The Sackur-Tetrode equation is a formula for calculating the entropy of an ideal gas .
S.
{\ displaystyle S}
It is:
S.
(
E.
,
V
,
N
)
=
k
B.
N
ln
[
(
V
N
)
(
E.
N
)
3
2
]
+
3
2
k
B.
N
(
5
3
+
ln
4th
π
m
3
H
2
)
{\ displaystyle S (E, V, N) = k _ {\ mathrm {B}} N \ ln \ left [\ left ({\ frac {V} {N}} \ right) \ left ({\ frac {E } {N}} \ right) ^ {\ frac {3} {2}} \ right] + {\ frac {3} {2}} k _ {\ mathrm {B}} N \ left ({\ frac {5 } {3}} + \ ln {\ frac {4 \ pi m} {3h ^ {2}}} \ right)}
With:
Otto Sackur and Hugo Tetrode set up this complex equation independently of one another.
Inferences
Since the entropy of the variables is known, temperature, pressure and chemical potential can be derived ( see microcanonical ensemble ):
E.
,
V
,
N
{\ displaystyle E, V, N}
1
T
(
1
p
-
μ
)
=
(
∂
E.
∂
V
∂
N
)
S.
(
E.
,
V
,
N
)
{\ displaystyle {\ frac {1} {T}} {\ begin {pmatrix} 1 \\ p \\ - \ mu \ end {pmatrix}} = {\ begin {pmatrix} \ partial _ {E} \\\ partial _ {V} \\\ partial _ {N} \ end {pmatrix}} S (E, V, N)}
The inverse temperature is thus obtained by deriving it from the energy:
1
T
=
(
∂
S.
∂
E.
)
V
,
N
=
3
2
k
B.
N
1
E.
{\ displaystyle {\ frac {1} {T}} = \ left ({\ frac {\ partial S} {\ partial E}} \ right) _ {V, N} = {\ frac {3} {2} } k _ {\ mathrm {B}} N {\ frac {1} {E}}}
This gives the caloric equation of state :
E.
=
3
2
k
B.
N
T
{\ displaystyle E = {\ tfrac {3} {2}} k _ {\ mathrm {B}} NT}
p
T
=
(
∂
S.
∂
V
)
E.
,
N
=
k
B.
N
1
V
{\ displaystyle {\ frac {p} {T}} = \ left ({\ frac {\ partial S} {\ partial V}} \ right) _ {E, N} = k _ {\ mathrm {B}} N {\ frac {1} {V}}}
This gives the thermal equation of state:
p
V
=
k
B.
N
T
{\ displaystyle pV = k _ {\ mathrm {B}} NT}
-
μ
T
=
(
∂
S.
∂
N
)
E.
,
V
=
k
B.
ln
[
(
V
N
)
(
E.
N
)
3
2
]
+
3
2
k
B.
ln
(
4th
π
m
3
H
2
)
=
k
B.
ln
(
V
N
λ
3
)
{\ displaystyle - {\ frac {\ mu} {T}} = \ left ({\ frac {\ partial S} {\ partial N}} \ right) _ {E, V} = k _ {\ mathrm {B} } \ ln \ left [\ left ({\ frac {V} {N}} \ right) \ left ({\ frac {E} {N}} \ right) ^ {\ frac {3} {2}} \ right] + {\ frac {3} {2}} k _ {\ mathrm {B}} \ ln \ left ({\ frac {4 \ pi m} {3h ^ {2}}} \ right) = k _ {\ mathrm {B}} \ ln \ left ({\ frac {V} {N \ lambda ^ {3}}} \ right)}
With the thermal De Broglie wavelength and the relationship for the internal energy , the Sackur-Tetrode equation can also be written as:
λ
=
H
2
π
m
k
B.
T
{\ displaystyle \ lambda = {\ tfrac {h} {\ sqrt {2 \ pi mk _ {\ mathrm {B}} T}}}}
E.
=
3
2
k
B.
N
T
{\ displaystyle E = {\ tfrac {3} {2}} k _ {\ mathrm {B}} NT}
S.
=
k
B.
N
ln
(
V
N
λ
3
)
+
k
B.
N
5
2
{\ displaystyle S = k _ {\ mathrm {B}} N \ ln \ left ({\ frac {V} {N \ lambda ^ {3}}} \ right) + k _ {\ mathrm {B}} N {\ frac {5} {2}}}
Derivation
An atomic ideal gas is located in a closed box (constant volume, no energy or particle exchange with the environment, no external fields). So it is to be described micro-canonically . Here the entropy is calculated from the sum of states over .
N
{\ displaystyle N}
S.
=
k
B.
ln
Z
m
{\ displaystyle S = k _ {\ mathrm {B}} \ ln Z_ {m}}
The micro-canonical partition function is:
Z
m
(
E.
0
)
=
1
N
!
(
2
π
ℏ
)
3
N
∫
R.
6th
N
d
3
x
1
d
3
p
1
...
d
3
x
N
d
3
p
N
δ
(
E.
0
-
H
(
x
→
1
,
p
→
1
,
...
