Question

A certain hydrate has the formula $MgS{O_4}.x{H_2}O$ . A quantity of 54.2g of the compound is heated in an oven to drive off the water. If the steam generated exerts a pressure of 24.8atm in a 2.0L container at ${120^\circ }C$, then calculate $x$.
A.2
B.5
C.6
D.7

Answer 1

The hydrate of $MgS{O_4}.x{H_2}O$ has $x$ number of water of crystallization molecules. The approach to solving this question is using a combination of the ideal gas law equation along with a basic stoichiometric formula.
Formula Used:
$PV = nRT$ , where
$P$ = pressure of a gas
$V =$ volume contained by the gas
$n$ = no. of moles
$R =$ Gas constant which depends on the units involved
$T =$ temperature of the gas

Complete step by step answer:
Given is the hydrate compound of molecular formula $MgS{O_4}.x{H_2}O$.
$\Rightarrow$It is given in the question that the Mass or weight of the hydrate is = $54.2g$
Hence, we know that the Molecular weight of $MgS{O_4}.x{H_2}O$ will be
mass of $Mg +$ mass of $S +$ mass of $O \times 4 + x \times ($ mass of $H \times 2 +$ mass of $O)$
Given molecular mass are:
$Mg = 24,S = 32,O = 16,Mass H = 1$
Substituting these values in the above equation we get:
$24 + 32 + 16 \times 4 + x \times (1 \times 2 + 16)$
Solving this equation we get:
$120 + 18x$
This will be the molecular mass of $MgS{O_4}.x{H_2}O$ .
$\therefore$ using the ideal gas equation, we calculate the value of x:
Ideal Gas Law equation is:
$PV = nRT$
Where:
$P$ = pressure of a gas
$V =$ volume contained by the gas
$n$ = no. of moles
$R =$ Gas constant which depends on the units involved
$T =$ temperature of the gas
Given, in the question:
$P = 24.8atm,V = 2L,T = {120^ \circ }C\,or\,120 + 273K,R = 8.314J{K^ - 1}mo{l^ - }$
Substituting these values we get:
$24.8 \times 2 = n \times 8.314 \times (120 + 273)$

We know that the number of moles can be given as:
$number\,of\,moles = \dfrac{{given\,mass}}{{molecular\,mass}}$
We know the given mass will be: $54.2x$
Substituting all these values in the equation, we get:
$\Rightarrow$ $24.8 \times = \dfrac{{54.2x}}{{120 + 18x}} \times 8.314 \times 393$
Solving the equation we get:
$\Rightarrow$ $208.4 =$ $\dfrac{{120 + 18x}}{{54.2x}}$ =$\dfrac{{32.265}}{{49.6}}$
On further simplification:
$\Rightarrow$ $120 + 18x = 54.2 \times 0.65x$
$\Rightarrow$ $x = 6.96 \simeq 7$
Hence, the correct option is (D).

Note:
We use the ideal gas law equation which means we are assuming it to be an ideal gas. Which means there is no interaction between the gas molecules. However, in reality, we know gases do not exhibit ideal gas behavior and hence observed data and theoretical values may have some discrepancies.