An ideal gas undergoes an adiabatic expansion such that its final volume is 27 times its initial volume. If the final absolute temperature of the gas is 1 9 of its initial absolute temperature, the ratio of specific heats ( ) for the gas is :

Step-by-step solution

For an adiabatic process, the relationship between temperature and volume is given by: T V^ -1 = constant T₁ V₁^ -1 = T₂ V₂^ -1 ( V₂ V₁ )^ -1 = T₁ T₂ Given that V₂ = 27 V₁ and T₂ = T₁ 9 , we substitute these values into the equation: (27 )^ -1 = 9 (3^3 )^ -1 = 3^2 3^ 3( -1) = 3^2 Equating the exponents of both sides: 3( -1) = 2 - 1 = 2 3 = 1 + 2 3 = 5 3 Answer: 5 3