Question

Find the rms speed of an argon molecule at ${27}^{0}C$ (Molecular weight of argon=$40$ gm/mol)
A.$234.2m/s$
B.$342.2m/s$
C.$432.2m/s$
D.$243.2m/s$

Answer 1

Hint: -First convert the temperature in Kelvin by adding $273.15$ to the given value. Then convert the molecular mass from gm to kg. Then use the relation between rms speed, temperature and molecular mass, which is given as-
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{3RT}{M}}}$ Where $v_{rms}$ is rms speed of the gas, T is the temperature, M is the molecular mass of the gas and R is universal gas constant. The value of R=$8.314$ J/molK. Now put the given values in the equation and solve it to get the answer.

Complete step by step answer:
Given temperature=${27}^{0}C$
We know that to convert degree Celsius into Kelvin we add $273.15$ to the value of temperature.
So temperature T=$273.15+27=300.15$ K
Also given the molecular mass of argon=$40$ gm/mol
And we know that $1kg=1000gm$ so –
$\Rightarrow 1gm={10}^{-3}kg$
Then the molecular mass of argon M=$40\times {10}^{-3}$kg/mol
Now we have to find the rms speed of argon.
We know that the relation between rms speed, temperature and molecular mass is given as-
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{3RT}{M}}}$ Where $v_{rms}$ is rms speed of the gas, T is the temperature, M is the molecular mass of the gas and R is universal gas constant. The value of R=$8.314$ J/molK
On putting the given values in the formula, we get-
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{3\times 8.314\times 300.15}{40\times {10}^{-3}}}}$
On solving, we get-
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{7486.3413}{40\times {10}^{-3}}}}$
On division, we get-
$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{187.1585325}{{10}^{-3}}}}$
On simplifying, we get-
$$\Rightarrow v_{rms}=\sqrt{{\displaystyle \frac{187158.5325\times {10}^{-3}}{{10}^{-3}}}}=\sqrt{187158.5325}$$
$\Rightarrow v_{rms}=432.2\text{ m/s}$

Hence the correct answer is C.

Note:
Argon is a noble gas which is colorless and odorless. It is an inert gas. It is used for-
-It is used in fluorescent tubes and low-energy light bulbs.
-It is used in the double-glazed windows to fill the space between panes.
-It is used in the tyres of luxury cars to reduce road noise and to protect the rubber.
-It is used in processing of titanium and in arc welding.
-The atmosphere created using this gas is used to grow crystals of silicon and germanium.