A diatomic gas ( = 7 5 ) is compressed adiabatically to volume v ₀ 32 , where V₀ is its initial volume. The initial temperature of the gas is T_i in kelvin and the final temperature is xT _ i in kelvin. The value of x is

Step-by-step solution

For an adiabatic process with diatomic gas ( = 7/5 ), the temperature-volume relation T V^ -1 = constant applies. Given initial conditions V_i = V₀ , T_i and final conditions V_f = V₀/32 , T_f = x T_i , the adiabatic relation becomes: T_i V₀^ 2/5 = x T_i (V₀/32)^ 2/5 Dividing both sides by T_i V₀^ 2/5 yields 1 = x / 32^ 2/5 . Since 32 = 2^5 , we have 32^ 2/5 = (2^5)^ 2/5 = 2^2 = 4 . Thus x = 4 , corresponding to option B.