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Hint: Van't Hoff factor i, determines the extent of association or dissociation, which can lead to the abnormal molar mass of a substance.

Van't Hoff equation: This equation gives the variation of the equilibrium constant of a reaction with temperature.

As we know that,

${ dlnK }_{ p }{ \div dT=-\Delta H }^{ - }{ \div R }{ T }^{ 2 }$

${ dlnK }_{ p }{ ={ \Delta H }^{ - } }{ \div R1/T^{ 2 } }{ \times dT }$

where, ${ K }_{ p }$ = Equilibrium constant

R = Ideal gas constant

T = temperature

${ \Delta H }$ = standard enthalpy change

Integrating both the sides, we get

$\int { dlnK_{ p } } { =\Delta H }^{ - }{ \div R }\int { 1\div { T }^{ 2 } } { .dT }

{ [lnK }_{ p }{ ] }_{ K_{ 1 } }^{ K_{ 2 } }{ =\Delta H^{ - } }{ \div R[1/T]_{ T_{ 1 } }^{ T_{ 2 } } }$

${ ln }_{ e }\dfrac { K_{ 2 } }{ K_{ 1 } } { \Delta H }^{ - }{ [1\div T }_{ 1 }{ -1/T }_{ 2 }{ ] }$

Now, convert ${ ln }_{ e }$ to ${ log }_{ 10 }$ by multiplying it with 2.303,we get

${ 2.303log }_{ 10 }\dfrac { K_{ 2 } }{ K_{ 1 } } { \Delta H }^{ - }{ \div R[{ T }_{ 2 }{ -T }_{ 1 } }{ /T }_{ 1 }{ T }_{ 2 }{ ] }$

Rearranging the values, we get

${ log }_{ 10 }\dfrac { K_{ 2 } }{ K_{ 1 } } { \Delta H }^{ - }{ \div R2.303[{ T }_{ 2 }{ -T }_{ 1 } }{ /T }_{ 1 }{ T }_{ 2 }{ ] }$

${ logK }_{ 2 }{ \div K }_{ 1 }{ =\Delta H\div 2.303[1\div T }_{ 2 }{ -1\div T }_{ 1 }{ ] }$

Hence, the correct option is B.

Additional Information:

The van 't Hoff equation also known as the Vukancic-Vukovic equation in chemical thermodynamics relates the change in temperature T to the change in the equilibrium constant K, given the standard enthalpy change ${ \Delta H }^{ \circ }$ for the process.

The van 't Hoff factor is a measure of the effect of a solute on colligative properties such as:

(I) Osmotic pressure,

(II) Relative lowering in vapor pressure,

(III) Elevation of boiling point and

(IV) Freezing point depression.

For substances which do not completely ionize in water for those substances, it is not an integer.

Compounds that completely ionize into solution, cations, and anions formed attract each other and form aggregates in concentrated solutions. ex- NaCl ionizes to ${ Na }^{ + }$ and ${ Cl }^{ - }$ , at high concentrations of NaCl , its ions forms aggregates in solution, therefore i value is less than 2 in such case, ${ 1 < i < 2 }$,

To increase the value of i we take a dilute solution. For dilute solutions aggregations between cations and anions is less ${ i = 2 }$.

In some cases, the substance added in the solvent forms self-aggregation, in that case also i < 1 as the number of moles dissolved in the solvent becomes less than the number of moles added due to self-association.

Note: The possibility to make a mistake is that you may choose option A. But there will be positive signs instead of negative signs as we when we integrate it will resolve.

__Complete step-by-step answer:__Van't Hoff equation: This equation gives the variation of the equilibrium constant of a reaction with temperature.

As we know that,

${ dlnK }_{ p }{ \div dT=-\Delta H }^{ - }{ \div R }{ T }^{ 2 }$

${ dlnK }_{ p }{ ={ \Delta H }^{ - } }{ \div R1/T^{ 2 } }{ \times dT }$

where, ${ K }_{ p }$ = Equilibrium constant

R = Ideal gas constant

T = temperature

${ \Delta H }$ = standard enthalpy change

Integrating both the sides, we get

$\int { dlnK_{ p } } { =\Delta H }^{ - }{ \div R }\int { 1\div { T }^{ 2 } } { .dT }

{ [lnK }_{ p }{ ] }_{ K_{ 1 } }^{ K_{ 2 } }{ =\Delta H^{ - } }{ \div R[1/T]_{ T_{ 1 } }^{ T_{ 2 } } }$

${ ln }_{ e }\dfrac { K_{ 2 } }{ K_{ 1 } } { \Delta H }^{ - }{ [1\div T }_{ 1 }{ -1/T }_{ 2 }{ ] }$

Now, convert ${ ln }_{ e }$ to ${ log }_{ 10 }$ by multiplying it with 2.303,we get

${ 2.303log }_{ 10 }\dfrac { K_{ 2 } }{ K_{ 1 } } { \Delta H }^{ - }{ \div R[{ T }_{ 2 }{ -T }_{ 1 } }{ /T }_{ 1 }{ T }_{ 2 }{ ] }$

Rearranging the values, we get

${ log }_{ 10 }\dfrac { K_{ 2 } }{ K_{ 1 } } { \Delta H }^{ - }{ \div R2.303[{ T }_{ 2 }{ -T }_{ 1 } }{ /T }_{ 1 }{ T }_{ 2 }{ ] }$

${ logK }_{ 2 }{ \div K }_{ 1 }{ =\Delta H\div 2.303[1\div T }_{ 2 }{ -1\div T }_{ 1 }{ ] }$

Hence, the correct option is B.

Additional Information:

The van 't Hoff equation also known as the Vukancic-Vukovic equation in chemical thermodynamics relates the change in temperature T to the change in the equilibrium constant K, given the standard enthalpy change ${ \Delta H }^{ \circ }$ for the process.

The van 't Hoff factor is a measure of the effect of a solute on colligative properties such as:

(I) Osmotic pressure,

(II) Relative lowering in vapor pressure,

(III) Elevation of boiling point and

(IV) Freezing point depression.

For substances which do not completely ionize in water for those substances, it is not an integer.

Compounds that completely ionize into solution, cations, and anions formed attract each other and form aggregates in concentrated solutions. ex- NaCl ionizes to ${ Na }^{ + }$ and ${ Cl }^{ - }$ , at high concentrations of NaCl , its ions forms aggregates in solution, therefore i value is less than 2 in such case, ${ 1 < i < 2 }$,

To increase the value of i we take a dilute solution. For dilute solutions aggregations between cations and anions is less ${ i = 2 }$.

In some cases, the substance added in the solvent forms self-aggregation, in that case also i < 1 as the number of moles dissolved in the solvent becomes less than the number of moles added due to self-association.

Note: The possibility to make a mistake is that you may choose option A. But there will be positive signs instead of negative signs as we when we integrate it will resolve.

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