Question
The speed of sound in oxygen $$\left( {{O_2}} \right)$$ at a certain temperature is $$460\,\,m\,{s^{ - 1}}.$$ The speed of sound in helium $$\left( {{He}} \right)$$ at the same temperature will be (assume both gases to be ideal)
A.
$$1421\,\,m\,{s^{ - 1}}$$
B.
$$500\,\,m\,{s^{ - 1}}$$
C.
$$650\,\,m\,{s^{ - 1}}$$
D.
$$330\,\,m\,{s^{ - 1}}$$
Answer :
$$1421\,\,m\,{s^{ - 1}}$$
Solution :
The speed of sound in a gas is given by $$v = \sqrt {\frac{{\gamma RT}}{M}} $$
$$\eqalign{
& \therefore \,\,\frac{{{v_{{O_2}}}}}{{{v_{He}}}} = \sqrt {\frac{{{\gamma _{{O_2}}}}}{{{M_{{O_2}}}}} \times \frac{{{M_{He}}}}{{{\gamma _{He}}}}} \cr
& = \sqrt {\frac{{1.4}}{{32}} \times \frac{4}{{1.67}}} \cr
& = 0.3237 \cr
& \therefore \,\,{v_{He}} = \frac{{{v_{{O_2}}}}}{{0.3237}} \cr
& = \frac{{460}}{{0.3237}} \cr
& = 1421\,\,m/s \cr} $$