Discussion Forum

One mole of nitrogen gas being initially  at a temperature of T₀ =300K  is adiabatically  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)

Answer:

Explanation:

given, initial  temperature of 1 mole of $N_{2}$ gas

   $T_{0}$ =300K

 initial pressure of gas , $p_{1}$=p

 and final pressure of gas , $p_{2}=10p$

 By adiabatic process  relation between p and T.

 $p_{1}^{1-\gamma}T_{1}^{\gamma}=p_{2}^{1-\gamma}T_{2}^{\gamma}$

$\left(\frac{p_{1}}{p_{2}}\right)^{1-\gamma}=\left(\frac{T_{2}}{T_{1}}\right)^{\gamma}\Rightarrow\left(\frac{p}{10p}\right)^{1-\gamma}=\left(\frac{T_{2}}{300}\right)^{\gamma}$

$10^{\gamma-1}=\frac{T_{2}^{\gamma}}{300^{\gamma}}\Rightarrow T_{2}^{\gamma}= 10^{\gamma-1}\times 300^{\gamma}$

 $\Rightarrow$     $T_{2}=10^{\frac{\gamma-1}{\gamma}}\times 300= 10^{1-\frac{1}{\gamma}}\times300$

 $= 10^{1-\frac{1}{7/5}}\times300\left[\because For diatomic,\gamma=\frac{7}{5}\right]$

 $= 10^{2/7}\times300= (100)^{1/7} \times 300$

 =$1.9 \times 300$              [$\because  100^{3/7}=1.9$]

 =570 K

Therefore, the final gas temperature after compression is 570 K