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_(2)/L_(1))^(2//3)`
C. `sqrt(L_(2)/L_(1))`
D. `sqrt(L_(1)/L_(2))`

Answer

A

Explanation

For a monoatomic gas, `gamma=5/3`
If A is the internal area of cross section of the cylinder, then initial volume `V_(1)=AL` and final volume `V_(2)=AL_(2)` For the adiabatic change, `PV^(gamma)` = constant
or `TV^(gamma-1)` = constant
`thereforeT_(1)V_(1)^(gamma-1)=T_(2)V_(2)^(gamma-1)`
`therefore(T_(2)/T_(1))=(V_(1)/V_(2))^(gamma-1)=[(AL_(1))/(AL_(2))]^(gamma-1)=[L_(1)/L_(2)]^(5//3-1)`
`thereforeT_(2)/T_(1)=[L_(1)/L_(2)]^(2//3)`