Two thermally insulated vessels of volume `V_(1) and V_(2)` are filled with oxygen and connected by a small tube equipped with a value. Pressure of oxygen in them is `P_(1) and P_(2)` respectivley and temperature is `T_(1) and T_(2)` , respectively. Calculate velocity of sound in oxygen when value is opened and thermal equilibrium is established.
(Given, molar mass of oxygen = M)

Explanation

Since, vessels are thermally insulated, therefore, no heat flow takes place and there is no net work done by system of gas in the two vessels. Hence, for whole gas Q = 0, W = 0.
But `Q = W + Delta U ` . Therefore increase in internal energy of the system is `Delta U = 0 ` Using this condition, final temperature T of the gas can be calculated. Using gas equation PV = nRT
Initial number of moles in first vessel , `n_(1) = (P_(1)V_(1))/(RT_(1))`
and initial number of moles in second vessel,
`n_(2) = (P_(2)V_(2))/(RT_(2))`
Let molar specific heat of oxygen at constant volume be `C_(u)`
Initial internal energy of oxygen in first vessel, `U_(2) n_(2) C_(u) T_(2)`
Total final internal energy of oxygen in the two vessels, `U = (n_(1) + n_(2)) C_(u) T `
But for the system `Delta U = 0 `
`:. U = U_(1) + U_(2)`
or, `T = (n_(1)T_(2) +n_(2) T_(2))/(n_(1) + n_(2)) = ((P_(1)V_(2)+P_(2)V_(2))/(P_(1)V_(1)T_(1)+P_(2)V_(2)T_(1)))T_(1)T_(2)`
Velocity of sound waves in a gas at temperature T is
`u = sqrt((gammaRT)/(M))`
Since , oxygen is diatomic, therefore its adiabatic
constant, `gamma = (7)/(5)`
`:.` Velocity of sound in oxygen, u ` = sqrt((7)/(5)*(RT)/(M))`
`= sqrt((7R)/(5M) ((P_(1)V_(2)+P_(2)V_(2))/(P_(1)V_(1)T_(1)+P_(2)V_(2)T_(1)))T_(1)T_(2))`