The pressure of a gas is raised from $${27^ \circ }C$$ to $${927^ \circ }C.$$ The root mean square speed
A.
is $$\sqrt {\left( {\frac{{927}}{{27}}} \right)} $$ times the earlier value
B.
remains the same
C.
gets halved
D.
gets doubled
Answer :
gets doubled
Solution :
$$RMS$$ speed is defined as the square root of the mean of the squares of the random velocities of the individual molecules of a gas. From Maxwellian distribution law, $$RMS$$ speed is given by $${c_{rms}} = \sqrt {\left( {\frac{{3kT}}{m}} \right)} $$
$$ \Rightarrow {c_{rms}} \propto \sqrt T $$
For two different cases i.e. at two different temperatures
$$\therefore \frac{{{{\left( {{c_{rms}}} \right)}_1}}}{{{{\left( {{c_{rms}}} \right)}_2}}} = \sqrt {\frac{{{T_1}}}{{{T_2}}}} $$
Here, $${T_1} = {27^ \circ }C = 300\,K$$
$$\eqalign{
& {T_2} = {927^ \circ }C = 1200\,K \cr
& \therefore \frac{{{{\left( {{c_{rms}}} \right)}_1}}}{{{{\left( {{c_{rms}}} \right)}_2}}} = \sqrt {\frac{{300}}{{1200}}} = \frac{1}{2} \cr
& \Rightarrow {\left( {{c_{rms}}} \right)_2} = 2{\left( {{c_{rms}}} \right)_1} \cr} $$
Hence, root mean square speed will be doubled.
Releted MCQ Question on Heat and Thermodynamics >> Kinetic Theory of Gases
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