2. Ideal Gas In this chapter w e study some basic properties of ideal gas of massive iden- tical particles. We start by considering a single partic le in a box. We then discuss the statistical pro perties of an ensemble of identical indistinguishable particles and introduce the concepts of Fermions and Bosons. In the rest of this chapter we mainly focus on the classical limit. For this case we derive expressions for the pressure, heat capacity, energy an d entropy and discuss how internal degrees of freedom may modify these results. In the last part of this chapter we discuss an example of an heat engine based on ideal gas (Carnot heat e ngine). We show t hat t he efficiency of such a heat engine, which employs a reversible process, obtains the largest possible value th at is allowed by the second law of ther mod ynamics. 2.1 A Particle in a Box Consider a particle having mass M in a box. For simplicity the b ox is assumed to have a cube shape with a volume V = L 3 . The corresponding potential energy is given by V (x, y, z)= ½ 00≤ x, y, z ≤ L ∞ else . (2.1) The quantum eigenstates and eigenenergies are determ ined by requiring that the wav efunction ψ (x, y, z)satisfies the Schr¨odinger equation − ~ 2 2M µ ∂ 2 ψ ∂x 2 + ∂ 2 ψ ∂y 2 + ∂ 2 ψ ∂z 2 ¶ + Vψ= Eψ . (2.2) In addition, we require that the wa vefunction ψ vanishes on the surfaces of the bo x. The normalized s olutions are given by ψ n x ,n y ,n z (x, y, z)= µ 2 L ¶ 3/2 sin n x πx L sin n y πy L sin n z πz L , (2.3) where n x ,n y ,n z =1, 2, 3, (2.4) Chapter 2. Ideal Gas The corresponding eigenenerg ies are given by ε n x ,n y ,n z = ~ 2 2M ³ π L ´ 2 ¡ n 2 x + n 2 y + n 2 z ¢ . (2.5) For simplicity we consider the case where the particle doe sn’t hav e any in- ternal degree of freedom (such as spin). Later we will release this assumption and generalize the results for particles having internal degrees of freedom. The partition function is given by Z 1 = ∞ X n x =1 ∞ X n y =1 ∞ X n z =1 exp ³ − ε n x ,n y ,n z τ ´ = ∞ X n x =1 ∞ X n y =1 ∞ X n z =1 exp ¡ −α 2 ¡ n 2 x + n 2 y + n 2 z ¢¢ , (2.6) where α 2 = ~ 2 π 2 2ML 2 τ . (2.7) The follo wing relation can be employed to e stimate the dimensionless param - eter α α 2 = 7.9 × 10 −17 M m p ¡ L cm ¢ 2 τ 300 K , (2.8) where m p is the proton mass. As can be seen from the last r esult, it is often the case that α 2 ¿ 1. In this limit the sum can be approximated by an integral ∞ X n x =1 exp ¡ −α 2 n 2 x ¢ ' ∞ Z 0 exp ¡ −α 2 n 2 x ¢ dn x . (2.9) By changing the integration variable x = αn x one finds ∞ Z 0 exp ¡ −α 2 n 2 x ¢ dn x = 1 α ∞ Z 0 exp ¡ −x 2 ¢ dx = √ π 2α , (2.10) thus Z 1 = µ √ π 2α ¶ 3 = µ ML 2 τ 2π~ 2 ¶ 3/2 = n Q V, (2.11) where we have introduced the quantum density Eyal Buks Thermodynamics and Statistical Physics 46 2.1. A Particle in a Box n Q = µ Mτ 2π~ 2 ¶ 3/2 . (2.12) The par tition function (2.11) together with Eq. (1.70) allows evaluating the average energy (recall that β =1/τ ) hεi = − ∂ log Z 1 ∂β = − ∂ log µ ³ ML 2 2π~ 2 β ´ 3/2 ¶ ∂β = − ∂ log β −3/2 ∂β = 3 2 ∂ log β ∂β = 3τ 2 . (2.13) This result can be written as hεi = d τ 2 , (2.14) where d = 3 is the number of degrees of freedom of the pa rticle. As we will see later, this is an example of the equipartition theorem of statistical m echanics. Similarly, the energy va riance can be evaluated using Eq . (1.71) D (∆ε) 2 E = ∂ 2 log Z 1 ∂β 2 = − ∂ hεi ∂β = − ∂ ∂β 3 2β = 3 2β 2 = 3τ 2 2 . (2.15) Thus, using Eq. (2.13) the standard de viation is