A monoatomic ideal gas, initially at temperature `T_1,` is enclosed in a cylinder fitted with a friction less piston. The gas is allowed to expand adiabatically to a temperature `T_2` by releasing the piston suddenly. If `L_1 and L_2` are the length of the gas column before expansion respectively, then `(T_1)/(T_2)` is given by

Options

A. `((L_(1))/(L_(2)))^(2//3)`
B. `((L_(1))/(L_(2)))`
C. `((L_(2))/(L_(1)))`
D. `((L_(2))/(L_(1)))^(2//3)`

Answer

D

Explanation

For an adiabatic process,
`T_(1)V_(1)^(gamma-1)=T_(2)V_(2)^(gamma-1)`
`therefore" " T_(1)L_(1)^(gamma-1)=T_(2)L_(2)(gamma-1)" " (thereforeA="constant")`
`"or " (T_(1))/(T_(2))=((L_(2))/(L_(1)))^(gamma-1)`
`therefore " " gamma =(5)/(3)` for a monoatomic gas,
`therefore " " gamma -1=(2)/(3)`
`"Thus "(T_(1))/(T_(2))=((L_(1))/(L_(2)))^(2//3)`