I've been assigned this question as homework and have been attempting it for hours but cannot seem to get anywhere. Any help would be greatly appreciated!
An airbag of volume $40\ \mathrm L$ is inflated adiabatically through the following reaction:
$$\ce{NaN3(s) -> Na(s) + 3/2N2(g)}$$
One mole of $\ce{NaN3}$ decomposed produces an enthalpy of $-23.1\ \mathrm{kJ}$. $60\ \%$ of this energy is deposited in the gaseous product; assume the decomposition is instantaneous. $C_{\mathrm m,p}(\ce{N2})=29.12\ \ \mathrm{J\ mol^{-1}\ K^{-1}}$. When fully inflated, the pressure of the $\ce{N2}$ gas inside the airbag is $2.5\ \mathrm{atm}$.
Estimate the mass of $\ce{NaN3}$ needed to operate the airbag and state any assumptions or approximations being made.
So far, I'm only sure that I've done the following correctly:
$H=U+PV=q+w+PV$ where $q=0$ since the process is adiabatic, so $H=w+PV$
For the expansion of $\ce{N2}$ gas:
$H=0.6n(23.1\ \mathrm{kJ})=n(13.86\ \mathrm{kJ})$ where $n$ is moles of $\ce{NaN3}$
$n(13.86\ \mathrm{kJ})=U+PV$ so $U=n(13.86\ \mathrm{kJ})-(2.5\ \mathrm{atm})(40\ \mathrm L)=n(13.86\ \mathrm{kJ}-RT)$
$q=0$, $P=2.5\ \mathrm{atm}$, $V=40\ \mathrm L$ and $C_{\mathrm m,p}(\ce{N2})= 29.12\ \mathrm{J\ mol^{-1}\ K^{-1}}$
I'm not sure what else to do from here. I'd really appreciate any advice or if anyone could point me in the right direction. Thank you!