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Dec 23, 2018

Van't Hoff Equation

Van't Hoff Equation - Effect on temperature on Equilibrium Constant:

Equilibrium Constant KP of a reaction is constant at a given temperature but when temperature is changed, the value of equilibrium constant also changed. The quantitative relation, relation, Known as Van't Hoff equation is derived by using Gibbs - Helmholtz equation.
The Gibbs - Helmholtz equation is 

ΔG0 = ΔH0 + T[d(ΔG0)/dT]P 

Zero superscript is indicating the stranded values.
or, - (ΔH0/T2 ) = -(ΔG0/T2 ) + 1/T[d(ΔG0)/dT]P 
or, - (ΔH0/T2 )  [d/dTG0/T)]P 
Again Van't Hoff isotherm is, - RT lnKP = ΔG0
or, - R lnKP = ΔG0/T
Differentiating with respect to T at constant P,
- R [dlnKP/dT]P = [d/dTG0/T)]P
Comparing the above two equation we have,
(dlnKP/dT) = (ΔH0/T2)
This is the differential form of Van't Hoff reaction equation.
Greater the value of ΔH0, the faster the equilibrium constant (KP) changes with temperature (T).
Separating the variables and integrating,
dlnKP  = (ΔH0/R) (dT/T2)
(Assuming that ΔH0 is independent of temperature)
or, lnKP  = - (ΔH0/R)(1/T) + C (integrating constant)
The integration constant can be evaluated and identified as ΔS0/R using the relation ΔG0 = ΔH0 - TΔS0.
Thus the Van't hoff Equation is 

lnKP  = - (ΔH0/R)(1/T) + ΔS0/

For a reaction 2A + B ⇆ 2C, ΔG0(500K) = 2 KJ mol-1 
Find the KP at 500K for the reaction A + ½B  C .

ΔG0(500K) for the reaction A + ½B  C  is 2 KJ mol-1/2 = 1 KJ mol-1/2
The relation is ΔG0 = - RT lnKP 
or, KJ mol-1  = - 8.31 × 10-3 KJ mol-1 K-1 × 500K lnKP 
or, lnKP  = 1/(8.31 × 0.5) = 0.2406

∴ KP = 1.27

Plot of lnKP vs 1/T :

For exothermic reaction, ΔH0 = (-) ve.
Examples are the formation of ammonia from Hand N2.
Van't Hoff Equation - Effect on temperature on Equilibrium Constant
Plot of lnKP vs 1/T
For endothermic reaction, ΔH0 = (+) ve.
Examples are the dissociation of HI into Hand I2.
For the reaction, ΔH0 = 0, lnKP is independent of T. Provided ΔS0 does not change much with T.
Since for idel system, H is not a function of P and ΔH0 = ΔH
and Van't Hoff equation is dlnKP/dT = ΔH/RTand the integrated equation is,
lnKP = (ΔH/R)(1/T) + ΔS/R
However S of an ideal gas depends strongly on P, so ΔS and ΔG per mole of reaction in the mixture differ quite substantially from ΔS0 and ΔG0.
If we consider the integrated Van't Hoff equation at two temperature then, it becomes,

ln(KP2/KP1) = (ΔH/R){(T2 -T1)/T1T2)}

Where KP1 and KP2 are the equilibrium constants of the reaction at two different temperature T1 and T2 respectively.
Thus determination of KP1 and KP2 at two temperature helps to calculate the value of ΔH of the reaction.
The above reaction is called Van't Hoff reaction isobar since P remains constant during the change of temperature.

Use Gibbs - Helmholtz equation to derive the Vant Hoff reaction isochore. Under What condition do you expect a linear relationship between logk and 1/T ?

Again KP = KC (RT)ΔƔ
or, lnKP = lnKC + ΔƔ lnR + ΔƔlnT
Differentiating with respect to T,
dlnKP/dT = dlnKC/dT + ΔƔ/T
But dlnKP/dT = ΔH0/RT2
Hence, dlnKC/dT = ΔH0/RT2 - ΔƔ/T = (ΔH0 - ΔƔRT)/RT2
or, dlnKC/dT = ΔU0/RT2
ΔU0 is standerd heat of reaction at constant volume. This is really the Van't Hoff reaction isochore in differential form.
The integrated form of the equation at two temperature is,

ln(KC2/KC1) = (ΔU0/R){(T2 -T1)/T1T2)}
For ideal system ΔU0 = ΔU.
Since the reaction occurs in an equilibrium box at constant Volume this is called Van't Hoff reaction isochore.
Two important assumptions are:
(i) The reacting system is assumed to behave ideally and 
(ii) ΔH is taken independent of temperature for small range of temperature change.
Due to assumption involved ΔH and ΔU do not produce precise value of these reactions.
As ΔG is independent of pressure for ideal system a and b also independent of pressure.
Hence ΔG0 = - RT lnKP 
or, [dlnKP/dT]T = 0

Show that the equilibrium condition for any chemical reaction is given by ΔG = 0.

Vant Hoff reaction isotherm is ΔG = - RT lnKa + RT lnQa
But when the reaction attain equilibrium Qa = Ka
Thus, ΔG = 0.

Expression of Le -Chatelier Principle from Van't Hoff Equation:

Van't Hoff equations gives quantitative expression of Le-Chatelier Principle. From the equation,

lnKP = -(ΔH/R)(1/T) + C

It is evident that for endothermic reaction (ΔH〉0), increase of T increases the value of a of the reaction. But for exothermic reaction (ΔHㄑ 0), with rise in temperature, a is decreased. 
This change of KP also provides the calculation of quantitative change of equilibrium yield of products.
This above statement is in accordance with Le - Chatelier Principle which states that whenever stress is placed on any system in a state of equilibrium, the system is always react in a direction to reduce the applied stress. Thus when T is increased, the system in equilibrium will try to move in a direction in which heat is absorbed that is endothermic reaction is favors.