Statistical Molecular Thermodynamicspollux.chem.umn.edu/4501/Lectures/ThermoVid_4_03.pdf · 2015....
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Statistical Molecular Thermodynamics Christopher J. Cramer Video 4.3 Ideal Monatomic Gas: Properties
Transcript of Statistical Molecular Thermodynamicspollux.chem.umn.edu/4501/Lectures/ThermoVid_4_03.pdf · 2015....
Statistical Molecular Thermodynamics
Christopher J. Cramer
Video 4.3
Ideal Monatomic Gas: Properties
Monatomic Ideal Gas Q
€
Q(N,V ,T) =qtrans(V ,T)qelec(T)[ ]N
N!
We have derived that for a monatomic ideal gas:
( ) VhTmkTVq
2/3
2B
trans2, !
"
#$%
&=π
€
qelec(T) ≈ g1 + g2e−βε 2
where
Monatomic Ideal Gas Internal Energy U
€
U = kBT2 ∂lnQ
∂T#
$ %
&
' ( N ,V
= NkBT2 ∂lnq
∂T#
$ %
&
' ( V
= NkBT2 ∂ln qtransqelec( )
∂T
)
* + +
,
- . . V
€
U = NkBT2 ∂∂Tln 2πmkBT
h2$
% &
'
( ) 3 / 2
V g1 + g2e−βε 2 +!( )
-
. /
0
1 2
3 4 5
6 5
7 8 5
9 5 V
=32NkBT +!
qtrans qelec
monatomic gas )(),(),( electrans TqTVqTVq =!
),(),,(NTVqTVNQ
N
=indistinguishable, non-interacting
molecules/atoms
?
Monatomic Ideal Gas Internal Energy U
€
U = kBT2 ∂lnQ
∂T#
$ %
&
' ( N ,V
= NkBT2 ∂lnq
∂T#
$ %
&
' ( V
= NkBT2 ∂ln qtransqelec( )
∂T
)
* + +
,
- . . V
qelec contribution typically small
TNkU B23
≈Dominated by translational contribution.
Fraction in excited electronic states usually very small at “low” temperatures
monatomic gas
€
U = NkBT2 ∂∂Tln 2πmkBT
h2$
% &
'
( ) 3 / 2
V g1 + g2e−βε 2 +!( )
-
. /
0
1 2
3 4 5
6 5
7 8 5
9 5 V
=32NkBT +
Ng2ε2e−βε 2
qelec+!
qtrans qelec
!),(),,(
NTVqTVNQ
N
= )(),(),( electrans TqTVqTVq =
indistinguishable, non-interacting
molecules/atoms
Monatomic Ideal Gas Internal Energy U
qelec contribution typically small
qtrans qelec
€
U ≈32NkBT for N = NA, T = 298.15 K, energy in kJ
3.719 3.719
2.081 x 10–26 0.332
€
32NkBT
€
Ng2ε2e−βε 2
qelec
Li ε2 = 14903.66 cm–1
g2 = 2
F ε2 = 404.0 cm–1
g2 = 2
€
U = NkBT2 ∂∂Tln 2πmkBT
h2$
% &
'
( ) 3 / 2
V g1 + g2e−βε 2 +!( )
-
. /
0
1 2
3 4 5
6 5
7 8 5
9 5 V
=32NkBT +
Ng2ε2e−βε 2
qelec+!
qelec=4.142
Monatomic Ideal Gas Heat Capacity CV
€
CV =∂U∂T#
$ %
&
' ( N ,V
=∂32NkBT + Ng2ε2e
−ε 2 / kBT /qelec +!#
$ %
&
' (
∂T
+
,
- - - -
.
/
0 0 0 0 N ,V
=32NkB +
Ng2ε22
kBT2qelec
e−ε 2 / kBT +!
€
U =32NkBT +
Ng2ε2e−ε 2 / kBT
qelec+!
12.47 12.47
5.020 x 10–27 2.172
for N = NA, T = 298.15 K, CV in J•K–1
€
32NkB
€
Ng2ε22
kBT2qelec
e−ε 2 / kBT +!
Li ε2 = 14903.66 cm–1
g2 = 2
F ε2 = 404.0 cm–1
g2 = 2
Monatomic Ideal Gas Pressure P
€
P = kBT∂lnQ∂V
#
$ %
&
' ( N ,T
= NkBT∂lnq∂V
#
$ %
&
' ( T
= NkBT∂ln qtransqelec( )
∂V
)
* + +
,
- . . T
monatomic gas
€
P = NkBT∂∂Vln 2πmkBT
h2$
% &
'
( ) 3 / 2
V g1 + g2e−βε 2 +!( )
-
. /
0
1 2
3 4 5
6 5
7 8 5
9 5 T
= NkBT∂∂VlnV
$
% &
'
( ) T
=NkBTV
!),(),,(
NTVqTVNQ
N
= )(),(),( electrans TqTVqTVq =
indistinguishable, non-interacting
molecules/atoms
the only function of V
VTNkP B= Ideal Gas Law (PV=nRT)
Summary: Monatomic Ideal Gas
€
Q(N,V ,T) =qtrans(V ,T)qelec(T)( )N
N!
( ) VhTmkTVq
2/3
2B
trans2, !
"
#$%
&=π
€
qelec(T) ≈ g1 + g2e−βε 2
Partition function:
TNkU B23
=
B23 NkCV =
VTNkP B=
Energy
Heat Capacity
Pressure
RTU23
=
RCV 23
=
€
P =RTV
(molar)
Next: Ideal Diatomic Gas: Part 1
