Partition Function, Entropy, Distinguishability, Planck Distribution, Heat Capacity of Platinum

Running Head: PVB 301 1
PVB 301 Assignment 2
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PVB 301 2
1. PARTITION FUNCTION OF A 2D GAS
The partition equation for a system of particles that obeys Boltzmann statistics has a
definition (Harris, 2012).
Z=∑
j
g j e−ε j/ kBT
....................................... (1)
Where there are ∆G j energy states within the macro level then
Z=∑
j
∆ G j e−ε j/ kB T
......................................... (2) And ε j= h2 A−1
8 m n j
2
......................................................... (3) For single particle in a box using the
principle of quantum mechanics. For a gas confined in a 2D area;
G j= 1
4 π n j
2....................................................... (4)
∆ G j= 1
2 π nj ∆ n j substituting ∆ G j in the second equation;
Z= π
2 ∑
j
n j ∆ nj e−ε j /kB T
............................... (5)
Substituting ε jin equation (3) into equation (5);
Z= π
2 ∑
j
n j ∆ nj e−¿ h2 A−1
8 mk B T nj
2 ¿
Integrating from 0 to ∞; Z= π
2 ∫
0
∞
nj exp (−h2 A−1
8 mk B T nj
2
)d n j;
Z=A ( 2 πm kB T
h2 )
2
2 = A
( h2
2 πm kB T )
= A
λth
2 Where λth= h
√ (2 πmk B T )
2. ENTROPY AND DISTINGUISHABILITY

PVB 301 3
a) Probability density for distributing N1 particles in a sub volume V1, and N2 in
V2, is given by the binomial distribution (Santos, Dorini, & Cunha, 2008);
W = N !
N1 ! N2 ! (V 1
V )N1
( V 2
V )N2
¿ W =¿ Ωc ( V 1 N1 ) +¿ Ωc ( V 2 N 2 ) −¿ Ωc ( V 1 N ) Ωc(V , N )=( V N
N ! )
Entire entropy;Ωp (U , N )=
( (2 πmU )
3 N
2
(3 N
2 )! )
Assuming N>>1 Ωp ( U , V , N ) =
( Ωc (V , N)Ωp (U , N )
h3 N )
Substituting : S=kInΩ ( U ,V , N )This gives the same Sackur-Tetrode equation for distinguishable
particles as for indistinguishable ones.
b) Helmholtz energy: F=U – TS
dA=−SdT − pdV =−SdT − pdV +∑
j
u j dN j
Where N j=no of particles
u j=corresponding chemical potentials
dA=−SdT −∑
i
ui dNi +∑
j
u j dN jCITATION Vid 08 ¿ 1033(Vidal , 2008)
Probability to find the system in some energy Eigen state r
Pr = e−βEr
Z Where β= 1
kT , Z=∑
r
e− βr
U = ⟨ E ⟩ =∑
r
Pr Xr= 1
β
∂ logz
∂ x