Question

The molecules of a given mass of a gas have rms velocity of \[200m/s\] at \[{27^ \circ }C\] and \[1.0 \times {10^5}N{m^{ - 2}}\] pressure. When the temperature and pressure of the gas are respectively, \[{127^ \circ }C\] and \[0.05 \times {10^5}N{m^{ - 2}}\], the rms velocity of the its molecules in m/s is:
A. \[100\sqrt 2 \]
B. \[\dfrac{{400}}{{\sqrt 3 }}\]
C. \[\dfrac{{100\sqrt 2 }}{3}\]
D. \[\dfrac{{100}}{3}\]

Answer 1

Hint: There are two formulas of rms velocity one involves temperature and molecular weight while other formula involves pressure and density, since density is not given and molecular weight for a particular gas remains constant unless there is some chemical reaction, you can use the formula involving temperature and molecular weight.

Complete step by step solution:
Rms velocity is square root of sum of squares of velocities of each molecules divided by the total number of molecules,
and rms is abbreviated as root mean square speed,
for an ideal gas, using kinetic theory of gas
we know that rms velocity \[{v_{rms}} = \sqrt {\dfrac{{3RT}}{M}} \]
where, R is gas constant, T is temperature of the gas and M is molecular weight of the gas,
now if we compare the rms velocity at two different temperature
we will get \[\dfrac{{{{[{v_{rms}}]}_1}}}{{{{[{v_{rms}}]}_2}}} = \dfrac{{\sqrt {\dfrac{{3R{T_1}}}{M}} }}{{\sqrt {\dfrac{{3R{T_2}}}{M}} }} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}} \]
now if we put the values of temperature and rms velocity in the above expression we will get,
\[\dfrac{{{{[{v_{rms}}]}_1}}}{{{{[{v_{rms}}]}_2}}} = \dfrac{{\sqrt {\dfrac{{3R{T_1}}}{M}} }}{{\sqrt {\dfrac{{3R{T_2}}}{M}} }} = \sqrt {\dfrac{{{T_1}}}{{{T_2}}}}
\Rightarrow \dfrac{{200}}{{{{[{v_{rms}}]}_2}}} = \sqrt {\dfrac{{300K}}{{400K}}} \]
\[ \Rightarrow {[{v_{rms}}]_2} = 200\sqrt {\dfrac{4}{3}} = \dfrac{{400}}{{\sqrt 3 }}\]
Because \[{T_1} = {27^ \circ }C = 300K\& {T_2} = {127^ \circ }C = 400K\]

So, the correct answer is “Option B”.

Note:
Here pressure is given in the question to confuse the student it has no use here although it could have been useful if density were mentioned but rms velocity all you need is temperature and molecular weight and if it given in the question you should not worry about pressure and volume, one more thing you should never forget to change the units of temperature in kelvin while putting the value of temperature in the formula.