given by r D (∆ε) 2 E = r 2 3 hεi . (2.16) What is the physical meaning of the quantum density? The de Broglie wavelength λ of a particle having mass M and velocity v is giv en by Eyal Buks Thermodynamics and Statistical Physics 47 Chapter 2. Ideal Gas λ = h Mv . (2.17) For a particle having energy equals to the average energy hεi =3τ/2 one has Mv 2 2 = 3τ 2 , (2.18) thus in this case the de-Broglie wavelength, w hich is denoted as λ T (the thermal waveleng th) λ T = h √ 3Mτ , (2.19) and therefore one has (recall that ~ = h/2π) n Q = Ã π h 2 2Mτ ! 3/2 = Ã 1 λ T r 2π 3 ! 3 . (2.20) Thus the quantum density is inversely proportional to the thermal wavelength cubed. 2.2 Gibbs Paradox In the previous section we have studied the case of a single particle. Let us now consider the case where t he box is occupied b y N particles of the same t ype. For simplicity, we consider the case where the density n = N/V is sufficiently small to safely allowing to neglect any interactio n between the particles. In this case the gas is said to be ideal. Definition 2.2.1. Idea l gas is an ensemble of non-interacting identical par- ticles. What is the partition function of the ideal gas? Recall that for the single particle case we have found that the par t ition function is giv en by [see Eq. (2.6)] Z 1 = X n exp (−βε n ) . (2.21) In this expression Z 1 is obtained by summing over a ll single particle orbital states, w hich are d e noted by the vector of quan tum num bers n =(n x ,n y ,n z ). These states are called orbitals . Since the total num ber of particles N is constrained we need to calculate the canonical partition function. For the case of distinguishable particles one may argue that the canonical partition function is given by Eyal Buks Thermodynamics and Statistical Physics 48 2.2. Gibbs Paradox Z c ? = Z N 1 = Ã X n exp (−βε n ) ! N . (2.22) However, as was demonstrated by Gibbs in his famous paradox, this answer is wrong. To see this, we employ Eqs. (1.72) and (2.11) and assume that the partition function is given by Eq. (2.22) above, thus σ −βU =logZ c ? =logZ N 1 = N log (n Q V ) , (2.23) orintermsofthegasdensity n = N V , (2.24) we find σ −βU ? = N log ³ N n Q n ´ . (2.25) What is wrong with this result? It su ggests that the quantity σ −βU is not simply proportional to the size of the system. In other words, for a given n and a given n Q , σ −βU is not proportional to N. As we will see below, such a behavior ma y lead to a v iolation of the second law of thermodynamics. To see this consider a box containing N identical particles h aving volume V .What happens when we div ide the box into two sections by introducing a partition? Let the number of particles in the first (second) section be N 1 (N 2 )whereas thevolumeinthefirst (second) section be V 1 (V 2 ). The following hold N = N 1 + N 2 , (2.26) V = V 1 + V 2 . (2.27) The density in eac h section is expected to be the same as the dens ity in the box before the partition was introduced n = N V = N 1 V 1 = N 2 V 2 . (2.28) Now we use Eq. (2.25) to evaluate the change in entropy ∆σ due to the process of dividing the box . Since no energy is required to add (or to rem ove) the partition one has ∆σ = σ tot − σ 1 − σ 2 ? = N log ³ N n Q n ´ − N 1 log ³ N 1 n Q n ´ − N 2 log ³ N 2 n Q n ´ = N log N −N 1 log N 1 − N 2 log N 2 . (2.29) Using the Stirling’s f ormula (1 .150) log N! ' N log N − N, (2.30) Eyal Buks Thermodynamics and Statistical Physics 49 Chapter 2. Ideal Gas … orbital 1 N 1 =0 orbital 2 N 2 =1 orbital 3 N 3 =2 orbital 4 N 4 =0 3 1 2 … orbital 1 N 1 =0 orbital 2 N 2 =1 orbital 3 N 3 =2 orbital 4 N 4 =0 33 11 22 Fig. 2. 