When `100J` of heat is given to an ideal gas it expands from `200cm^(3)` to `400cm^(3)` at a constant pressure of `3 xx 10^(5) Pa`. Calculate (a) the change in internal energy of the gas (b) the number of moles in the gas if the initial temperature is `400K`, (c ) the molar heat capacity `C_(p)` at constant pressure and (d) the molar heat capacity `C_(v)` at constant volume. `[R = (25)/(3)J//mol-K]`

Answer

(a) `40J` (b) `(9)/(500) moles` (c ) `(125)/(9) J//mol-K` (d) `(50)/(9)J//mol-K`

Explanation

(a) `DeltaU = DeltaQ - W`
`= 100 - 3 xx 10^(5) xx 200 xx 10^(-6) = 100 - 60 = 40J`
(b) `n = (PV)/(RT) = (3xx10^(5)xx200xx10^(-6))/((25)/(3)xx400) =(9)/(500)`moles
(c ) `PdV = nRdT`
`dT = (3xx10^(5)xx200xx10^(-6))/((9)/(500)xx(25)/(3)) rArr dT = 400K`
As `C_(P) = (Q)/(ndT) = (100)/((9)/(500)xx400) = (125)/(9) J//mol-K`
(d) `C_(P) -C_(V) = R`
`C_(V) = C_(P) - R`
`= (125)/(9) - (25)/(3) = (50)/(9) J//mol-K`.