Question

Derive van’t Hoff general solution equation.

Derive van’t Hoff general solution equation for ‘n’ moles of solute. 

Derive van’t Hoff equation for dilute solutions.

Answer 1

According to van’t Hoff-Boyle’s law, osmotic pressure of a dilute solution is inversely proportional to the volume containing 1 mole of solute at constant temperature and according to van’t Hoff-Charles’ law, osmotic pressure of a dilute solution is directly proportional to the absolute temperature at constant concentration.

If π is the osmotic pressure, V is the volume of the solution and T is the absolute temperature, then

π ∝ `1/V`    ...(i) ...[van’t Hoff-Boyle’s law at constant temperature] 

∴ πV = constant

π ∝ T     .....(ii) ...[van’t Hoff-Charles’ law at constant concentration]

∴ `π/T` = constant

Combining (i) and (ii) we get,

π ∝ `T/V`

∴ π = Constant × `T/V`

∴ πV = R'T, where R' is a constant.

This equation is parallel to the ideal gas equation PV = RT (n = 1)
Since the calculated value of R' is almost same as R, the equation can be written as πV = RT (for 1 mole of solute )

This equation was derived for 1 mole of solute dissolved in V dm³. If n moles of solute are dissolved in V dm³ of solution, the equation becomes

πV = nRT

∴ `pi = (nRT)/V`

`C = n/V`

∴ π = CRT

where,

π = osmotic pressure,

C = concentration of solution in moles/litre

R = gas constant = 0.082 L atm mol⁻¹ K⁻¹ or 8.314 J mol⁻¹ K⁻¹

T = absolute temperature

n = number of moles of solute,

V = volume of the solution.

Answer 2

We have already seen that according to Boyle-van’t Hoff law

`pi prop 1/V` (at constant temperature)    ...(i)

and according to Charles’-van’t Hoff law

π ∝ T (at constant concentration)    ...(ii)

Combining Eqs. (i) and (ii), we have

`pi prop T/V`

or `pi = S T/V`

or πV = ST    ...(iii)

In this equation, S is a constant known as the solution constant. van’t Hoff showed that the value of solution constant (S) is equal to that of the gas constant, R, i.e.,

S = R

Hence, equation (iii) can be written as

πV = RT    ...(iv)

This equation is similar to the gas equation (PV = RT) and is known as the van’t Hoff equation.

For n moles of the solute dissolved in V litres of the solution, van’t Hoff equation can be written as

πV = nRT