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# A sample of ethane gas $\left( {{C}_{2}}{{H}_{6}} \right)$ has the same mass as $1.5\times {{10}^{20}}$ molecules of methane $\left( C{{H}_{4}} \right)$ . How many ethane molecules does the sample of gas contain?

Last updated date: 11th Sep 2024
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Hint: Avogadro’s number is the number of molecules present in 1 mole which is equal to $6.023\times {{10}^{23}}$ entities. It is denoted by the symbol ${{N}_{A}}$ . This value is directly used to find the number of atoms in certain different types of molecules which have different representational units.

Complete Solution :
-Everything present in this universe is made up of atoms. The mass of an atom cannot be measured truly due to its very small size. So a relative term is used for it called relative atomic mass. It is based on the mass of 1 carbon-12 atom. It was adopted internationally in 1961 and according to it, relative atomic is given by
$Relative\text{ }atomic\text{ }mass\text{ }=\text{ }\dfrac{mass\text{ of 1 atom of element}}{\dfrac{1}{12}\text{x mass of 1 C-12 atom}}$

-Mole is the amount of substance that contains the atoms, molecules or other particles in an entity equal to the atoms present in 12 g of C-12 isotope. It defines the basis of physical chemistry and so its number is called Avogadro's number which is equal to 6.023 x ${{10}^{23}}$ entities. It is denoted by the symbol ${{N}_{A}}$ .
-Some standard terms used in mole concept are
1 gram-atom = 1 mole atom
1 gram-molecule = 1 mole molecule
1 gram-ion = 1 mole ion

-Here we are given that the number of molecules of methane are $1.5\times {{10}^{20}}$. We know that the number of 1 mole of atoms for any substance is equal to avogadro’s number.
It means that for methane, 16 grams of it will have molecules equal to $6.023\times {{10}^{23}}$. So the weight of $1.5\times {{10}^{20}}$ molecules will be given as
$\dfrac{1.5\times {{10}^{20}}\times 16}{{{N}_{A}}}$

-Also we are given that the sample of ethane gas $\left( {{C}_{2}}{{H}_{6}} \right)$ has the same mass as $1.5\times {{10}^{20}}$molecules of methane $\left( C{{H}_{4}} \right)$. Molecular weight of ethane is 30 grams. Mass of 1 molecule of ethane will be equal to $\dfrac{30}{{{N}_{A}}}$ grams. The weight of ethane molecules is equal to those of $1.5\times {{10}^{20}}$ molecules of methane $\left( C{{H}_{4}} \right)$.

-Thus the number of molecules of ethane will be given as
$\dfrac{1.5\times {{10}^{20}}\times 16}{{{N}_{A}}}\times \dfrac{{{N}_{A}}}{30}$
= $0.8\times {{10}^{20}}$
Therefore the number of ethane molecules will be $0.8\times {{10}^{20}}$.

Note: Mole is not only related to mass but also to volume as any matter that has a certain mass will occupy a certain volume. 1 mole is the amount of molecules present in 22.4 litres of a gas calculated at standard temperature and pressure.