Question

How many atoms are in a gram of air?

Answer 1

Hint: The number of atoms of gases present in the air would correspond to the percentage composition of each of the gases individually and since air is a homogeneous mixture of all the gases, the number of atoms would be assumed to be distributed homogeneously in the air.

Complete step-by-step solution:
At first we will discuss the major components which constitute the air, and then we will discuss their percentage composition. We know that air is a mixture of various gases, and a major part of those gases consist of nitrogen which is $78.08%$ of the total mixture, and $20.95%$ consists of the gas oxygen, and noble gas like argon constitutes $0.93%$ and carbon dioxide has $0.04%$.

Now, we need to calculate the molar mass of air, which would be equal to the sum of molar masses of each of these gases. So we will multiply the molar masses of each of these gases with the percentage compositions.

Gas	Molar mass	Percentage	Contribution
${{N}_{2}}$	$28.01$	$78.08$	$21.870$
${{O}_{2}}$	$32$	$20.95$	$6.704$
$Ar$	$39.95$	$0.93$	$0.372$
$C{{O}_{2}}$	\[44.01\]	$0.04$	$0.018$
			Total $=28.964$

So, now the relative molar mass came out to be $28.964$.
Then, now we will calculate the number of moles for one gram of air. And we know that the number of moles are given mass present per molar mass of the compound or substance.
\[Moles{ }of{ }air=\dfrac{1g{ }air\times 1{ }mol{ }air}{28.964g{ }air}=0.034mol{ }air\]
So, we can see that the number of moles of air came out to be $0.034$ moles.
Now, we will calculate the atoms of all the gases by using the number of moles of air and number of moles of each gas.
The calculation of one of the gases, say nitrogen, will be done by us and then we would write the rest of the values in the table directly.
\[{{n}_{molecules}}=\dfrac{0.034mol{ }air\times 0.7808{ }mol{{N}_{2}}{ }}{1mol{ }air}=0.026{ }96{ }mol{ }{{N}_{2}}\]
Now we will calculate the number of atoms present in these molecules as each molecule of nitrogen contains two atoms of nitrogen and so we will multiply the number of moles with two, so we get,
\[{{n}_{atoms}}=\dfrac{0.026molecules{ }{{N}_{2}}\times 2{ }atoms{ }N}{1molecule{ }{{N}_{2}}}=0.053molN\]
Number of atoms nitrogen could be calculated by multiplying this number with Avogadro’s number, and we get,
\[Atoms{ }of{ }N=0.053{ }92mol{ }N\times \dfrac{6.022\times {{10}^{23}}atoms{ }N}{1mol{ }N}\]
\[=3.247\times {{10}^{22}}atoms{ }N\]
So, the number of atoms comes out to be \[3.247\times {{10}^{22}}atoms{ }N\]
Now, in the similar way we would calculate the number of atoms in each gas, but in this question we would just write the values in the table directly rather than doing the calculation for each gas.

Gas	Percentage	Number of molecules	Number of atoms	Atoms$/{{10}^{22}}$
${{N}_{2}}$	$78.08$	\[0.026{ }96\]	\[0.053{ }92\]	\[3.247\]
${{O}_{2}}$	$20.95$	\[0.007{ }234\]	\[0.014{ }47\]	\[0.871\]
$Ar$	$0.93$	\[0.000{ }321\]	\[0.000{ }321\]	\[0.0193\]
$C{{O}_{2}}$	$0.04$	\[0.000{ }014\]	\[0.000{ }041\]	\[0.0025\]
	Total	\[0.034{ }53\]	\[0.068{ }752\]	\[4.140\]

So, from the above table we can say that the total number of atoms in a gram of air is, \[4.140\times {{10}^{22}}\] atoms, which is the required answer.

Note: The number of atoms which are present in one gram of air is the sum of the number of atoms of each of the gases present in molecular form. At first we calculated the number of moles of each of the gases present in the air by the help of the percentage composition of those gases and then we calculated the total number of atoms using the number of moles.