Introduction to Thermodynamics and Statistical Physics phần 4

2. Ideal Gas In this chapter we study some basic properties of ideal gas of massive iden-tical particles. We start by considering a single particle in a box. We then discuss the statistical properties 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 and 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 engine). We show that the efficiency of such a heat engine, which employs a reversible process, obtains the largest possible value that is allowed by the second law of thermodynamics. 2.1 A Particle in a Box Consider a particle having mass M in a box. For simplicity the box is assumed to have a cube shape with a volume V = L3. The corresponding potential energy is given by V (x,y,z) = ½ 0 0 ≤ x,y,z ≤ L . (2.1) The quantum eigenstates and eigenenergies are determined by requiring that the wavefunction ψ(x,y,z) satisfies the Schr¨odinger equation 2 µ 2 2 2 ¶ −2M ∂x2 + ∂y2 + ∂z2 +Vψ = Eψ . (2.2) In addition, we require that the wavefunction ψ vanishes on the surfaces of the box. The normalized solutions are given by ψnx,ny,nz (x,y,z) = µL¶3/2 sin nxπx sin nyπy sin nzπz , (2.3) where nx,ny,nz = 1,2,3,... (2.4) Chapter 2. Ideal Gas The corresponding eigenenergies are given by εnx,ny,nz = 2M ³L´2 ¡n2 +n2 +n2¢ . (2.5) For simplicity we consider the case where the particle doesn’t have 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 Z1 = X X X exp³−εnx,ny,nz ´ nx=1 ny=1 nz=1 = X X X exp¡−α2 ¡n2 +n2 +n2¢¢ , nx=1 ny=1 nz=1 where 2 ~2π2 2ML2τ (2.6) (2.7) The following relation can be employed to estimate the dimensionless param-eter α α2 = 7.9×10−17 , (2.8) mp cm 300K where mp is the proton mass. As can be seen from the last result, it is often the case that α2 ¿ 1. In this limit the sum can be approximated by an integral ∞ exp¡−α2n2¢ ` Z exp¡−α2n2¢dnx . (2.9) nx=1 0 By changing the integration variable x = αnx one finds ∞exp¡−α2n2¢dnx = α ∞exp¡−x2¢dx = 2α , (2.10) 0 0 thus µ√ ¶3 µ 2 ¶3/2 Z1 = 2α = 2π~2 = nQV , (2.11) where we have introduced the quantum density Eyal Buks Thermodynamics and Statistical Physics 46 µ ¶3/2 nQ = 2π~2 . 2.1. A Particle in a Box (2.12) The partition function (2.11) together with Eq. (1.70) allows evaluating the average energy (recall that β = 1/τ) hεi = −∂logZ1 ∂logµ³2π~2β´3/2¶ = − ∂β ∂logβ−3/2 ∂β 3 ∂logβ 2 ∂β 3τ 2 (2.13) This result can be written as hεi = dτ , (2.14) where d = 3 is the number of degrees of freedom of the particle. As we will see later, this is an example of the equipartition theorem of statistical mechanics. Similarly, the energy variance can be evaluated using Eq. (1.71) D(∆ε)2E = ∂2 logZ1 ∂hεi ∂β = −∂β 2β 3 2β2 = 3τ2 . (2.15) Thus, using Eq. (2.13) the standard deviation is given by rD E r (∆ε) = 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 given by Eyal Buks Thermodynamics and Statistical Physics 47 Chapter 2. Ideal Gas λ = Mv . (2.17) For a particle having energy equals to the average energy hεi = 3τ/2 one has Mv2 3τ 2 2 (2.18) thus in this case the de-Broglie wavelength, which is denoted as λT (the thermal wavelength) λT = √3Mτ , (2.19) and therefore one has (recall that ~ = h/2π) Ã !3/2 Ã r !3 nQ = = . (2.20) 2Mτ T 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 the box is occupied by N particles of the same type. For simplicity, we consider the case where the density n = N/V is sufficiently small to safely allowing to neglect any interaction between the particles. In this case the gas is said to be ideal. Definition 2.2.1. Ideal 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 partition function is given by [see Eq. (2.6)] X Z1 = exp(−βεn) . (2.21) n In this expression Z1 is obtained by summing over all single particle orbital states, which are denoted by the vector of quantum numbers n = (nx,ny,nz). These states are called orbitals . Since the total number 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 ... - tailieumienphi.vn