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Question: Find the rms speed of an argon molecule at \(27^0C\) (Molecular weight of argon=\(40\) gm/mol) A.\...

Find the rms speed of an argon molecule at 270C27^0C (Molecular weight of argon=4040 gm/mol)
A.234.2 m/s234.2{\text{ }}m/s
B.342.2 m/s342.2{\text{ }}m/s
C.432.2 m/s432.2{\text{ }}m/s
D.243.2 m/s243.2{\text{ }}m/s

Explanation

Solution

-First convert the temperature in Kelvin by adding 273.15273.15 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-
⇒vrms=3RTM\Rightarrow {v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} Where vrms{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=8.3148.314 J/molK. Now put the given values in the equation and solve it to get the answer.

Complete step by step answer:
Given temperature=270C27^0C
We know that to convert degree Celsius into Kelvin we add 273.15273.15 to the value of temperature.
So temperature T=273.15+27=300.15273.15 + 27 = 300.15 K
Also given the molecular mass of argon=4040 gm/mol
And we know that 1kg=1000gm1kg = 1000gm so –
⇒1gm=10−3kg\Rightarrow 1gm = {10^{ - 3}}kg
Then the molecular mass of argon M=40×10−340 \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-
⇒vrms=3RTM\Rightarrow {v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} Where vrms{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=8.3148.314 J/molK
On putting the given values in the formula, we get-
⇒vrms=3×8.314×300.1540×10−3\Rightarrow {v_{rms}} = \sqrt {\dfrac{{3 \times 8.314 \times 300.15}}{{40 \times {{10}^{ - 3}}}}}
On solving, we get-
⇒vrms=7486.341340×10−3\Rightarrow {v_{rms}} = \sqrt {\dfrac{{7486.3413}}{{40 \times {{10}^{ - 3}}}}}
On division, we get-
⇒vrms=187.158532510−3\Rightarrow {v_{rms}} = \sqrt {\dfrac{{187.1585325}}{{{{10}^{ - 3}}}}}
On simplifying, we get-
⇒vrms=187158.5325×10−310−3=187158.5325\Rightarrow {v_{rms}} = \sqrt {\dfrac{{187158.5325 \times {{10}^{ - 3}}}}{{{{10}^{ - 3}}}}} = \sqrt {187158.5325}
⇒vrms=432.2 m/s\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.