1. A figure describing an example term in the expansion (2.22) for a gas containing N = 3 identical particles. one finds ∆σ ' log N! N 1 !N 2 ! > 0 . (2.31) Thus we c ame to the conclusion that the process of dividing the bo x leads to reduction in the total entropy! This paradoxical result violates the second law of thermodynamics. According to this law we expect no chang e in the entropy since the process of dividing the box is a reversible one. What is wrong with the partition function given by Eq. (2.22)? Ex - panding this partition function yields a sum of terms each having the form exp µ −β P n N n ε n ¶ ,whereN n is the nu mber of particles occupyin g orbital n.Letg (N 1 ,N 2 , )bethenumberoftermsinsuchanexpansionassociated with a given set of occupa tion numbers {N 1 ,N 2 , }. S ince the partition func- tion (2.22) treats the particles as b eing distinguishable, g (N 1 ,N 2 , )mayin general be larger than unity. In fact, it is easy to see that g (N 1 ,N 2 , )= N! N 1 !N 2 ! × . (2.32) For example, consider the state that is described by Fig. 2.1 below for a gas containing N = 3 particles. The expansion (2.22) contains 3!/1!/2! = 3 terms having the same occupation numbers (N n =1ifn =2,N n =2 if n =3,andN n =0forallothervaluesofn). However, for identical particles these 3 states are indistinguishable. Therefore, only a sing le term in the partition function should represent such a configuration. In general, the partition function should include a single term only for each given set of occupation numbers {N 1 ,N 2 , }. 2.3 Fermions and Bosons As we saw in the previous section the canonical partition function given by Eq. (2.22) is incorrect. For indistinguishable particles ea c h set of orbital Eyal Buks Thermodynamics and Statistical Physics 50 2.3. Fermions and Bosons occupation numbers {N 1 ,N 2 , } should be counted only once. In this section we take another approach and instead of evaluating the canonical partition function of the system we consider the grandcanonical partition function. This is done by considering ea ch orbita l a s a s ubsystem and by evaluating its grandcanonical partition function, whic h w e denote below as ζ.Todothis correctly, however, it is impor tant to take into accou nt the exclusio n rul es imposes by quantum mechanics upon the po ssible values of the occupation numbers N n . The particles in nature ar e divided into two type: Fermion and Bosons. While Fermions have half in teger spin Bosons have integer spin. According to quantum mech an i cs the orbital occupation numbers N n can take the following values: • For Fermions: N n =0or1 • For Bosons: N n can be any integer. These rules are employed below to evalua te the grandcanonical partition function of an orbital. 2.3.1 Fermi-Dirac Distribution In this case the occup ation number can take only two possible values: 0 or 1. Thus, by Eq. (1.79) the grandcanon ical partition function of a n orbital hav ing en ergy is ε is given by ζ =1+λ exp (−βε) , (2.33) where λ =exp(βµ)(2.34) is the fugacity [see Eq. (1.95)]. The avera ge occupation of the orbital, which is denoted by f FD (ε)=hN (ε)i, is found using Eq . (1.94) f FD (ε)=λ ∂ log ζ ∂λ = λ exp (−βε) 1+λ exp (−βε) = 1 exp [β (ε − µ)] + 1 . (2.35) The function f FD (ε) is called the Fermi-Dirac function . Eyal Buks