From the ideal gas law equation (PV = nRT), you can derive the following equation to calculate the molecular mass of a gas (unit: g/mol).
MM = gRT / PV
where g is the mass of the gas, R is the value for the gas constant, T is the Kelvin temperature, P is the pressure, and V is the volume of the gas.
From the data given in your question, we need to figure out the values of g, R, T, P and V.
Mass of empty vessel = 134.74 g
Mass of vessel + gas Y = 137.28 g
Mass of gas Y = (Mass of vessel + gas Y) − (Mass of empty vessel)
= 137.28 − 134.74 = 2.54 g
The value of the gas constant (R) that we will be using is 8.314 L kPa mol-1 K-1. As a result, we need to convert the unit for Pressure (P) from kNm-2 to kPa. Conveniently, the conversion factor is 1 kNm-2 = 1 kPa, therefore, P = 99.3 kPa.
T = °C + 273 K = 31 + 273 = 304 K
The volume that the gas occupies in the vessel will be obtained from this part of your question...."The glass vessel was then filled completely with water and the mass was 1067.90g."
Mass of water = (Mass of water + vessel) − Mass of empty vessel
= 1067.90 − 134.74 = 933.16 g of water
We will assume density of water (d) = 1.00 g/mL. Using the density and mass of water from above, we will calculate the volume of water it occupies in the vessel. This calculated volume of water WILL BE THE SAME as the volume of the gas because the gas also occupies the same volume within the vessel. In other words, Vwater = Vgas !!
d = m / V. Rearrange to solve for V.
V = m / d = 933.16 / 1.00 = 933.16 mL
Almost done! Volume has to be converted to Liters since the value of R has volume expressed in Liters (L).
V = 933.16 mL x (1 L / 1000 mL) = 0.93316 L
Now plug in the values of g, R, T, P and V in the ideal gas law equation.
MM = ------
2.54 g x 8.314 L kPa mol-1 K-1 x 304 K
99.3 kPa x 0.93316 L
= 65.1 g/mol