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Physics-

General

Easy

Question

The Speed of sound in a mixture of 1 mole of helium and 2 moles of oxygen at $27 to the power of 0 end exponent C$ is

$800 m s^{- 1}$
$400.8 m s^{- 1}$
$600 m s^{- 1}$
$1200 m s^{- 1}$

The correct answer is: $400.8 m s to the power of negative 1 end exponent$

Molecular weight of mixture
$M subscript m i x end subscript equals fraction numerator n subscript 1 end subscript M subscript 1 end subscript plus n subscript 2 end subscript M subscript 2 end subscript over denominator n subscript 1 end subscript plus n subscript 2 end subscript end fraction equals fraction numerator 1 cross times 4 plus 2 cross times 32 over denominator 1 plus 2 end fraction equals fraction numerator 68 over denominator 3 end fraction$
$equals fraction numerator 68 over denominator 3 end fraction cross times 10 to the power of 3 end exponent k g blank m o l to the power of negative 1 end exponent$
For helium $C subscript v subscript 1 end subscript end subscript equals fraction numerator 3 over denominator 2 end fraction R$
For oxygen $C subscript v subscript 2 end subscript end subscript equals fraction numerator 5 over denominator 2 end fraction R$
$open parentheses C subscript v end subscript close parentheses subscript m i x end subscript equals fraction numerator n subscript 1 end subscript C subscript v subscript 1 end subscript end subscript plus n subscript 2 end subscript C subscript v subscript 2 end subscript end subscript over denominator n subscript 1 end subscript plus n subscript 2 end subscript end fraction$
$equals fraction numerator 1 cross times fraction numerator 3 R over denominator 2 end fraction plus 2 cross times fraction numerator 5 R over denominator 2 end fraction over denominator 1 plus 2 end fraction equals fraction numerator 13 R over denominator 6 end fraction$
$open parentheses C subscript p end subscript close parentheses subscript m i x end subscript equals open parentheses C subscript v end subscript close parentheses subscript m i x end subscript plus R$
$equals fraction numerator 13 R over denominator 6 end fraction plus R equals fraction numerator 19 R over denominator 6 end fraction$
$y subscript m i x end subscript equals square root of fraction numerator open parentheses C subscript p end subscript close parentheses subscript m i x end subscript over denominator open parentheses C subscript v end subscript close parentheses subscript m i x end subscript end fraction end root equals fraction numerator 19 over denominator 13 end fraction$
$v equals square root of fraction numerator gamma subscript m i x end subscript R T over denominator M subscript m i x end subscript end fraction end root$
$equals square root of fraction numerator 19 over denominator 13 end fraction cross times fraction numerator 8.31 cross times 300 over denominator fraction numerator 68 over denominator 3 end fraction cross times 10 to the power of negative 3 end exponent end fraction end root equals 400.9 blank m s to the power of negative 1 end exponent$

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