Thermodynamics Question 262
Question: A mass of diatomic gas $ (\gamma =1.4) $ at a pressure of 2 atmospheres is compressed adiabatically so that its temperature rises from $ 27{}^\circ C $ to $ 927{}^\circ C $ . The pressure of the gas in final state is
Options:
A) 28 atm
B) 68.7 atm
C) 256 atm
D) 8 atm
Show Answer
Answer:
Correct Answer: C
Solution:
[c] $ T _{1}=273+27=300K $
$ T _{2}=273+927=1200K $
For adiabatic process, $ {{P}^{1-\gamma }}{{T}^{\gamma }}=\text{constant}\Rightarrow \text{P} _{1}^{1-\gamma }T _{1}^{\gamma }=\text{P} _{2}^{1-\gamma }T _{2}^{\gamma } $
$ \Rightarrow {{( \frac{P _{2}}{P _{1}} )}^{1-\gamma }}={{( \frac{T _{1}}{T _{2}} )}^{\gamma }}\Rightarrow {{( \frac{P _{1}}{T _{2}} )}^{1-\gamma }}={{( \frac{T _{2}}{T _{1}} )}^{\gamma }} $
$ {{( \frac{P _{2}}{P _{1}} )}^{1-1.4}}={{( \frac{1200}{300} )}^{1.4}}\Rightarrow {{( \frac{P _{1}}{T _{2}} )}^{-0.4}}={{( 4 )}^{1.4}} $
$ {{( \frac{P _{2}}{P _{1}} )}^{0.4}}={{4}^{1.4}}\text{or, }P _{2}=P _{1}{{4}^{( \frac{1.4}{0.4} )}}=P _{1}{{4}^{( \frac{7}{2} )}} $
$ =P _{1}( 2^{7} )=2\times 128=256atm $
