A monoatomic ideal gas, initially at temperature T₁ , is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature T₂ by releasing the piston suddenly. If L and 2 L are the lengths of the gas column before and after expansion respectively, then T₁ T₂ is

Step-by-step solution

For an adiabatic process of an ideal gas, the relationship between temperature and volume is given by T V^ - 1 = constant . For a monoatomic gas, the adiabatic index = 5 3 . Thus, - 1 = 5 3 - 1 = 2 3 . The volume of the gas is proportional to the length of the gas column, V L . Therefore, T₁ L₁^ 2/3 = T₂ L₂^ 2/3 . Given L₁ = L and L₂ = 2L , we have T₁ L^ 2/3 = T₂ (2L)^ 2/3 . Rearranging to find the ratio T₁ T₂ : T₁ T₂ = (2L)^ 2/3 L^ 2/3 = ( 2L L )^ 2/3 = 2^ 2/3 . Answer: 2^ 2 / 3