Thermodynamics and Statistical Physics 51 Chapter 2. Ideal Gas 2.3.2 Bos e-Einstein Distribution In this case the occupation num ber can take any integer value. Thus, by Eq. (1.79) the grandcanonical partition function of an orbital ha ving energy is ε is given by ζ = ∞ X N=0 λ N exp (−Nβε) = ∞ X N=0 [λ exp (−βε)] N = 1 1 − λexp (−βε) . (2.36) The average occupation of the orbital, which is denoted by f BE (ε)=hN (ε)i, is found using Eq. (1.94) f BE (ε)=λ ∂ log ζ ∂λ = λ exp (−βε) 1 − λexp (−βε) = 1 exp [β (ε − µ)] − 1 . (2.37) The function f BE (ε) is called the Bose-Einstein function . 2.3.3 Classical Limit The classical limit occurs when exp [β (ε − µ)] À 1 . (2.38) As can be seen from Eqs. (2.35) and (2.37) the following holds f FD (ε) ' f BE (ε) ' exp [β (µ −ε)] ¿ 1 , (2.39) and ζ ' 1+λ exp (−βε) . (2.40) Th us the classical limit corresponds to the c ase where the a verage occupation of an orbital is close to zero, namely the orbital is on average almost emp ty. The main results of the above discussed cases (Fermi-Dirac distribution, Bose- Einstein distribution and the c lassical limit) are sum ma rized in table 2.1 below. Eyal Buks Thermodynamics and Statistical Physics 52 2.4. Ideal Gas in the Classical Limit Table 2.1 . Fermi-Dirac, Bose-Einstein and classical distributions. orbital partition function average occupation Fermions 1 + λ exp (−βε) 1 exp[β(ε−µ)]+1 Bosons 1 1−λ exp(−βε) 1 exp[β(ε−µ)]−1 classical limit 1 + λ exp (−βε)exp[β (µ − ε)] 0 0.5 1 1.5 2 -2 2468101214161820 energy 2.4 Ideal Gas in the Classical Limit The rest of this chapter is devoted to t he classical limit. The grandcanonical partition function ζ n of orbital n having energy ε n is given by Eq. (2.40) abov e. The grandcanonical partition function of the en tire system Z gc is found by multiplying ζ n of all orbitals Z gc = Y n (1 + λ exp (−βε n )) . (2.41) Each term in the expansion of the above expression represents a set of orbital occupation num bers, where each occupation number can tak e one of the po ssible va lues: 0 or 1. We exploit the fact that in the cla ssical lim it λ exp ( −βε) ¿ 1(2.42) and employ the first order expansion log (1 + x)=x + O ¡ x 2 ¢ (2.43) to obtain Eyal Buks Thermodynamics and Statistical Physics 53 [...]... and employing Eqs (2 .47 ) and (2 .48 ) one finds that U= 3N τ 2 (2 .49 ) Namely, the total energy is N hεi, where hεi is the average single particle energy that is given by Eq (2.13) The entropy is evaluate using Eq (1.86) σ = log Zgc + βU + ηN µ ¶ 3 µ = N 1+ − 2 τ ¶ µ 5 − µβ =N 2 Eyal Buks Thermodynamics and Statistical Physics (2.50) 54 2 .4 Ideal Gas in the Classical Limit Furthermore, using Eqs (2 .44 ),... Eqs (2 .44 ), (2 .48 ), (2.11) and (1.95) one finds that n , (2.51) µβ = log nQ where n = N/V is the density This allows expressing the entropy as µ ¶ nQ 5 + log σ=N (2.52) 2 n Using the definition (1.116) and Eqs (2 .49 ) and (2.52) one finds that the Helmholtz free energy is given by ¶ µ n −1 (2.53) F = Nτ log nQ 2 .4. 1 Pressure The pressure p is defined by ¶ µ ∂F p=− ∂V τ ,N (2. 