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Physics-

General

Easy

Question

The ratio of speed of sound in nitrogen and helium gas at 300 K is

$\sqrt{\frac{2}{7}}$
$\frac{\sqrt{1}}{7}$
$\frac{\sqrt{3}}{5}$
$\frac{\sqrt{6}}{5}$

The correct answer is: $fraction numerator square root of 3 over denominator 5 end fraction$

According to Laplace, the speed of sound in a gas is
$v equals square root of fraction numerator gamma p over denominator d end fraction end root equals square root of fraction numerator gamma R T over denominator M end fraction end root$
Where R is gas constant, T the temperature and M the molecular weight.
For monatomic gas helium, $gamma subscript 1 end subscript equals fraction numerator 5 over denominator 3 end fraction comma M subscript 1 end subscript equals 4$
For diatomic gas nitrogen, $gamma subscript 2 end subscript equals fraction numerator 7 over denominator 5 end fraction comma M subscript 2 end subscript equals 28$
$therefore blank fraction numerator v subscript N subscript 2 end subscript end subscript over denominator v subscript H subscript e end subscript end subscript end fraction equals square root of fraction numerator gamma subscript 1 end subscript M subscript 2 end subscript over denominator gamma subscript 2 end subscript M subscript 1 end subscript end fraction end root$
$equals square root of fraction numerator 5 divided by 3 over denominator 7 divided by 5 end fraction cross times fraction numerator 28 over denominator 4 end fraction end root equals square root of fraction numerator 5 cross times 5 over denominator 3 end fraction end root equals fraction numerator 5 over denominator square root of 3 end fraction$
Hence,
$fraction numerator v subscript h end subscript subscript 2 end subscript over denominator v subscript H e end subscript end fraction equals fraction numerator square root of 3 over denominator 5 end fraction$

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