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

Hint: 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,
\[\begin{array}{l}
{V_O} = \sqrt {\dfrac{{1.4TR}}{{{M_O}}}} \\
{V_O} = \sqrt {\dfrac{{1.4TR}}{{16{M_H}}}}
\end{array}\]
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.