Question

How much heat energy should be added to the gaseous mixture consisting of 1 g of hydrogen and 1g of helium to raise it s temperature from `0^(@)C` to `100^(@)C`
(a) at constant volume,
b. at constant pressure `(R = 2 cal//mol K)`?

Answer 1

As hydrogen is diatomic and has molecular weight 2,
`(C_(V)) H = (5)/(2) R`
and `mu_(H) = (1)/(2)`
While He is monatomic and has molecular wieght 4,
`(C_(V))_(He) = (3)/(2) R` and `mu_(He) = (1)/(4)`
So, by conservation of energy, we get
`(C_(V))_(mix) = (mu(C_(V))_(1) + mu_(2) (C_(V))_(2))/(mu_(1) + mu_(2)) = ((1)/(2) xx (5)/(2) R + (1)/(4) xx (3)/(2) R)/((1)/(2) + (1)/(4))`
`(C_(V))_(mix) = (13)/(8) xx (4)/(3) R = (13)/(6) R`
a. So
`(Delta Q)_(V) = mu C_(V) Delta T = ((1)/(4) + (1)/(4)) xx (13)/(6) xx 2 xx (100 - 0) = 325 cal`
b. Now as `(C_(P))_(mix) = (C_(V))_(mix) + R = (13)/(6) R + R = (19)/(6) R`
`(Delta Q)_(P) = mu C_(P) Delta T = ((1)/(4) + (1)/(4)) xx (19)/(6) xx 2 xx (100 - 0) = 475 cal`