An ideal gas `(gamma=3//2)` is compressed adiabatically form volume `400cm^(3)` to `100cm^(3)`. The initial pressure and temperature are `100 kPa` and `400K`. Find (a) the number of moles of the gas (b) the molar heat capacity at constant pressure, (c ) the final pressure and temperature, (d) the work done by the gas in the process and (e) the change in internal energy of the gas.`(R = (25)/(3)J//mol-k)`

Answer

(a) `0.021 moles`,
(b) `25J//mol-K`,
(c ) `800 KN//m^(2), 800K`,
(d) `-20 J`,
(e) `80J`

Explanation

(a) no of moles of gas `n = (PV)/(RT)`
`n = (100 xx 10^(3) xx 400 xx 10^(-6))/((25)/(3)xx400) = (3)/(250)`
`= 0.12 "moles"`
(b) `C_(P) = (R gamma)/((gamma-1)) = ((25)/(3)xx(3)/(2))/(((3)/(2)-1)) = 25 J//mol-K`
(c ) Final pressure `P_(1)V_(1)^(gamma) = P_(2)V_(2)^(gamma)`
`P_(2) = P_(1) ((V_(1))/(V_(2)))^(gamma)`
`P_(2) = 100 xx 10^(3) ((400)/(100))^(3//2) = 100 xx 10^(3) xx (2)^(3//2)`
`P_(2) xx 8 xx 10^(5) Pa = 800 kPa`
Final Temperature `T_(1)V_(1)^(gamma-1) = T_(2)V_(2)^(gamma-1)`
`T_(2)=T_(1) ((V_(1))/(V_(2)))^(gamma-1)`
`T_(2) = 400 ((400)/(100))^((3)/(2)-1) = 400 xx 2^(2xx(1)/(2))`
`T_(2) = 800 K`
(d) Work done by gas
`= (P_(1)V_(1)-P_(2)V_(2))/((gamma-1))`
`= (100 xx 10^(3)xx400 xx 10^(-6)-800 xx 10^(3) xx 100 xx 10^(-6))/(((3)/(2)-1))`
`= ((49=80))/((1)/(2))=-80J`
(e) `DeltaU = nC_(v)DeltaT = (nRDeltaT)/((gamma-1))`
`= (3)/(250) xx (25)/(3) xx ((800-400))/(((3)/(2)-1))=(1)/(10) xx(400)/(1//2) = 80J`.