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Derive van’t Hoff general solution equation

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

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#### Solution

- 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"` ...(1) ...[ van't Hoff-Boyle's law at constant temperature]

∴ πV = constant

π ∝ T .....(2) ...[ van't Hoff-Charles' law at constant concentration ]

∴ `π/"T"` = constant - Combining (1) and (2) we get,

π ∝ `"T"/"V"`

∴ π = Constant x `"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 dm
^{3}. If n moles of solute are dissolved in V dm^{3 }of solution, the equation becomes

πV = nRT

∴ π = ` "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.

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