Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Statistical Mechanics - Physics at Oregon State University
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
3.7. IDEAL GAS IN FIRST APPROXIMATION. 59<br />
the ideal gas laws are simply the limits of the correct functional dependencies to<br />
zero density. It is just fortun<strong>at</strong>e th<strong>at</strong> almost every gas is close to ideal, otherwise<br />
the development of thermal physics would have been completely different!<br />
The quantum mechanical model of an ideal gas is simple. Take a box of<br />
dimensions L × L × L, and put one particle in th<strong>at</strong> box. If L is large, this is<br />
certainly a low density approxim<strong>at</strong>ion. The potential for this particle is defined<br />
as 0 inside the box and ∞ outside. The energy levels of this particle are given<br />
by<br />
ɛ(nx, ny, nz) = 2<br />
<br />
π<br />
2 (n<br />
2M L<br />
2 x + n 2 y + n 2 z) (3.85)<br />
with nx ,ny, and nz taking all values 1,2,3,..... The partition function is easy to<br />
calcul<strong>at</strong>e in this case<br />
<br />
nx<br />
2 − 2Mk e B T ( π<br />
L) 2 n 2<br />
x<br />
Z1 = <br />
<br />
ny<br />
nxnynz<br />
e −βɛ(nxnynz) =<br />
2 − 2Mk e B T ( π<br />
L) 2 n 2<br />
y<br />
<br />
nz<br />
2 − 2Mk e B T ( π<br />
L) 2 n 2<br />
z (3.86)<br />
Define α2 =<br />
2 π 2<br />
(2ML2kBT ) and realize th<strong>at</strong> the three summ<strong>at</strong>ions are independent.<br />
This gives<br />
<br />
∞<br />
Z1 = e −α2n 2<br />
n=1<br />
3<br />
(3.87)<br />
We cannot evalu<strong>at</strong>e this sum, but if the values of α are very small, we can use<br />
a standard trick to approxim<strong>at</strong>e the sum. Hence, wh<strong>at</strong> are the values of α?<br />
If we take ≈ 10 −34 Js, M ≈ 5 × 10 −26 kg (for A around 50), and L = 1m,<br />
we find th<strong>at</strong> α 2 ≈ 10−38 J<br />
kBT , and α ≪ 1 transl<strong>at</strong>es into kBT ≫ 10 −38 J. With<br />
kB ≈ 10 −23 JK −1 this transl<strong>at</strong>es into T ≫ 10 −15 K, which from an experimental<br />
point of view means always. Keep in mind, though, th<strong>at</strong> one particle per m 3 is<br />
an impossibly low density!<br />
In order to show how we calcul<strong>at</strong>e the sum, we define xn = αn and ∆x = α<br />
and we obtain<br />
∞<br />
e −α2n 2<br />
n=1<br />
= 1<br />
α<br />
∞<br />
n=1<br />
e −x2n∆x<br />
(3.88)<br />
In the limit th<strong>at</strong> ∆x is very small, numerical analysis shows th<strong>at</strong><br />
∞<br />
n=1<br />
e −x2n∆x<br />
≈<br />
∞<br />
e −x2<br />
0<br />
dx − 1<br />
<br />
1 −<br />
∆x + O e ∆x<br />
2<br />
(3.89)<br />
where the integral is equal to 1√<br />
2 π. The term linear in ∆x is due to the fact th<strong>at</strong><br />
in a trapezoidal sum rule we need to include the first (n=0) term with a factor