The Van-'t-Hoff equation or Van-'t-Hoff'sche or van-'t-Hoffsche reaction isobaric (after Jacobus Henricus van 't Hoff ) describes in thermodynamics the connection between the position of the equilibrium of a chemical Reaction and temperature (at constant pressure):
(
∂
ln
K
∂
T
)
p
=
Δ
r
H
0
(
T
)
R.
T
2
{\ displaystyle {\ Bigl (} {\ frac {\ partial \ ln K} {\ partial T}} {\ Bigr)} _ {p} = {\ frac {\ Delta _ {r} H ^ {0} ( T)} {RT ^ {2}}}}
in which
K
{\ displaystyle K}
the equilibrium constant ,
T
{\ displaystyle T}
the temperature ,
Δ
r
H
0
(
T
)
{\ displaystyle \ Delta _ {r} H ^ {0} (T)}
the standard molar enthalpy of reaction as a function of temperature (the standard condition pressure is fulfilled) and
T
{\ displaystyle T}
p
0
=
1
bar
{\ displaystyle p_ {0} = 1 \, {\ text {bar}}}
R.
{\ displaystyle R}
is the general gas constant .
The index p stands for the constant pressure. Another formulation of the Van't Hoff equation for the inverse temperature with the Boltzmann constant is:
β
=
(
k
B.
T
)
-
1
{\ displaystyle \ beta = (k _ {\ mathrm {B}} T) ^ {- 1}}
k
B.
{\ displaystyle k _ {\ mathrm {B}}}
(
∂
ln
K
∂
β
)
p
=
-
Δ
r
H
0
(
β
)
N
A.
,
{\ displaystyle \ left ({\ frac {\ partial \ ln K} {\ partial \ beta}} \ right) _ {p} = - {\ frac {\ Delta _ {r} H ^ {0} (\ beta )} {N _ {\ mathrm {A}}}},}
where is Avogadro's constant .
N
A.
{\ displaystyle N _ {\ mathrm {A}}}
Derivation
The following generally applies to the equilibrium constant :
K
{\ displaystyle K}
ln
K
=
-
Δ
r
G
0
(
T
)
R.
T
{\ displaystyle \ ln K = - {\ frac {\ Delta _ {r} G ^ {0} (T)} {RT}}}
Their partial derivative according to the temperature at constant pressure results in:
(
∂
ln
K
∂
T
)
p
=
+
Δ
r
G
0
(
T
)
R.
T
2
-
(
∂
Δ
r
G
0
(
T
)
∂
T
)
p
R.
T
{\ displaystyle {\ Bigl (} {\ frac {\ partial \ ln K} {\ partial T}} {\ Bigr)} _ {p} = + {\ frac {\ Delta _ {r} G ^ {0} (T)} {RT ^ {2}}} - {\ frac {{\ Bigl (} {\ frac {\ partial \ Delta _ {r} G ^ {0} (T)} {\ partial T}} { \ Bigr)} _ {p}} {RT}}}
The derivation of the molar, free enthalpy of reaction according to the temperature at constant pressure is calculated as follows:
(
∂
G
m
0
∂
T
)
p
=
-
S.
m
0
(
T
)
{\ displaystyle {\ Bigl (} {\ frac {\ partial G_ {m} ^ {0}} {\ partial T}} {\ Bigr)} _ {p} = - S_ {m} ^ {0} (T )}
→
(
∂
ln
K
∂
T
)
p
=
+
Δ
r
G
0
(
T
)
R.
T
2
+
T
Δ
r
S.
0
(
T
)
R.
T
2
{\ displaystyle \ rightarrow {\ Bigl (} {\ frac {\ partial \ ln K} {\ partial T}} {\ Bigr)} _ {p} = + {\ frac {\ Delta _ {r} G ^ { 0} (T)} {RT ^ {2}}} + {\ frac {T \ Delta _ {r} S ^ {0} (T)} {RT ^ {2}}}}
With the Gibbs-Helmholtz equation
Δ
r
G
0
(
T
)
=
Δ
r
H
0
(
T
)
-
T
Δ
r
S.
0
(
T
)
{\ displaystyle \ Delta _ {r} G ^ {0} (T) = \ Delta _ {r} H ^ {0} (T) -T \ Delta _ {r} S ^ {0} (T)}
surrendered:
(
∂
ln
K
∂
T
)
p
=
Δ
r
H
0
(
T
)
-
T
Δ
r
S.
0
(
T
)
+
T
Δ
r
S.
0
(
T
)
R.
T
2
=
Δ
r
H
0
(
T
)
R.
T
2
{\ displaystyle {\ Bigl (} {\ frac {\ partial \ ln K} {\ partial T}} {\ Bigr)} _ {p} = {\ frac {\ Delta _ {r} H ^ {0} ( T) -T \ Delta _ {r} S ^ {0} (T) + T \ Delta _ {r} S ^ {0} (T)} {RT ^ {2}}} = {\ frac {\ Delta _ {r} H ^ {0} (T)} {RT ^ {2}}}}
Van't Hoff reaction isochore
If the volume is kept constant during a reaction, the reaction is described by the change in the standard free energy . The result is the van-'t-Hoff'sche reaction isochore :
ln
K
=
-
Δ
r
F.
(
N
i
,
V
,
T
)
0
/
(
R.
T
)
{\ displaystyle \ ln K = - \ Delta _ {r} F (N_ {i}, V, T) ^ {0} / (RT)}
(
∂
ln
K
∂
T
)
V
=
Δ
r
U
0
R.
T
2
{\ displaystyle {\ Bigl (} {\ frac {\ partial \ ln K} {\ partial T}} {\ Bigr)} _ {V} = {\ frac {\ Delta _ {r} U ^ {0}} {RT ^ {2}}}}
solution
The formal solution of the Van't Hoff equation is
K
(
T
)
=
K
(
T
0
)
exp
(
∫
T
0
T
d
T
Δ
r
H
0
(
T
)
R.
T
2
)
{\ displaystyle K (T) = K (T_ {0}) \ exp \ left (\ int _ {T_ {0}} ^ {T} \ mathrm {d} T {\ frac {\ Delta _ {r} H ^ {0} (T)} {RT ^ {2}}} \ right)}
In Ulich's approximation , one assumes a standard reaction enthalpy that is constant - at least in a certain temperature interval.
This results in:
K
(
T
)
=
K
(
T
0
)
exp
(
-
Δ
r
H
0
R.
T
+
Δ
r
H
0
R.
T
0
)
{\ displaystyle K (T) = K (T_ {0}) \ exp \ left (- {\ frac {\ Delta _ {r} H ^ {0}} {RT}} + {\ frac {\ Delta _ { r} H ^ {0}} {RT_ {0}}} \ right)}
See also
Web links
Individual evidence
↑ Peter W. Atkins, Julio de Paula: Physical chemistry . 4th, completely revised edition. Wiley-VCH, 2006, ISBN 3-527-31546-2 , pp. 237 .
↑ Derivation of Van't Hoff's reaction isochors - Chemgapedia. Retrieved February 8, 2019 .
↑ Ulich approximations . In: Lexicon of Chemistry. Retrieved July 25, 2014.
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