Question

a metal, m, of atomic mass 56 amu reacts with chlorine to form a salt that can be represented as mclₓ. a boiling - point elevation experiment is performed to determine the subscript x, and therefore, the formula of the salt. a 24.7 g sample of the salt is dissolved in 100.0 g of water and the boiling point of the solution is found to be 376.14 k. find the formula of the salt. assume complete dissociation of the salt in solution.

Explanation:

Step1: Calculate the boiling - point elevation

The boiling - point of pure water is $T_{b}^0=373.15\ K$. The boiling - point elevation $\Delta T_b=T_b - T_{b}^0$.
$\Delta T_b=376.14\ K - 373.15\ K = 2.99\ K$.

Step2: Use the boiling - point elevation formula

The boiling - point elevation formula is $\Delta T_b = iK_bm$, where $i$ is the van't Hoff factor, $K_b$ is the ebullioscopic constant of water ($K_b = 0.512\ K\cdot kg/mol$), and $m$ is the molality of the solution.
Since the salt $MCl_x$ dissociates completely as $MCl_x
ightarrow M^{x +}+xCl^-$, the van't Hoff factor $i=x + 1$.
The molality $m=\frac{n_{solute}}{m_{solvent}(kg)}$, where $n_{solute}$ is the number of moles of solute and $m_{solvent}=100.0\ g=0.1000\ kg$.
Let the molar mass of $MCl_x$ be $M = 56+35.5x\ g/mol$. The number of moles of solute $n_{solute}=\frac{24.7\ g}{(56 + 35.5x)\ g/mol}$.
So, $m=\frac{\frac{24.7\ g}{(56 + 35.5x)\ g/mol}}{0.1000\ kg}=\frac{247}{56 + 35.5x}\ mol/kg$.

Step3: Substitute into the boiling - point elevation formula

Substitute $\Delta T_b = 2.99\ K$, $i=x + 1$, $K_b = 0.512\ K\cdot kg/mol$, and $m=\frac{247}{56 + 35.5x}\ mol/kg$ into $\Delta T_b = iK_bm$.
$2.99=(x + 1)\times0.512\times\frac{247}{56 + 35.5x}$.
First, simplify the right - hand side: $(x + 1)\times0.512\times\frac{247}{56 + 35.5x}=\frac{(x + 1)\times0.512\times247}{56 + 35.5x}=\frac{(x + 1)\times126.964}{56 + 35.5x}$.
So, $2.99(56 + 35.5x)=(x + 1)\times126.964$.
Expand both sides: $167.44+106.145x=126.964x+126.964$.
Move the $x$ terms to one side and the constants to the other side: $106.145x-126.964x=126.964 - 167.44$.
$-20.819x=-40.476$.
Solve for $x$: $x=\frac{-40.476}{-20.819}\approx2$.

Answer:

The formula of the salt is $MCl_2$.