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Questions
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.
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Solution 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.
Solution 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
Notes
Students can refer to the provided solutions based on their preferred marks.
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