This equation gives the quantitative temperature dependence of equilibrium constant (K). The relation between standard free energy change (\(\triangle\)G^O) and equilibrium constant is
\(\Delta { G }^{ 0 }=-RTln\ K\)    ...(1)
We know that
\(\Delta { G }^{ 0 }=\Delta { H }^{ 0 }-T\Delta { S }^{ 0 }\)
Substituting (1) in equation (2)
\(-RTln\ K\ =\Delta { H }^{ 0 }-T\Delta { s }^{ 0 }\)
Rearranging
In \(K=\cfrac { -\Delta H^{ 0 } }{ RT } +\cfrac { { \Delta S }^{ 0 } }{ R } \)     ...(3)
Differentiating equation (3) with respect to temperature
\(\cfrac { d\left( In\quad K \right) }{ dT } =\cfrac { \Delta { H }^{ 0 } }{ { RT }^{ 2 } } \)       ...(4)
Equation 4 is known as differential form of Van't Hoff equation.
On integrating the equation 4, between T₁ and T₂ with their respective equilibrium constants K₁ and K_2.
\(\int _{ { k }_{ 1 } }^{ { K }_{ 2 } }{ d\left( In\ K \right) =\cfrac { \Delta { H }^{ 0 } }{ R } \int _{ { T }_{ 2 } }^{ { { T }_{ 2 } } }{ \cfrac { dT }{ { T }^{ 2 } } } } \)
\(\left[ In\quad K \right] _{ { K }_{ 1 } }^{ { K }_{ 2 } }=\cfrac { \Delta { H }^{ 0 } }{ R } \left[ -\cfrac { 1 }{ T } \right] ^{ { T }_{ 2 } }_{ { T }_{ 1 } }\)
\(In\quad { K }_{ 2 }-In\quad { K }_{ 1 }=\cfrac { \Delta { H }^{ 0 } }{ R } -\left[ \cfrac { 1 }{ { T }_{ 2 } } +\cfrac { 1 }{ { T }_{ 2 } } \right] \)
\(In\quad \cfrac { { K }_{ 2 } }{ { K }_{ 1 } } =\cfrac { \Delta { H }^{ 0 } }{ R } \left[ \cfrac { { T }_{ 2 }-{ T }_{ 1 } }{ { T }_{ 2 }{ T }_{ 1 } } \right] \)
\(log\quad \cfrac { { K }_{ 2 } }{ { K }_{ 1 } } =\cfrac { \Delta { H }^{ 0 } }{ 2.303R } \left[ \cfrac { { T }_{ 2 }-{ T }_{ 1 } }{ { T }_{ 2 }{ T }_{ 1 } } \right] \)  ...(5)
Equation (5) is known as integrated form of Van't Hoff equation.

Addition of an inert gas to a reaction at equilibrium, at constant volume has no effect.

If a system at equilibrium is disturbed, then the system shifts itself in a direction that nullifies the effect of that disturbance.

At any instant, the rate of a chemical reaction, at a given temperature is directly proportional to the product of the active masses of the reactants at that instant.
Rate of the reaction \(\alpha \) [Reactant]^x

Let us consider the formation of ammonia in which, 'a' moles nitrogen and 'b' moles hydrogen gas are allowed to react in a container of volume V. Let 'x' moles of nitrogen react with 3x moles of hydrogen to give 2x moles of ammonia.
\({ N }_{ 2 }\left( g \right) +3{ H }_{ 2 }\left( g \right) \rightleftharpoons { 2NH }_{ 3 }\left( g \right) \)

Applying law of mass action
\({ K }_{ C }=\cfrac { \left[ { NH }_{ 3 } \right] ^{ 2 } }{ \left[ { N }_{ 2 } \right] \left[ { H }_{ 2 } \right] ^{ 3 } } \)
= \(\cfrac { \left( \cfrac { 2x }{ V } \right) ^{ 2 } }{ \left( \cfrac { a-x }{ V } \right) \left( \cfrac { b-3x }{ V } \right) ^{ 3 } } \)
= \(\cfrac { \left( \cfrac { 4x }{ V } \right) ^{ 2 } }{ \left( \cfrac { a-x }{ V } \right) \left( \cfrac { b-3x }{ V } \right) ^{ 3 } } \)
\({ K }_{ C }=\cfrac { 4{ x }^{ 2 }{ V }^{ 2 } }{ \left( a-x \right) \left( b-3x \right) ^{ 2 } } \)
The equilibrium constant K_p can also be calculated as follows:
\({ K }_{ p }={ K }_{ C }\left( RT \right) ^{ \left( \Delta { n }_{ g } \right) }\)
\(\Delta \)n_g =n_p - n_r = 2 - 4 = -2
\({ K }_{ p }=\cfrac { { 4x }^{ 2 }{ V }^{ 2 } }{ \left( a-x \right) \left( b-3x \right) ^{ 3 } } \left( RT \right) ^{ -2 }\)
Total number of moles at equilibrium,
n = a - x + b - 3x + 2x = a + b - 2x
\({ K }_{ p }=\cfrac { { 4x }^{ 2 }{ V }^{ 2 } }{ \left( a-x \right) \left( b-3x \right) ^{ 3 } } \times \left[ \cfrac { PV }{ n } \right] ^{ -2 }\)
\({ K }_{ p }=\cfrac { { 4x }^{ 2 }{ V }^{ 2 } }{ \left( a-x \right) \left( b-3x \right) ^{ 3 } } \times \left[ \cfrac { n }{ PV } \right] ^{ 2 }\)
\({ K }_{ p }=\cfrac { { 4x }^{ 2 }{ V }^{ 2 } }{ \left( a-x \right) \left( b-3x \right) ^{ 3 } } \times \left[ \cfrac { a+b-2x }{ PV } \right] ^{ 2 }\)
\({ K }_{ p }=\cfrac { 4{ x }^{ 2 }\left( a+b\quad -2x \right) ^{ 2 } }{ { P }^{ 2 }\left( a-x \right) \left( b-3x \right) ^{ 3 } } \)