54) Using Eq (2.53) and keeping... Taking into account internal degrees of freedom the grandcanonical partition function of an orbital having orbital energy εn for the case of Fermions becomes Y (1 + λ exp (−βεn ) exp (−βEl )) , (2.71) ζ FD,n = l Eyal Buks Thermodynamics and Statistical Physics 57 Chapter 2 Ideal Gas where {El } are the eigenenergies of a particle due to internal degrees of freedom, and where λ = exp (βµ) and β = 1/τ... Zgc = log Zgc (2 .48 ) hN i = − ∂η β In what follows, to simplify the notation we remove the diagonal brackets and denote hU i and hNi by U and N respectively As was already pointed out earlier, probability distributions in statistical mechanics of macroscopic parameters are typically extremely sharp and narrow Consequently, in many cases no distinction is made between a parameter and its expectation... box containing the gas due to collisions between the particles and the walls To see that this is indeed the case consider a gas of N particles contained in a box having cube shape and volume V = L3 One of the walls is chosen to lie on the x = 0 plane Consider and elastic collision between this wall and a particle having momentum p = (px , py , pz ) After the collision py and pz remain unchanged, however,... (−βεn ) n = λZ1 , (2 .44 ) where Z1 = V µ Mτ 2π~2 ¶3/2 (2 .45 ) [see Eq (2.11)] is the single particle partition function In terms of the Lagrange multipliers η = −µ/τ and β = 1/τ the last result can be rewritten as µ ¶3/2 M −η (2 .46 ) log Zgc = e V 2π~2 β The average energy and average number of particle are calculated using Eqs (1.80) and (1.81) respectively µ ¶ 3 ∂ log Zgc log Zgc , = (2 .47 ) hU i = − ∂β... fn = λ The total grandcanonical partition function is given by Y Zgc = ζn , (2.77) (2.78) n thus Eyal Buks Thermodynamics and Statistical Physics 58 2 .4 Ideal Gas in the Classical Limit X log Zgc = log ζ n n X = log [1 + λZint exp (−βεn )] n ' λZint X exp (−βεn ) n = λZint Z1 , (2.79) where we have used the fact that in the classical limit λZint exp (−βεn ) ¿ 1 Furthermore, using Eq (2.11) and recalling... 3 +τ , cV = N 2 ∂τ 2 cp = cV + N (2. 84) (2.85) (2.86) (2.87) (2.88) (2.89) Proof See problem 3 of set 2 Eyal Buks Thermodynamics and Statistical Physics 59 Chapter 2 Ideal Gas 2.5 Processes in Ideal Gas The state of an ideal gas is characterized by extensive parameters (by definition, parameters that are proportional to the system size) such as U , V , N and σ and by intensive parameters (parameters... process) yields ∆U = Q − W , (2. 94) We discuss below some specific examples for processes for which dN = 0 The initial values of the pressure, volume and temperature are denoted as p1 , V1 and τ 1 respectively, whereas the final values are denoted as p2 , V2 and τ 2 respectively In all these processes we assume that the gas remains in Eyal Buks Thermodynamics and Statistical Physics 60 ... finds that ­ 2® px =τ m (2.58) Using this result and Eq (2.56) one finds that the pressure due to a single particle is p = τ /V , thus the total pressure is p= Nτ V (2.59) 2 .4. 2 Useful Relations In this section we derive some useful relations between thermodynamical quantities Claim The following holds µ ¶ ∂U p=− ∂V σ,N Proof Using the definition (2. 54) and recalling that F = U − τ σ one finds µ µ µ ¶ . N µ 5 2 − µβ ¶ . (2.50) Eyal Buks Thermodynamics and Statistical Physics 54 2 .4. Ideal Gas in the Classical Limit Furthermore, using Eqs. (2 .44 ), (2 .48 ), (2.11) and (1.95) one finds that µβ =log n n Q ,. ssical lim it λ exp ( −βε) ¿ 1(2 .42 ) and employ the first order expansion log (1 + x)=x + O ¡ x 2 ¢ (2 .43 ) to obtain Eyal Buks Thermodynamics and Statistical Physics 53 Chapter 2. Ideal Gas log. ' N log N − N, (2.30) Eyal Buks Thermodynamics and Statistical Physics 49 Chapter 2. Ideal Gas … orbital 1 N 1 =0 orbital 2 N 2 =1 orbital 3 N 3 =2 orbital 4 N 4 =0 3 1 2 … orbital 1 N 1 =0 orbital