Question

A gas container contains \[{\rm{35}}\,\,{\rm{g}}\] nitrogen gas, \[{\rm{64}}\,{\rm{g}}\]\[{{\rm{O}}_2}\] gas and \[{\rm{0}}{\rm{.112}}\,\,{\rm{lit}}\]\[{\rm{C}}{{\rm{H}}_4}\] of at STP, what is the average molecular weight of the gas mixture?
A. \[32\]
B. \[30.44\]
C. \[28\]
D. \[16\]

Answer 1

Hint: STP stands for standard temperature and pressure. At STP, the standard temperature is taken as \[{\rm{273}}\,{\rm{K}}\] and the standard pressure is taken as \[{\rm{1}}\]atmospheric pressure. One mole of gaseous substances contains \[{\rm{22}}{\rm{.4}}\,\,{\rm{L}}\] at STP.

Formula used Average molecular weight of the gas sample can be calculated by the relationship as shown below:
\[{{\rm{M}}_{{\rm{avg}}}} = \,\,{{\rm{M}}_{{{\rm{N}}_{\rm{2}}}}}{{\rm{X}}_{{{\rm{N}}_{\rm{2}}}}}\,{\rm{ + }}\,{{\rm{M}}_{{{\rm{O}}_{\rm{2}}}}}{{\rm{X}}_{{{\rm{O}}_{\rm{2}}}}}{\rm{ + }}{{\rm{M}}_{{\rm{C}}{{\rm{H}}_{\rm{4}}}}}{{\rm{X}}_{{\rm{C}}{{\rm{H}}_{\rm{4}}}}}\]
where, \[{{\rm{M}}_{{\rm{avg}}}}\] = average molecular weight
\[\,{{\rm{M}}_{{{\rm{N}}_{\rm{2}}}}}\] = molecular weight of \[{{\rm{N}}_2}\]
\[{{\rm{X}}_{{{\rm{N}}_{\rm{2}}}}}\] = mole fraction of \[{{\rm{N}}_2}\]
\[{{\rm{M}}_{{{\rm{O}}_{\rm{2}}}}}\] = molecular weight of \[{{\rm{O}}_{\rm{2}}}\]
\[{{\rm{X}}_{{{\rm{O}}_{\rm{2}}}}}\] = mole fraction of \[{{\rm{O}}_{\rm{2}}}\]
\[{{\rm{M}}_{{\rm{C}}{{\rm{H}}_{\rm{4}}}}}\] = molecular weight of \[{\rm{C}}{{\rm{H}}_{\rm{4}}}\]
\[{{\rm{X}}_{{\rm{C}}{{\rm{H}}_{\rm{4}}}}}\] = mole fraction of \[{\rm{C}}{{\rm{H}}_{\rm{4}}}\]

Complete Step by Step Solution:
A mole is the amount of substance that contains as many particles (atoms, molecules, ions, etc.) as there are atoms exactly in \[{\rm{0}}{\rm{.012}}\,{\rm{kg}}\]\[{\rm{(i}}{\rm{.e}}{\rm{. 12 g)}}\] of carbon \[({}^{{\rm{12}}}{\rm{C)}}\]. It must be remembered that the mass of one mole of an atom is equal to the gram atomic mass of the element, the mass of one mole of a molecule is equal to a gram molecular mass of the substance and the mass of one mole of formula units in case of an ionic compound is equal to gram formula mass of the ionic compound.

As per the given data,
‘Y’ atom is \[0.1809\]times heavier than that of an atom of an oxygen
Mass of one mole of an atom of oxygen is known to be \[{\rm{16}}\,{\rm{g}}\].
Therefore,

Again, element ‘X’ is \[1.02\] times heavier than that of ‘Y’ atom
So,

Hence, the atomic mass of ‘X’ is found to be \[{\rm{2}}{\rm{.952}}\,{\rm{g}}\].
Therefore, option A is correct.

Additional information: Gram atomic mass of an element is defined as the atomic mass of an element in grams or it is the mass in grams which is numerically equal to the atomic mass of an element. For example, the atomic mass of oxygen is \[{\rm{16}}\,{\rm{amu}}\]. Therefore,
\[{\rm{1}}\,\,{\rm{gram}}\,\,{\rm{atom}}\,\,{\rm{of}}\,\,{\rm{oxygen}}\,\,{\rm{ = }}\,{\rm{gram}}\,\,{\rm{atomic}}\,\,{\rm{mass}}\,\,{\rm{of}}\,\,{\rm{oxygen}}\,\,{\rm{ = }}\,\,{\rm{16}}\,{\rm{g}}\].
Number of gram atoms and the mass in grams of an element are related to each other by the relation:
\[{\rm{Number}}\,\,{\rm{of}}\,\,{\rm{gram}}\,\,{\rm{atoms}}\,\,{\rm{ = }}\,\,\dfrac{{{\rm{mass}}\,\,{\rm{in}}\,\,{\rm{grams}}}}{{{\rm{gram}}\,\,{\rm{atomic}}\,\,{\rm{mass}}}}\]

Note: Mole concept is used to calculate the absolute weight of an atom or a molecule, the number of atoms or molecules present in the given mass of a substance, the mass of a given number of atoms or molecules and size of individual atoms or molecules etc.