Question 6.7: Determining a Molar Mass with the Ideal Gas Equation Propyle...

First determine the mass of water required to fill the vessel.

mass of water to fill vessel = 138.2410 g – 40.1305 g = 98.1105 g

Use the density of water in a conversion factor to obtain the volume of water (and hence, the volume of the glass vessel).

\text{volume of water (volume of vessel)} = 98.1105  g  H_2 O \times \frac{1  mL  H_2 O}{0.9970  g  H_2 O} = 98.41  mL = 0.09841  L

The mass of the gas is the difference between the weight of the vessel filled with propylene gas and the weight of the empty vessel.

mass of gas = 40.2959 g – 40.1305 g = 0.1654 g

The values of temperature and pressure are given.

T = 24.0 °C + 273.15 = 297.2 K

Substitute data into the rearranged version of equation (6.13).

M = \frac{m R T}{P V} =\frac{0.1654  g  \times  0.08206  \sout{atm}  \sout{L}  mol ^{-1}  \sout{K ^{-1}}  \times  297.2  \sout{K }}{0.9741  \sout{atm}  \times  0.09841  \sout{L} }

= 42.08  g  mol^{-1}

Assess
Cancellations leave the units g and mol^{-1}. The unit g mol^{-1} or g/mol is that for molar mass, the quantity we are seeking. We can use another approach to solving this problem. We can substitute the pressure (0.9741 atm), temperature (297.2 K), and volume (0.09841 L) into the ideal gas equation to calculate the number of moles in the gas sample (0.003931 mol). Because the sample contains 0.003931 mol and has a mass of 0.165 g, the molar mass is 0.165  g/0.003931  mol = 42.0  g  mol^{-1}. The advantage of this alternative approach is that it makes use of only the ideal gas equation; you do not have to memorize or derive equation (6.13) for the cases when you might need it.