JEE MAIN - Physics (2008 - No. 24)
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 $$(He)$$ at the same temperature will be (assume both gases to be ideal)
$$1421\,\,m{s^{ - 1}}$$
$$500\,\,m{s^{ - 1}}$$
$$650\,\,m{s^{ - 1}}$$
$$300\,\,m{s^{ - 1}}$$
Explanation
The speed of sound in a gas is given by $$v = \sqrt {{{\gamma RT} \over M}} $$
$$\therefore$$ $${{{v_{{O_2}}}} \over {{v_{He}}}} = \sqrt {{{{\gamma _{{O_2}}}} \over {{M_{{O_2}}}}} \times {{{M_{He}}} \over {{\gamma _{He}}}}} $$
$$ = \sqrt {{{1.4} \over {32}} \times {4 \over {1.67}}} = 0.3237$$
$$\therefore$$ $${v_{He}} = {{{v_{{O_2}}}} \over {0.3237}}$$
$$ = {{460} \over {0.3237}}$$
$$ = 1421\,m/s$$
$$\therefore$$ $${{{v_{{O_2}}}} \over {{v_{He}}}} = \sqrt {{{{\gamma _{{O_2}}}} \over {{M_{{O_2}}}}} \times {{{M_{He}}} \over {{\gamma _{He}}}}} $$
$$ = \sqrt {{{1.4} \over {32}} \times {4 \over {1.67}}} = 0.3237$$
$$\therefore$$ $${v_{He}} = {{{v_{{O_2}}}} \over {0.3237}}$$
$$ = {{460} \over {0.3237}}$$
$$ = 1421\,m/s$$
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