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## Q. 5.9

Achmist has synthesized a greenish-yellow gaseous compound of chlorine and oxygen and finds that its density is 7.71 g/L at 36°C and 2.88 atm. Calculate the molar mass of the compound and determine its molecular formula.

Strategy  Because Equations (5.11) and (5.12) are rearrangements of each other, we can calculate the molar mass of a gas if we know its density, temperature, and pressure. The molecular formula of the compound must be consistent with its molar mass. What temperature unit should we use?

$d=\frac{m}{V}=\frac{P \mathscr{M}}{R T}$       (5.11)

$\mathscr{M}=\frac{d R T}{P}$      (5.12)

## Verified Solution

From Equation (5.12)

$\mathscr{M}=\frac{dRT}{P}$

$=\frac{(7.71 g/L)(0.0821 L . atm/K . mol)(36+273) K}{2.88 atm}$

=67.9  g/mol

Alternatively, we can solve for the molar mass by writing

molar mass of compound =$\frac{mass of compound}{moles of compound}$

From the given density we know there are 7.71 g of the gas in 1 L. The number of moles of the gas in this volume can be obtained from the ideal gas equation

$n=\frac{PV}{RT}$ $=\frac{(2.88 atm)(1.00 L)}{(0.0821 L . atm/K . mol)(309 K)}$

=0.1135  mol

Therefore, the molar mass is given by

$\mathscr{M}=\frac{mass}{number of moles}=\frac{7.71 g}{0.1135 mol}=67.9 g/mol$

We can determine the molecular formula of the compound by trial and error, using only the knowledge of the molar masses of chlorine (35.45 g) and oxygen (16.00 g). We know that a compound containing one Cl atom and one O atom would have a molar mass of 51.45 g, which is too low, while the molar mass of a compound made up of two Cl atoms and one O atom is 86.90 g, which is too high. Thus, the compound must contain one Cl atom and two O atoms and have the formula $ClO_{2}$, which has a molar mass of 67.45 g.