible outcomes are known, but not definite. For exa mple, tossing a die leads to oneof the 6 numbers 1, 2, 3, 4, 5, 6 turning up, but which number will occur is not knownin advance. Presumably, a set of elementary outcomes can be defined and all possibleoutcomes can be specified by saying which elementary outcomes must occur. For example,the tossing of the die resulting in an even number would be made up of the elementaryevents: the toss is 2 or the toss is 4 or the toss is 6. A set of elementary events is suchthat one and only one event can occur in any repetition of the experiment. For example,the events (1) the toss results in a prime number and (2) the toss gives an even numbercould not both be part of a set of elementary events, because if the number 2 comes up,both events have o ccurred!One imagines that a very large number of tosses of the die take place. Furthermore,in each toss, an attempt is made to ensure that there i s no memory of the previ ous toss.(This is anot her way of saying successive tosses are independent.) Then the probabilityof an event is just the fraction of times it occurs in this large set of experiments, thatis, ne/N, where neis the number of times event e occurs and N is the total number ofexperiments. In principle, we should take the limit as the number of trials goes to ∞.From this definition it is easy to see that the probabilities of a set of elementary eventsmust satisfypi≥ 0 ,andXipi= 1 ,where piis the probability of event i and i is an index that ranges over the