,
x
→
N
,
p
→
N
)
)
{\ displaystyle Z_ {m} (E_ {0}) = {\ frac {1} {N! (2 \ pi \ hbar) ^ {3N}}} \ int _ {\ mathbb {R} ^ {6N}} d ^ {3} x_ {1} d ^ {3} p_ {1} \ ldots d ^ {3} x_ {N} d ^ {3} p_ {N} \; \ delta (E_ {0} -H ( {\ vec {x}} _ {1}, {\ vec {p}} _ {1}, \ ldots, {\ vec {x}} _ {N}, {\ vec {p}} _ {N} ))}
The gas particles are said to be individual atoms (no rotations or vibrations, only translation possible) that do not interact with one another. The associated Hamilton function is:
H
(
x
→
1
,
p
→
1
,
...
,
x
→
N
,
p
→
N
)
=
∑
i
=
1
N
p
→
i
2
2
m
{\ displaystyle H ({\ vec {x}} _ {1}, {\ vec {p}} _ {1}, \ ldots, {\ vec {x}} _ {N}, {\ vec {p} } _ {N}) = \ sum _ {i = 1} ^ {N} {\ frac {{\ vec {p}} _ {i} ^ {\; 2}} {2m}}}
Inserted in the sum of states:
Z
m
(
E.
0
)
=
1
N
!
(
2
π
ℏ
)
3
N
∫
R.
3
N
d
3
x
1
...
d
3
x
N
⏟
V
N
∫
R.
3
N
d
3
p
1
...
d
3
p
N
δ
(
E.
0
-
∑
i
=
1
N
p
→
i
2
2
m
)
{\ displaystyle Z_ {m} (E_ {0}) = {\ frac {1} {N! (2 \ pi \ hbar) ^ {3N}}} \ underbrace {\ int _ {\ mathbb {R} ^ { 3N}} d ^ {3} x_ {1} \ ldots d ^ {3} x_ {N}} _ {V ^ {N}} \ int _ {\ mathbb {R} ^ {3N}} d ^ {3 } p_ {1} \ ldots d ^ {3} p_ {N} \; \ delta \ left (E_ {0} - \ sum _ {i = 1} ^ {N} {\ frac {{\ vec {p} } _ {i} ^ {\; 2}} {2m}} \ right)}
The location integrations could be carried out easily. Now one goes over to -dimensional spherical coordinates in order to simplify the momentum integration. The radius is , so a volume element is written as a radius element times a surface element .
3
N
{\ displaystyle 3N}
p
=
(
∑
i
=
1
N
p
→
i
2
)
1
/
2
{\ displaystyle p = (\ sum \ nolimits _ {i = 1} ^ {N} {\ vec {p}} _ {i} ^ {\; 2}) ^ {1/2}}
d
p
{\ displaystyle dp}
p
3
N
-
1
d
Ω
3
N
{\ displaystyle p ^ {3N-1} d \ Omega _ {3N}}
Z
m
(
E.
0
)
=
V
N
N
!
(
2
π
ℏ
)
3
N
∫
d
Ω
3
N
∫
0
∞
d
p
p
3
N
-
1
δ
(
E.
0
-
p
2
/
2
m
)
{\ displaystyle Z_ {m} (E_ {0}) = {\ frac {V ^ {N}} {N! (2 \ pi \ hbar) ^ {3N}}} \ int d \ Omega _ {3N} \ int _ {0} ^ {\ infty} dp \, p ^ {3N-1} \, \ delta (E_ {0} -p ^ {2} / 2m)}
The integral over is the surface ( sphere ) of a 3N-dimensional unit sphere and is:
d
Ω
3
N
{\ displaystyle d \ Omega _ {3N}}
S.
3
N
-
1
=
2
π
3
N
2
Γ
(
3
N
2
)
=
2
π
3
N
2
(
3
N
2
-
1
)
!
{\ displaystyle S_ {3N-1} = {\ frac {2 \ pi ^ {\ frac {3N} {2}}} {\ Gamma ({\ frac {3N} {2}})}} = {\ frac {2 \ pi ^ {\ frac {3N} {2}}} {({\ frac {3N} {2}} - 1)!}}}
The delta function can be rewritten as:
δ
(
E.
0
-
p
2
/
2
m
)
=
m
2
m
E.
0
[
δ
(
2
m
E.
0
-
p
)
+
δ
(
2
m
E.
0
+
p
)
]
{\ displaystyle \ delta (E_ {0} -p ^ {2} / 2m) = {\ frac {m} {\ sqrt {2mE_ {0}}}} \ left [\ delta ({\ sqrt {2mE_ {0 }}} - p) + \ delta ({\ sqrt {2mE_ {0}}} + p) \ right]}
If inserted into the sum of conditions, this gives:
Z
m
(
E.
0
)
=
V
N
N
!
(
2
π
ℏ
)
3
N
2
π
3
N
2
(
3
N
2
-
1
)
!
m
2
m
E.
0
∫
0
∞
d
p
p
3
N
-
1
[
δ
(
2
m
E.
0
-
p
)
+
δ
(
2
m
E.
0
+
p
)
]
⏟
2
m
E.
