What is the molar mass of a gas if \(0.281\;{\rm{g}}\) of the gas occupies a volume of \(125\;{\rm{mL}}\)at a temperature \({126^\circ }{\rm{C}}\)and a pressure of \({\bf{777}}{\rm{ }}{\bf{torr}}\)?

A substance's volume is the amount of space it occupies

Let us solve the given problem.

We have,

\(PV = nRT\)

where,

\(P = \)pressure of the gas.

\(V = \)volume of the gas.

\(n = \)number of moles of the gas.

\(T = \)temperature of the gas.

\(\begin{aligned}{}R &= {\rm{ Ideal gas constant}}\\ & = 0.08206\;{\rm{L atm mo}}{{\rm{l}}^{ - 1}}{K^{ - 1}}\end{aligned}\)

If, amount of the gas\((n)\)is in moles , temperature\((T)\)is in Kelvin\((K)\), pressure\((P)\)is in atm.

\(\begin{aligned}{}\frac{P}{{RT}} & = \frac{n}{V}\\\frac{{P \cdot (\mu )}}{{RT}} & = \frac{{n \cdot (\mu )}}{V}\\\frac{{P \cdot (\mu )}}{{RT}} & = \frac{m}{V}\\\frac{{P \cdot (\mu )}}{{RT}} & = \rho \end{aligned}\)

Therefore,

Density\((\rho ) = \frac{{P \cdot (\mu )}}{{RT}}\)

where \(\mu = \)molar mass.

Consider the given problem.

mass \((m) = 0.281\) grams.

Volume \((V) = 125mL\).

Pressure \((P) = 777torr\).

Temperature \((T) = {126^\circ }C\)

Converting the temperature in degree Celsius to kelvins.

We have, \({0^\circ }C = (0) + 273.15K\)

So,

\(\begin{aligned}{}{28^\circ }C & = (126) + 273.15K\\ &= 399.15K\end{aligned}\)

Converting pressure in torr into atm.

We have,\(1torr = 0.0013\;{\rm{atm}}\).

So,

\(\begin{aligned}{}777{\rm{ torr}} &= (777) \cdot (0.0013){\rm{atm}}.\\ &= 1.01\;{\rm{atm}}.\end{aligned}\)

Converting the volume in\(mL\). to\(L\).

We have,\(1\;{\rm{mL}}. = 0.001\;{\rm{L}}\).

Therefore,

\(\begin{aligned}{}125mL. &= (125) \cdot (0.001)L.\\ &= 0.125\;{\rm{L}}.\end{aligned}\)

Calculate total molar mass.

\(\begin{aligned}{}\frac{{P \cdot (\mu )}}{{RT}} &= \frac{m}{V}{\rm{ }}\\{\rm{molar mass }}(\mu ) &= \frac{{mRT}}{{PV}}\\ &= \frac{{(0.281) \cdot (0.08206) \cdot (399.15)}}{{(1.01) \cdot (0.125)}}\\& = \frac{{9.20}}{{0.12625}}\\ &= 72.87\frac{{{\rm{ gram }}}}{{{\mathop{\rm mol}\nolimits} }}\end{aligned}\)

Therefore, molar mass \((\mu ) = 72.87\frac{{{\rm{ gram }}}}{{{\rm{ mol }}}}\).