Question

Calculate the speed of sound at 288 K in hydrogen, helium, and nitrogen. Under what conditions will the speed of sound in hydrogen be equal to that in helium?

Answer 1

Step-by-step explanation:

The sped of sound is given as follows.

C =
`√(\gamma RT)`

It is known that for hydrogen,

R = 4124 J/kg K

T = 288 k

`\gamma` = 1.41

Therefore, calculate the value of
`C_(hydrogen)` as follows.

`C_(hydrogen) = √(\gamma RT)`

=
`√(1.41 * 4124 J/kg K * 288)`

= 1294.1 m/s

For helium,

R = 2077 J/kg K

T = 288 k

`\gamma` = 1.66

Therefore, calculate the value of
`C_(helium)` as follows.

`C_(helium) = √(\gamma RT)`

=
`√(1.66 * 2077 J/kg K * 288)`

= 996.48 m/s

For nitrogen,

R = 296.8 J/kg K

T = 288 k

`\gamma` = 1.4

Therefore, calculate the value of
`C_(hydrogen)` as follows.

`C_(hydrogen) = √(\gamma RT)`

=
`√(1.4 * 296.8 J/kg K * 288)`

= 345.93 m/s

So, speed of sound in hydrogen is calculated as follows.

=
`\sqrt{1.41 * 4124 * T_(H)}`

=
`76.26 \sqrt{T_(H)}`

Speed of sound in helium is as follows.

=
`\sqrt{1.66 * 2077 * T_(He)}`

=
`58.72 \sqrt{T_(He)}`

For both the speeds to be equal,

`76.26 \sqrt{T_(H)}` =
`58.72 \sqrt{T_(He)}`

`(T_(H))/(T_(He))` = 0.593

Therefore, we can conclude that the temperature of hydrogen is 0.593 times the temperature of helium.