When $25\, g$ of a non-volatile solute is dissolved in $100\,g$ of water, the vapour pressure is lowered by $2.25 \times 10^{-1}\, mm$. If the vapour pressure of water at $20^{\circ} C$ is $17.5\, mm$, what is the molecular weight of the solute?
- 206
- 302
- 350
- 276
The Correct Option is C
Approach Solution - 1
Weight of non-volatile solute, $w=25\, g$
Weight of solvent, $W=100\, g$
Lowering of vapour pressure,
$p^{\circ}-p_{s}=0.225\, mm$
Vapour pressure of pure solvent,
$p^{\circ}=17.5\, mm$
Molecular weight of solvent
$\left(H_{2} O\right), M=18\, g$
Molecular weight of solute, $m=?$
According to Raoult's law
$p^{\circ}-p_{s}$
${p^{\circ}}=\frac{w \times M}{m \times W}$
$\frac{0.225}{17.5}=\frac{25 \times 18}{m \times 100}$
$m=\frac{25 \times 18 \times 17.5}{22.5}$
$=350\, g$
Approach Solution -2
To find the molecular weight of the solute, we can use the formula for the relative lowering of vapor pressure in a solution. This formula is given by Raoult's law:
\[\frac{\Delta P}{P^0} = \frac{n_2}{n_1}\]
where:
- \(\Delta P\) is the lowering of vapor pressure (\(2.25 \times 10^{-1} \, mm\))
- \(P^0\) is the vapor pressure of the pure solvent (water at \(20^{\circ} C\), \(17.5 \, mm\))
- \(n_2\) is the number of moles of solute
- \(n_1\) is the number of moles of solvent (water)
First, calculate the moles of water (\(n_1\)):
\[n_1 = \frac{\text{mass of water}}{\text{molar mass of water}} = \frac{100 \, g}{18 \, g/mol} \approx 5.56 \, mol\]
Next, using Raoult's law, we can rearrange the formula to solve for \(n_2\):
\[\frac{\Delta P}{P^0} = \frac{n_2}{n_1}\]
\[n_2 = n_1 \times \frac{\Delta P}{P^0}\]
Substitute the given values:
\[n_2 = 5.56 \, mol \times \frac{2.25 \times 10^{-1} \, mm}{17.5 \, mm}\]
\[n_2 = 5.56 \times 0.012857 \approx 0.0714 \, mol\]
Now, we know the number of moles of the solute, and we also know the mass of the solute (\(25 \, g\)). We can find the molar mass (\(M\)) of the solute using the formula:
\[M = \frac{\text{mass of solute}}{\text{number of moles of solute}} = \frac{25 \, g}{0.0714 \, mol} \approx 350 \, g/mol\]
Therefore, the molecular weight of the solute is approximately \(350 \, g/mol\) so the correct Answer is option 3.
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Concepts Used:
Solutions
A solution is a homogeneous mixture of two or more components in which the particle size is smaller than 1 nm.
For example, salt and sugar is a good illustration of a solution. A solution can be categorized into several components.
Types of Solutions:
The solutions can be classified into three types:
- Solid Solutions - In these solutions, the solvent is in a Solid-state.
- Liquid Solutions- In these solutions, the solvent is in a Liquid state.
- Gaseous Solutions - In these solutions, the solvent is in a Gaseous state.
On the basis of the amount of solute dissolved in a solvent, solutions are divided into the following types:
- Unsaturated Solution- A solution in which more solute can be dissolved without raising the temperature of the solution is known as an unsaturated solution.
- Saturated Solution- A solution in which no solute can be dissolved after reaching a certain amount of temperature is known as an unsaturated saturated solution.
- Supersaturated Solution- A solution that contains more solute than the maximum amount at a certain temperature is known as a supersaturated solution.
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