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Question: An evacuated glass vessel weighs \(50g\) when empty, \(148g\) when filled with a liquid density \(0....

An evacuated glass vessel weighs 50g50g when empty, 148g148g when filled with a liquid density 0.98gml−10.98gm{l^{ - 1}} and 50.5g50.5g when filled with ideal gas at 760mmHg760mmHg 300K300K. Determine the molar mass of the gas. [Write your answer to the nearest integer]..

Explanation

Solution

We know that the mass per unit volume is the density of a material and it is used to calculate the rigidity of the material.
Density=MassVolume{\text{Density}} = \dfrac{{{\text{Mass}}}}{{{\text{Volume}}}}

Complete step by step answer:
Given,
The mass of an empty vessel is 50g50g.
The mass of a vessel filled with gas is 50.5g50.5g.
The mass of the liquid filled in the vessel is148g148g.
The density of the liquid is 0.98gml−10.98gm{l^{ - 1}}
The pressure of the ideal gas is 760mmHg760mmHg.
The temperature is 300K300K.
Now, calculate the mass of the ideal gas as,
Mass of ideal gas=50.5−50=0.5g{\text{Mass of ideal gas}} = 50.5 - 50 = 0.5g
The mass of the liquid is calculated as,
Mass of liquid=148−50=98g{\text{Mass of liquid}} = 148 - 50 = 98g
Let us calculate the volume using the density formula.
Volume=MassDensity{\text{Volume}} = \dfrac{{{\text{Mass}}}}{{{\text{Density}}}}
Volume=980.98=100ml=0.1L{\text{Volume}} = \dfrac{{98}}{{0.98}} = 100ml = 0.1L
We can calculate the molar mass of a substance liberated from the ideal gas equation.
The ideal gas equation is,
PV=nRTPV = nRT
We know that, the amount of moles in given amount of any substance is equal to the grams of the substance divided by its molecular mass of the substance,
Moles(n)=Mass(W)Molecular mass(M){\text{Moles}}\left( {\text{n}} \right) = \dfrac{{{\text{Mass}}\left( W \right)}}{{{\text{Molecular mass}}\left( M \right)}}
Thus, the ideal gas equation becomes.
PV=WMRTPV = \dfrac{W}{M}RT
(760760)×0.1=0.5M(0.0821Latmmol−1K−1)(300K)\left( {\dfrac{{760}}{{760}}} \right) \times 0.1 = \dfrac{{0.5}}{M}\left( {0.0821Latmmo{l^{ - 1}}{K^{ - 1}}} \right)\left( {300K} \right)
The molar mass of substance = 123.15≈123g123.15 \approx 123g.

Note:
Now we can discuss about the details of molar mass as,
Molar mass of an element is defined as the atomic mass of an element present in Avogadro’s number of atoms. To find the molar mass, one must have to change the units of atomic mass from the atomic mass unit to grams.
For example, sulfur has an atomic mass of 32amu32amu so one mole of sulfur has a molar mass of 32g32g and contains Avogadro’s number of atoms.
Remember to convert the pressure in millimeter mercury (mmHg)\left( {mmHg} \right) to the standard atmosphere unit by dividing the given value by 760760.