Question

214.2g of sugar syrup contains 34.2g of sugar. Calculate
(A) Molality of the solution.
(B) Mole fraction of sugar in the syrup.

Answer 1

Hint: Molality is defined as the ratio of number of moles of solute to the number of kilograms of the solvent. So first, we have to calculate the number of moles of sugar, find molality and then we have to calculate the number of moles of water to find the mole fraction.

Complete answer:
(A) The solute used is sugar - that is sucrose. Molecular formula of sucrose is ${C_{12}}{H_{22}}{O_{11}}$ . Using the atomic masses of the constituent elements, $C = 12u, H = 1u, O = 16u$ we can calculate the molecular mass as follows:
(12 \times 12) + (22 \times 1) + (11 \times 16)
= 144 + 22 + 176
= 342u
For any substance, molar mass = molecular mass expressed as grams. So the molar mass of sugar is 342g. Given mass of sugar is 34.2g. Now, let us find the number of moles:
$n = \dfrac{m}{M}$ Where n is the number of moles, m is the given mass and M is the molar mass.
$n = \dfrac{{34.2}}{{342}} = 0.1 $...........................................Equation (1)
In order to find molality, we need to find the mass of the solvent in terms of kilograms.
Mass of solution = mass of solute + mass of solvent
Given: Mass of solution = 214.2g and mass of solute (sugar) = 34.2g
Therefore, mass of solvent = mass of solution – mass of solute = 214.2 - 34.2 = 180g
Mass of solvent = $180 \times {10^{ - 3}} = 0.18kg $........................................Equation (2)
Now that we have the information required, let us calculate molality.
Molality $ = \dfrac{{0.1}}{{0.18}} = 0.555molK{g^{ - 1}}$
(B) Moving on to the next sub question where we have to find mole fraction of sugar in the sugar syrup. For this, we need to calculate the number of moles of the solvent, in this case, water. The molecular mass of water $\left( {{H_2}O} \right)$ is $\left( {2 \times 1} \right) + \left( {1 \times 16} \right) = 18u$ . Therefore its molar mass will be 18g. We have already calculated the given mass of water to be 180g. Thus, number of moles can be determined as
$n = \dfrac{m}{M} = \dfrac{{180}}{{18}} = 10 $...................................................Equation (3)
Mole fraction of sugar can be calculated as (no.of moles of sugar/total no.of moles)
Total no. of moles = No. of moles of sugar + no. of moles of water $ = 0.1 + 10 = 10.1$
Therefore, mole fraction of sugar $ = \dfrac{{0.1}}{{10.1}} = 9.90 \times {10^{ - 3}}$

Note: Be cautious while calculating the molecular formula of sugar. The correct molecular formula of sucrose is ${C_{12}}{H_{22}}{O_{11}}$ and not ${C_6}{H_{12}}{O_6}$