A monoatomic ideal gas initially at temperature ' T ₁ ' is enclosed in a cylinder fitted with massless, frictionless piston. By releasing the piston suddenly the gas is allowed to expand to adiabatically to a temperature ' T ₂ '. If ' L ₁ ' and ' L ₂ ' are the lengths of the gas columns before and after expansion respectively, then T₂ T₁ is

Step-by-step solution

For an adiabatic process aligned & T ₁ ~V ₁^ -1 = T ₂ ~V ₂^ -1 & T ₂ ~T ₁ = ( V ₁ ~V ₂ )^ -1 aligned For a monoatomic gas, r= 5 3 aligned & -1= 5 3 -1= 2 3 & ~V ₁= AL ₁ and V ₂= AL ₂ & T ₂ ~T ₁ = [ AL ₁ AL ₂ ]^ 2 / 3 = [ L ₁ ~L ₂ ]^ 2 / 3 & -1= 2 3 aligned For an adiabatic process, aligned & T ₂ ~T ₁ = ( V ₁ ~V ₂ )^ -1 = ( V ₁ ~V ₂ )^ 2 3 ~V & L T ₂ ~T ₁ & = ( L ₁ ~L ₂ )^ 2 3 aligned