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Section 11: Ideal Gas in the Low Density Limit In the last lecture the calculation of Z(1) for the ideal gas was begun. In this lecture we ﬁnish the calculation and discuss when the semi-classical treatment is valid. The concept of a density of states is also introduced. 11. 1. Calculation of Z(1) We ﬁrst ﬁnish the job begun in the last lecture 2 2 π ∞ 2 2 π Z(1) = exp −an n dn where a = 2 0 2kT ML2 The gaussian integral that should be familiar is ∞ π 1/2 exp(−an2 )dn = −∞ a and we see √ 2 1 d ∞ 2 1 d π 1/2 π Z(1) = − exp(−an )dn = − = π 2 da −∞ 2 da a 4a3/2 (the factor half comes from the limit 0 to inﬁnity in the integral for Z(1)). Inserting the expression for a gives 3/2 3/2 π 3/2 2ML2 2πMkT Z(1) = =V 8 β 2π 2 h2 where h = 2π has been used. 11. 2. Density of states Consider again the approximation of a sum by an integral 1 ∞ A(n) −→ A(n)4πn2 dn nx ,ny ,nz 8 0 where A(n) is any function of n. Now change variables to k (see 10.3) L dn L n= k dn = dk = dk π dk π and we ﬁnd 1 L2 k 2 L A(n) −→ A(k)4π dk = A(k)Γ(k)dk nx ,ny ,nz 8 π2 π 2 k where Γ(k)dk = V dk is the number states with k between k and k + dk. 2π 2 Γ(k) is known as the ‘density of states’ (here, in k space). 41 Similarly we can change variables to ǫ 2 2 1/2 k 2M ǫ= k= ǫ1/2 2M 2 1/2 dk 2M 1 dk = dǫ = dǫ dǫ 2 2ǫ1/2 Now to obtain the density of states in energy space, denoted here by g(ǫ), equate Γ(k)dk = g(ǫ)dǫ and change variables 3/2 2M V 1/2 g(ǫ)dǫ = ǫ dǫ 2 4π 2 and the meaning of the density of states g(ǫ) is that g(ǫ)dǫ = is the number of states with energy between ǫ and ǫ + dǫ. If you unsure of this change of variables from Γ(k) to g(ǫ), one can carry it out as was done in the lecture, ‘under the integral sign’ i.e. consider for any function A(k) dk A(k)Γ(k)dk = A(k)Γ(k) dǫ = A(ǫ)g(ǫ) dǫ dǫ Using the density of states one can write e.g. Z(1) as ∞ Z(1) = exp(−βǫ)g(ǫ)dǫ 0 Notice that the density of states increases with energy, that is at higher energies there are more states available to the particle (more ways to distribute the energy amongst the original quantum numbers nx , ny , nz ). 4 3 g(ε) 2 1 0 0 2 4 6 8 10 ε Figure 1: The density of states as a function of energy Here we have derived the density of states for a particular volume, a cube. However it turns out that the density of states has the same expression for any macroscopic volume V (e.g. sphere, rectangular box . . . ). 42 11. 3. Thermodynamic variables In the semi-classical treatment we have developed over this and lecture 10 3N/2 Z(1)N VN 2πMkT Z= = N! N! h2 We now proceed to calculate the usual thermodynamic variables F = −kT ln Z 1 3 2πMkT = NkT ln − ln + ln N − 1 V 2 h2 N 3 2πMkT = NkT ln −1− ln V 2 h2 where in the second line we used Stirling’s approximation for ln N! ∂ ln Z ∂ 3N 3NkT E = kT 2 = kT 2 ln T + · · · = ∂T ∂T 2 2 3/2 E −F V 2πMkT 5 S= = Nk ln + T N h2 2 ∂F P = − (see question 5.5) ∂V T NkT = (ideal gas law) V 3Nk CV = (equipartition) 2 Note that • The ideal gas law and equipartition of energy are recovered • The recovery of the ideal gas law ﬁnally identiﬁes our statistical mechanical deﬁnition of temperature (see lecture 4 key point 9) with the thermodynamic temperature. 11. 4. Examination of entropy First note that in the formula for S in the previous subsection takes the form S(N, V ) = Ns(N/V ) i.e. once we take out the overall factor of N the remaining dependence on N, V enters through the density N/V . This means the entropy is additive (or extensive) which is to say it scales as N (see question 6.1 for how this property comes from the N! factor of the semi-classical treatment). Other variables which should be additive are the energy, and free energy. 43 N, V N, V 2N, 2V S S 2S Figure 2: Additivity of the entropy However an undesirable property of the entropy is that it can become negative when 3/2 V 2πmkT 1 N h2 1/3 V h or N √ ♯ 2πmkT To interpret ♯ consider ﬁrst the left hand side: it is the cube root of the volume per particle which is a measure of the typical particle separation, and we call the lhs dtyp To interepret the rhs recall the average kinetic energy of a molecule is 3kT /2 thus the typical momentum ∼ (3mkT )1/2 . From quantum mechanics you will have met the de Broglie wavelength h h de Broglie wavelength = = . p (3mkT )1/2 Therefore we may identify the rhs of ♯ as the typical de Broglie wavelength of a gas molecule λtyp (for our purposes 3 is approximately 2π). Thus the results of the semi-classical treatment become unphysical when dtyp λtyp To understand this we refer to ﬁgure 3. λ typ d typ λ typ Figure 3: Picture of low density regime where wavepackets of individual particles do not overlap. (After Baierlein ﬁg 5.5) If dtyp > λtyp the wavepackets of individual particles do not overlap and the particles are ‘partly’ or ‘eﬀectively’ distinguishable i.e. we can identify individual particles although we cannot keep track of their labels during scattering events (see lecture 10). This regime is the low density limit. If dtyp < λtyp then the wavepackets overlap and we must consider a many-particle wavefunc- tion i.e. a full quantum treatment. This is the high density regime. 44

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Section 11 Ideal Gas in the Low Density Limit In the last lecture

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