One mole of nitrogen gas being initially at a temperature of `T_(0)=300` K is adiabayically compressed to increase its pressure 10 times. The final gas temperature after compression is (Assume, nitrogen gas molecules as rigid diatomic and `100^(1//7)=1.9`)

Options

A. 120 K
B. 750 K
C. 650 K
D. 570 K

Answer

D

Explanation

Given, initial temeperature of 1 mole of `N_(2)` gas
`T_(0)=300 K`
initial pressure of gas, `p_(1)=p`
and final pressure of gas, `p_(2)=10 p`
By adiabatic process relation between p and T,
`p_(1)^(1-gamma)T_(1)^(gamma)=p_(2)^(1-gamma).T_(2)^(gamma)`
`(p_(1)/p_(2))^(1-gamma)=(T_(2)/T_(1))^(gamma)implies (p/(10p))^(1-gamma)=(T_(2)/300)^(gamma)`
`10^(gamma-1)=T_(2)^(gamma)/300^(gamma)implies T_(2)^(gamma)=10^(gamma-1)xx300^(gamma)`
`implies T_(2)=10^((gamma-1)/gamma)xx300=10^(1-1/gamma)xx300`
`=10^(1-1/(7//5))xx300 [ :' "For diatomic, "gamma=7/5]`
`=10^(2//7)xx300=(100)^(1//7)xx300`
`=1.9xx300" "[ :' 100^(1//7)=1.9]`
`=570 K`
Therefore, the final gas temperature after compression is 570 K.