Problem 3.2

Given: \(T= \) 288 K.

To calculate:
Speed of sound in H₂.
If, speed in hydrogen = speed of sound in helium, then \( T= \)?.

The speed of sound can be calculated for any gas using the temperature, the gas constant, \( R \), and the ratio of specific heats, \( \gamma \), using the formula,

\[ a=\sqrt{\gamma\,R\,T}\ . \]

Speed of sound in hydrogen, H₂:

The molar mass is,

\[ \hat{m}=2.016\,\text{kg/kmol}\ , \]

and the ratio of specific heats is,

\[ \gamma=1.407\ . \]

The gas constant is,

\[ R=\frac{R_{u}}{\hat{m}}=\frac{8314}{\hat{m}}, \text{J/kg-K}\ . \]

Therefore,

\[ a=\sqrt{\gamma\,R\,T}=\sqrt{\gamma\,\frac{R_{u}}{\hat{m}}\,T}=\sqrt{1.407\times\frac{8314}{2.016}\times288} \]

\[ \boxed{a=1292.72\,\text{m/s}}\ . \]

If, speed of sound in hydrogen = speed of sound in helium,

\[ \sqrt{1.407\times\frac{R_{u}}{2.016}\times T_{H_{2}}}=\sqrt{1.667\times\frac{R_{u}}{4.003}\times T_{He}} \]

\[ T_{H_{2}}=\frac{2.016\times1.667}{1.407\times4.003}\times T_{He} \]

\[ \boxed{T_{H_{2}}=0.597\,T_{He}}\ . \]

\[ \boxed{T_{H_{2}}=0.597\times288= 171.94\,\text{K}}\ . \]

The speed of sound in hydrogen will be equal to that in helium when, the temperature (measured in Kelvin) in hydrogen is about 0.597 times of that in helium. So, the temperature of hydrogen needs to be about 171.94 K for the speed of sound in hydrogen to be equal to that in helium.

Problem 3.2:

Solution:

Speed of sound in hydrogen, H₂:

Problem 3.2:

Solution:

Speed of sound in hydrogen, H2:

Speed of sound in hydrogen, H₂: