A monoatomic ideal gas is compressed adiabatically to ( 1 27 ) of its initial volume. If initial temperature of the gas is ' T ' K and final temperature is ' xT ' K , the value of ' x ' is

Step-by-step solution

The temperature change in an adiabatic process for an ideal gas follows TV^ -1 = constant . For a monatomic gas, = 5 3 . Given T₁ = T , T₂ = xT , V₁ , and V₂ = V₁ 27 , the adiabatic relation yields: T V₁^ 2/3 = xT ( V₁ 27 )^ 2/3 Dividing both sides by T and V₁^ 2/3 gives 1 = x 27^ 2/3 . Since 27^ 2/3 = (3^3)^ 2/3 = 3^2 = 9 , the equation simplifies to 1 = x 9 . The solution is therefore x = 9 .