Question

The constant γ for oxygen as well as for hydrogen is 1.40 . If the speed of sound in oxygen is 450m/s, what will be the speed of hydrogen at the same temperature and pressure?

Answer 1

In the solution, the molecular mass of hydrogen is 1 and the oxygen is 16, equate both terms.

Complete step by step solution:
Given:
The adiabatic constant of the hydrogen is ${\gamma _H} = 1.4$.
The speed of the sound in oxygen is ${V_O} = 450\;{\rm{m/s}}$.
The molecular mass of Hydrogen is ${M_H} = 1$
The molecular mass of Oxygen is ${M_O} = 16$
Let us assume ${M_O} = 16{M_H}$
The equation of the speed of the sound in Hydrogen is,
${V_H} = \sqrt {\dfrac{{1.4TR}}{{{M_H}}}}$
The equation of the speed of the sound in Oxygen is,

{V_O} = \sqrt {\dfrac{{1.4TR}}{{{M_O}}}} \\\ {V_O} = \sqrt {\dfrac{{1.4TR}}{{16{M_H}}}} \end{array}$$ Here, T is the Temperature of the molecule. Now, taking the ratios of the speed of sound in hydrogen to the oxygen is, $$\begin{array}{l} \dfrac{{{V_H}}}{{{V_O}}} = \sqrt {\dfrac{{1.4TR \times 16{M_H}}}{{1.4TR \times {M_H}}}} \\\ = \sqrt {\dfrac{{16}}{1}} \\\ = \dfrac{4}{1} \end{array}$$ Now, $$\begin{array}{c} {V_H} = {V_O} \times 4\\\ = 4 \times 450\\\ = 1800\;{\rm{m/s}} \end{array}$$ **Therefore, the speed of the sound in the hydrogen is $$1800\;{\rm{m/s}}$$.** **Note:** The $\gamma $ is the gas constant whereas it is a constant value and remains the same 1.4 for all the gases. Be sure about the molecular mass of the hydrogen and the oxygen.