Question

If 224ml of a triatomic gas has a mass of $1g$ at $273K$ and $1atm$ pressure, then the mass of one atom is:
A: $8.30 \times {10^{ - 23}}gm$
B: $2.08 \times {10^{ - 23}}gm$
C: $5.53 \times {10^{ - 23}}gm$
D: $6.24 \times {10^{ - 23}}gm$

Answer 1

Hint: We know that triatomic molecules are made up of three atoms. For ideal gas we can use the ideal gas equation that is $PV = nRT$.Here $P = $pressure,$V = $volume,$n = $number of moles,$R = $ideal gas constant,$T = $ temperature.In one mole there are approximately $6.022 \times {10^{23}}$ particles.This number is known as Avogadro number as it was discovered by the famous scientist Amedeo Avogadro.

Complete step by step answer:
As in the question we are given the volume, temperature , mass and pressure is given. It is given that the molecule is triatomic that means it is made of three atoms. We can directly apply the ideal gas equation to solve the question $PV = nRT$ . Here $P = $pressure,$V = $volume,$n = $number of moles,$R = $ideal gas constant,$T = $temperature.
$
  P = 1atm \\
  V = 224ml \\
  T = 273K \\
  Mass = 1g \\
 $
Let the molecular mass of the triatomic gas be $M$.
$1atm \times \dfrac{{224L}}{{1000}} = \dfrac{1}{M} \times 0.0821Latm{K^{ - 1}}mo{l^{ - 1}} \times 273K$
$ \Rightarrow M = 100.05gm$
Now we found the molecular mass of the triatomic gas. As it is made up of three atoms, to find the atomic mass of the atom we will have to divide the molecular mass by $3$ .
So the Atomic mass $ = \dfrac{{100.05}}{3} = 33.35gm$ . If we go by the definition of atomic mass, it means that the mass of $6.022 \times {10^{23}}$ particles is equal to the atomic mass. So to find out the mass of one atom we need to divide the atomic mass by Avogadro number.
The mass of one atom$ = \dfrac{{33.35}}{{6.022 \times {{10}^{23}}}}gm = 5.53 \times {10^{ - 23}}gm$

So, the correct answer is Option C.

Note:
In the above question we have assumed the gas to be ideal gas. Only ideal gases follow the ideal gas equation. In ideal gas the volume of the actual atom is considered to be zero. The molecules feel no attraction or repulsion. The collisions between molecules are considered to be elastic.
Always remember that diatomic molecules are made up of two atoms , triatomic molecules are made up of three atoms and so on. Always use an ideal gas equation when the gas is ideal to solve questions where we have to find a relation in between volume,pressure,temperature and number of moles. Always be careful and avoid calculation errors.