Correct expressions among the following are: $(ρ$ - density of solution; $M$ - molarity of solution, $M_{2}$ - Molar mass of solute; $m - m o l a l i t y; X_{2}$ - Mole fraction of solute).

m = \frac{1000 M}{1000 ρ - M M_{2}}

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X_{2} = \frac{m M_{1}}{1 + m M_{1}}

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X_{2} = \frac{M M_{1}}{M (M_{1} - M_{2}) + ρ}

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m = \frac{ρ - M M_{2}}{M}

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Solution

The correct option is A $m = \frac{1000 M}{1000 ρ - M M_{2}}$
$X_{2} = \frac{n_{2}}{n_{1} + n_{2}} = \frac{n_{2}}{m_{1} / M_{1} + n_{2}} = \frac{n_{2} M_{1}}{m_{1} + n_{2} M_{1}} = \frac{n_{2} M_{1}}{(m_{1} + m_{2}) + n_{2} M_{1} - m_{2}} = \frac{n_{2} M_{1}}{V ρ + n_{2} (M_{1} + M_{2})}$
$= \frac{(n_{2} / V) M_{1}}{ρ + (\frac{n_{2}}{V}) (M_{1} + M_{2})} = \frac{M M_{1}}{ρ + M (M_{1} - M_{2})}$
Also,
$m = \frac{1000 M}{1000 ρ - M \times M_{2}}$
Let, mass of solute be $W_{2}$ and solvent be $W_{1}$
$M = \frac{W_{2}}{M_{2}} \times \frac{1000}{V}$ (Equation $01$ )
$m = \frac{W}{M_{2}} \times \frac{1000}{W_{1}}$ (Equation $02$ )
$ρ = \frac{W + W_{2}}{V}$
$⟹ V = \frac{W_{2} \times 1000}{M M_{2}}$ $m L$ (From $01$ )
So,
$W_{1} + W_{2} = W_{2} + \frac{1000 W_{2}}{m M_{2}}$
$W_{1} + W_{2} = \frac{1000}{M_{2}} (\frac{m M_{2}}{1000 m} + \frac{1}{m}) W_{1}$
$⟹ \frac{ρ}{M} = \frac{1}{m} + \frac{M_{2}}{1000}$
$m = \frac{M \times 1000}{1000 ρ - M M_{2}}$

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