0
3
N
-
1
=
V
N
N
!
(
2
π
ℏ
)
3
N
(
2
π
m
E.
0
)
3
N
2
(
3
N
2
)
!
3
N
2
E.
0
{\ displaystyle {\ begin {aligned} Z_ {m} (E_ {0}) & = {\ frac {V ^ {N}} {N! (2 \ pi \ hbar) ^ {3N}}} {\ frac {2 \ pi ^ {\ frac {3N} {2}}} {({\ frac {3N} {2}} - 1)!}} {\ Frac {m} {\ sqrt {2mE_ {0}}} } \ underbrace {\ int _ {0} ^ {\ infty} dp \, p ^ {3N-1} \, \ left [\ delta ({\ sqrt {2mE_ {0}}} - p) + \ delta ( {\ sqrt {2mE_ {0}}} + p) \ right]} _ {{\ sqrt {2mE_ {0}}} ^ {3N-1}} \\ & = {\ frac {V ^ {N}} {N! (2 \ pi \ hbar) ^ {3N}}} {\ frac {(2 \ pi mE_ {0}) ^ {\ frac {3N} {2}}} {({\ frac {3N} { 2}})!}} {\ Frac {3N} {2E_ {0}}} \ end {aligned}}}
In the limit of large numbers of particles, the factorial can be expanded to the second order using the Stirling formula :
N
!
≈
N
N
e
-
N
2
π
N
{\ displaystyle N! \ approx N ^ {N} e ^ {- N} {\ sqrt {2 \ pi N}}}
Z
m
(
E.
0
)
=
V
N
N
N
e
-
N
2
π
N
(
2
π
ℏ
)
3
N
(
2
π
m
E.
0
)
3
N
2
(
3
N
2
)
3
N
2
e
-
3
N
2
3
π
N
3
N
2
E.
0
=
(
V
N
)
N
(
4th
π
m
E.
0
3
N
(
2
π
ℏ
)
2
)
3
N
2
e
5
N
2
3
2
6th
π
E.
0
{\ displaystyle Z_ {m} (E_ {0}) = {\ frac {V ^ {N}} {N ^ {N} e ^ {- N} {\ sqrt {2 \ pi N}} (2 \ pi \ hbar) ^ {3N}}} {\ frac {(2 \ pi mE_ {0}) ^ {\ frac {3N} {2}}} {({\ frac {3N} {2}}) ^ {\ frac {3N} {2}} e ^ {- {\ frac {3N} {2}}} {\ sqrt {3 \ pi N}}}} {\ frac {3N} {2E_ {0}}} = \ left ({\ frac {V} {N}} \ right) ^ {N} \ left ({\ frac {4 \ pi mE_ {0}} {3N (2 \ pi \ hbar) ^ {2}}} \ right) ^ {\ frac {3N} {2}} e ^ {\ frac {5N} {2}} {\ frac {3} {2 {\ sqrt {6}} \ pi E_ {0}}}}
The entropy now results from:
S.
=
k
B.
ln
Z
m
(
E.
0
)
=
k
B.
N
ln
(
V
N
)
+
k
B.
3
N
2
ln
(
4th
π
m
E.
0
3
N
(
2
π
ℏ
)
2
)
+
k
B.
5
N
2
+
k
B.
ln
(
3
2
6th
π
E.
0
)
{\ displaystyle S = k _ {\ mathrm {B}} \ ln Z_ {m} (E_ {0}) = k _ {\ rm {B}} N \ ln \ left ({\ frac {V} {N}} \ right) + k _ {\ rm {B}} {\ frac {3N} {2}} \ ln \ left ({\ frac {4 \ pi mE_ {0}} {3N (2 \ pi \ hbar) ^ { 2}}} \ right) + k _ {\ mathrm {B}} {\ frac {5N} {2}} + k _ {\ mathrm {B}} \ ln \ left ({\ frac {3} {2 {\ sqrt {6}} \ pi E_ {0}}} \ right)}
The last summand can be neglected for large ones. Re-sorting provides the Sackur-Tetrode equation:
N
{\ displaystyle N}
S.
=
k
B.
N
ln
[
(
V
N
)
(
E.
0
N
)
3
2
]
+
3
2
k
B.
N
[
ln
(
4th
π
m
3
(
2
π
ℏ
)
2
)
+
5
3
]
{\ displaystyle S = k _ {\ mathrm {B}} N \ ln \ left [\ left ({\ frac {V} {N}} \ right) \ left ({\ frac {E_ {0}} {N} } \ right) ^ {\ frac {3} {2}} \ right] + {\ frac {3} {2}} k _ {\ mathrm {B}} N \ left [\ ln \ left ({\ frac { 4 \ pi m} {3 (2 \ pi \ hbar) ^ {2}}} \ right) + {\ frac {5} {3}} \ right]}
The case of a harmonic trap potential is discussed as an extension in FIG.
Individual evidence
↑ Martin Ligare: Classical thermodynamics of particles in harmonic traps . In: American Journal of Physics . 78, No. 8, 2010, p. 815. doi : 10.1119 / 1.3417868 .
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