A gas of density `0.00125 g cdot cm^(3)` , volume 8L at `0^(@)C` temperature and 1 atm pressure is supplied with 30 cal of heat to raise its temperature to `15^(@)C` at constant pressure. Determine the specific heat of the gas at constant pressure and at constant volume. Given, `R = 2 cal cdot mol^(-1) cdot^(@)C^(-1)`.

Explanation

Mass of the gas,
`m = (8 xx 10^(3) cm^(3)) xx 0.00125g cdot cm^(-3) = 10 g`
Heat gained by the gas at constant pressure,
`Q = mc_(p)(t_(f)-t_(i))`
or, `c_(p) = (Q)/(m(t_(f) - t_(i))) = (30)/(10 xx (15 - 0)) = 0.2 cal cdot g^(-1) cdot^(@)C^(-1)`
Now, mass of 8 L of the gas at STP = 10 g.
`:.` Mass of 22.4 L of the gas at STP = `(10 xx 22.4)/(8) = 28 g`
`:.` Molecular mass of the gas, M = 28
So, molar specific heat at constant pressure,
`C_(p) = mc_(p) = 28 xx 0.2 cal cdot mol^(-1) cdot^(@)C^(-1)`
Molar specific heat at constant volume,
`c_(v) = (C_v)/(M) = (28 xx 0.2 - 2)/(28) = 0.2 - 2/28`
`=0.1286 cal cdot g^(-1) cdot^(@)C^(-1)`.