A monoatomic ideal gas is compressed adiabatically to 12.5 % of its initial volume. if the initial temperature of the gas is ' T ₁ ^ K and the final temperature is ' xT ₁ ^ K , the value of x is

Step-by-step solution

For an adiabatic process in an ideal gas, the relation T V^ - 1 = constant holds, where = C_p / C_v is the adiabatic index. Given that the gas is monoatomic, = 5/3 . The initial conditions are temperature T₁ and volume V₁ , and the final temperature is T₂ = x T₁ with volume reduced to V₂ = V₁ / 8 . Applying the adiabatic relation: T₁ V₁^ 2/3 = T₂ V₂^ 2/3 Substitute the known values: T₁ V₁^ 2/3 = (x T₁) (V₁ / 8)^ 2/3 Simplify by canceling T₁ V₁^ 2/3 from both sides: 1 = x / 8^ 2/3 Since 8^ 2/3 = (2^3)^ 2/3 = 2^2 = 4