Question

The speed of sound in hydrogen at STP is \[V\]. The speed of sound in a mixture containing $3$ parts of hydrogen and $2$ parts of oxygen by volume at STP is $V\sqrt x $ then $x$ is
A. 7
B. 9
C. 13
D. 10

Answer 1

Hint: The density of oxygen is equal to $2\rho $. The density of hydrogen is equal to $32\rho $. We will multiply the density with the number of molecules. Later, divide the total value by the total number of molecules.

Complete step by step answer:
We are given with the information that the speed of sound in hydrogen at STP is \[V\].
We also know that The speed of sound in a mixture containing $3$ parts of hydrogen and $2$ parts of oxygen by volume at STP is $V\sqrt x $. Now, we have to find the value of $x$.
Let us first find the $\rho $ of the mixture. We can find ${\rho _{mix}}$ by taking the sum of the product of the 3 parts of molecular mass of hydrogen and the 3 parts of molecular mass of oxygen divided by total number of parts. Here, the number of oxygen molecules is $2$, the number of hydrogen molecules is $3$, the molecular mass of oxygen is $32$, and the molecular mass of hydrogen is $2$.
$
{\rho _{mix}} = \dfrac{{3(2) + 2(32)}}{5} \\
\Rightarrow {\rho _{mix}} = \dfrac{{70}}{5} \\
\Rightarrow {\rho _{mix}} = 14\rho $
Where, density of ${H_2} = 2\rho $.
The density of ${O_2} = 32\rho $.
Along with this, we have $V \propto \dfrac{1}{{\sqrt 7 }}V$.
Now, on the basis of the above values, let us find the value of $x$.
\[\dfrac{{{V^n}}}{V} = \sqrt {\dfrac{{2\rho }}{{14\rho }}} = \dfrac{1}{{\sqrt 7 }}\]
\[\therefore{V^n} = \dfrac{1}{{\sqrt 7 }}\]
Therefore, the value of $x$ is $7$.

So, option A is the correct answer.

Additional Information:
The volume occupied by a gas depends on the amount of the substance (the gas) as well as the temperature and the pressure’ states the ideal gas law. STP, that is, Standard Temperature and Pressure are 0 degree Celsius and 1 atmosphere of pressure. The parameters of the gases which are important for many calculations in physics as well as chemistry are usually calculated at STP.The ideal gas law can be written as \[V = \dfrac{{nRT}}{P}\] where $P$ is the pressure, $V$is the volume, $n$ is the number of moles of a gas, $R$ is the molar gas constant, and $T$ is the temperature.

Note: Students should know that can be calculated by multiplying the molecular mass with the number of molecules. Further they need to divide the total value obtained with the total number of molecules. Students often forget to multiply the molecular mass with the number of molecules.