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

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Section 11 Ideal Gas in the Low Density Limit In the last lecture Powered By Docstoc
					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.




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