Question

Derive van’t Hoff general solution equation 

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

Answer 1

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.
2. 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
3. Combining (1) and (2) we get,
   π ∝ `"T"/"V"`
   ∴ π = Constant x `"T"/"V"`
   ∴ πV = R'T, where R' is a constant.
4. 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 )
5. 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
   ∴ π = ` "nRT"/"V"`
6.  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.