For an adiabatic process `PV^gamma` = constant. Evaluate this "constant" for an adiabatic process in which 2 mol of an ideal gas is filled at 1.0 atm pressure and 300 K temperature. Consider the ideal gas to be diatomic rigid rotator.

Explanation

`mu`=2 mol , `P=1.01xx10^5`,T=300 K
For ideal gas `PV=muRT`
`therefore V=(muRT)/P`
Now, `PV^gamma`=constant (k)
`therefore P((muRT)/P)^gamma=k`
`therefore P^(1-gamma)(muRT)^gamma=k`
But for diatomic (rigid rotator) molecule
Since `C_V=(5R)/2` and `C_P=(7R)/2`
`gamma=C_P/C_V=7/5`
`k=(1.01xx10^5)^(1-7/5)xx(2xx8.31xx300)^(7/5)`
`=(1.01xx10^5)^((-2)/5)xx(4980)^7/5`
`=(4980)^(7/5)/(1.01xx10^5)^(2/5)`
`=7/5 log 4980 - 2/5 (log 1.01xx10^5)`
`=log[1.4(3.6972)-2/5 [ 5.0043]]`
=log [5.1761-2.0017]
=log 3.1744
`therefore k=1.49xx10^3`