Question

The density of nitrogen and oxygen at NTP are $1.25\text{ kg/}{{\text{m}}^{3}}\text{ and 1}\text{.43 kg/}{{\text{m}}^{3}}$ respectively. If the speed of sound in oxygen at NTP is 320 m/s. Calculate the speed of sound in nitrogen, under the same conditions of temperature and pressure. (y for both the gases is 1.4)

Answer 1

Hint: The speed of sound for any gas can be expressed as \[\sqrt{\dfrac{\gamma \text{RT}}{\text{M}}}\]. Here, \[\gamma \] is the specific heat ratio which is fixed for the monatomic and diatomic gas, R is the gas constant, T is the absolute temperature and M is the molar mass.

Complete answer:
- In the given question, we have to find the speed of sound in nitrogen gas under the same conditions of temperature and pressure.
- It is given that the density of nitrogen and oxygen is $1.25\text{ kg/}{{\text{m}}^{3}}\text{ and 1}\text{.43 kg/}{{\text{m}}^{3}}$
 At NTP conditions and the speed of sound in oxygen at 320 m/s.
- Also, the value of specific heat ratio is given for the diatomic gas that is 1.4.
- So, firstly, we will apply the formula of the speed of sound in oxygen and we will get the value of the molar mass of oxygen:
Speed of sound = \[\sqrt{\dfrac{\gamma \text{RT}}{\text{M}}}\]
320 = \[\sqrt{\dfrac{\text{1}\text{.4 }\times \text{ 8}\text{.314 }\times \text{ 293}}{{{\text{M}}_{{{\text{O}}_{2}}}}}}\]
\[{{\text{M}}_{{{\text{O}}_{2}}}}\] = 33.30 kg/K mol.
- Now, the value of density and mass is given so, we can calculate the volume of oxygen by applying the density formula:
$\text{D = }\dfrac{\text{Mass}}{\text{Volume}}$
$\text{1}\text{.43 = }\dfrac{33.30}{\text{Volume}}$
Volume = 23.25 mL.
- Now, for nitrogen, it is given that the density and volume of nitrogen are $1.25\text{ kg/}{{\text{m}}^{3}}$ and 23.28 mL, so the mass will be:
$\text{D = }\dfrac{\text{Mass}}{\text{Volume}}$ or \[\text{Mass = volume }\times \text{ density}\]
\[\text{Mass = 23}\text{.28 }\times \text{ 1}\text{.43 = 29}\text{.1 kg/k mole}\]
- Now, we will calculate the speed of sound in nitrogen:
\[\sqrt{\dfrac{\gamma \text{RT}}{\text{M}}}\] = \[\sqrt{\dfrac{\text{1}\text{.4 }\times \text{ 8}\text{.314 }\times \text{ 293}}{29.1\text{ }\times \text{ 1}{{\text{0}}^{-3}}}}\] = 342.33 m/s.
Therefore, the speed of sound in nitrogen is 342.33 m/s.

Note:
Gamma is also known as the adiabatic index. The speed of sound in a gas depends on two factors i.e. density of the medium and temperature of the medium. With increase in the density, the speed of the sound decreases. Whereas with the increase in the temperature, the speed of the sound also increases.