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Energy Distribution of an Ideal-Gas with n-degrees of Freedom
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Γ
(
x
)
{\displaystyle \Gamma (x)}
and
γ
(
x
)
{\displaystyle \gamma (x)}
, are the upper and lower incomplete gamma function .
With the value
Γ
(
3
2
)
=
π
2
{\displaystyle \Gamma ({\frac {3}{2}})={\frac {\sqrt {\pi }}{2}}}
The PDF and the CDF of the Chi-squared distribution are:
χ
P
D
F
2
(
x
)
=
1
2
k
2
Γ
(
k
2
)
x
k
2
−
1
exp
(
−
x
2
)
{\displaystyle \chi _{PDF}^{2}(x)={\frac {1}{{2}^{\frac {k}{2}}~\Gamma ({\frac {k}{2}})}}~{x}^{{\frac {k}{2}}-1}~\exp(-{\frac {x}{2}})}
χ
C
D
F
2
(
x
)
=
1
Γ
(
k
2
)
γ
(
k
2
,
x
)
{\displaystyle \chi _{CDF}^{2}(x)={\frac {1}{\Gamma ({\frac {k}{2}})}}~\gamma ({\frac {k}{2}},x)}
CALCULUS TRY 1: Velocity Distribution of Ideal-Gas
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The distribution for a velocity in a single direction (Degrees of Freedom n=1)
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f
(
V
i
)
=
m
2
π
k
T
exp
[
−
m
V
i
2
2
k
T
]
{\displaystyle f(V_{i})={\sqrt {\frac {m}{2~\pi ~kT}}}~\exp \left[-{\frac {m~V_{i}^{2}}{2~kT}}\right]}
The distribution for absolute velocity (Degrees of Freedom n=3)
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f
(
v
)
=
(
m
2
π
k
T
)
3
2
4
π
v
2
exp
[
−
m
v
2
2
k
T
]
{\displaystyle f(v)={({\frac {m}{2~\pi ~kT}})}^{\frac {3}{2}}~4~\pi ~v^{2}\exp \left[-{\frac {m~v^{2}}{2~kT}}\right]}
The Energy Distribution
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using:
ϵ
=
1
2
m
v
2
=
1
2
m
(
V
x
2
+
V
y
2
+
V
y
2
)
{\displaystyle \epsilon ={\frac {1}{2}}~m~v^{2}={\frac {1}{2}}~m~(V_{x}^{2}+V_{y}^{2}+V_{y}^{2})}
k
=
3
{\displaystyle k=3}
and a scale parameter
a
=
k
T
m
{\displaystyle a={\sqrt {\frac {kT}{m}}}}
f
(
v
)
=
(
1
2
π
)
1
2
2
v
2
a
3
exp
[
−
1
2
(
v
a
)
2
]
{\displaystyle f(v)={({\frac {1}{2~\pi }})}^{\frac {1}{2}}~2~{\frac {v^{2}}{a^{3}}}~\exp \left[-{\frac {1}{2}}~{({\frac {v}{a}})}^{2}\right]}
f
ϵ
(
ϵ
k
T
)
=
2
π
ϵ
k
T
exp
[
−
ϵ
k
T
]
{\displaystyle f_{\epsilon }({\frac {\epsilon }{kT}})={\frac {2}{\sqrt {\pi }}}~{\sqrt {\frac {\epsilon }{kT}}}~\exp \left[-{\frac {\epsilon }{kT}}\right]}
Rewriting the Energy Distribution as Maxwell-Boltzmann Distribution
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Using
k
=
3
{\displaystyle k=3}
and
x
=
2
ϵ
k
T
{\displaystyle x=2~{\frac {\epsilon }{kT}}}
.
Equation (x) can be rewritten as
f
ϵ
(
x
2
)
=
1
Γ
k
2
x
k
2
−
1
exp
(
x
2
)
=
χ
P
D
F
2
(
x
)
{\displaystyle f_{\epsilon }({\frac {x}{2}})={\frac {1}{\Gamma {\frac {k}{2}}}}~{x}^{{\frac {k}{2}}-1}~\exp({\frac {x}{2}})=\chi _{PDF}^{2}(x)}
So the Energy Cumulative Distribution Function (CDF) of three degree of freedom
n
=
3
{\displaystyle n=3}
is:
F
ϵ
(
x
2
)
=
χ
C
D
F
2
(
x
)
=
1
Γ
k
2
γ
(
k
2
,
x
)
{\displaystyle F_{\epsilon }({\frac {x}{2}})=\chi _{CDF}^{2}(x)={\frac {1}{\Gamma {\frac {k}{2}}}}~\gamma ({\frac {k}{2}},x)}
F
ϵ
(
ϵ
k
T
)
=
1
Γ
3
2
γ
(
3
2
,
2
ϵ
k
T
)
{\displaystyle F_{\epsilon }({\frac {\epsilon }{kT}})={\frac {1}{\Gamma {\frac {3}{2}}}}~\gamma ({\frac {3}{2}},2~{\frac {\epsilon }{kT}})}
Energy Distribution of n-Degrees of Freedom
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With the energy of a single degree
E
n
=
1
{\displaystyle E_{n=1}}
ϵ
=
3
E
n
=
1
{\displaystyle \epsilon =3~E_{n=1}}
We have
E
n
=
1
=
ϵ
×
1
3
{\displaystyle E_{n=1}=\epsilon \times {\frac {1}{3}}~}
I believe the following is NOT VALID?
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the energy in a diatomic gas with n=5 is
E
n
=
5
=
5
×
E
n
=
1
=
5
3
×
ϵ
{\displaystyle E_{n=5}=5\times E_{n=1}={\frac {5}{3}}\times \epsilon }
ϵ
=
E
n
=
5
×
3
5
{\displaystyle \epsilon =E_{n=5}\times {\frac {3}{5}}}
F
ϵ
(
ϵ
k
T
)
=
1
Γ
3
2
γ
(
3
2
,
2
3
5
E
n
=
5
k
T
)
{\displaystyle F_{\epsilon }({\frac {\epsilon }{kT}})={\frac {1}{\Gamma {\frac {3}{2}}}}\gamma ({\frac {3}{2}},2{\frac {3}{5}}~{\frac {E_{n=5}}{kT}})}