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Show that the velocity of sound in a gas...

Show that the velocity of sound in a gas is given by `v_t = v_(0) + (v_(0)//546) t`, where `v_(0)` is the velocity at `0^@C` and t is the temperature of the gas in `""^(@)C`. Assume `t lt lt 273`.

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The velocity of sound in a gas at temperature .t. is `v_t=sqrt((gamma RT)/(M))" "`...(1)
Now, `T = 273 + t = 273(1+ (t)/(273))`
In our case, `(t)/(273) lt lt 1`, so that
`sqrt(T)=sqrt(273(1+(t)/(273)))=sqrt(273)(1+(t)/(546))`
Putting the value of `sqrt(T)` in Eq. (1), we get
`v_t=sqrt((gamma)/(M)R(273))xx(1+(t)/(546))`
But the quantity under the square root symbol is the velocity at `0^@C` (i.e., `v_(0)`)
`:. v_t=v_(0)xx(1+(t)/(546))` or `v_t=v_(0)+(v_(0))/(546)t`
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