Temperature of air coming out of a tyre due to a burst

Question

I saw several questions where I had to calculate the temperature of the air coming out of a tyre due to a burst, given the external atmospheric pressure and the initial state of the gas present in the tyre.

All these questions utilised the fact that as the air comes out rapidly, it can be assumed to be adiabatic.

All this seems fine, until they apply the formula $$PV^\gamma=constant$$

According to me the air comes out rapidly so the process should be an irreversible one, and hence the formula $PV^\gamma=constant$ should not be valid, as it is only applicable for a reversible adiabatic process.

However all the questions apply this formula, so please clarify if this is faulty, or I have understood something incorrectly, or the difference on considering this to be reversible won't have a significant error over considering it irreversible.

Note:The pressure change of the gas is not small compared to the pressure state of the gas.

Answer 1

That equation holds at all equilibrium states through which the ideal gas passes during a quasi-static adiabatic process. Since the gas is rushing into the environment it passes through nonequilibrium states before finally achieving equilibrium and $PV = K$ cannot be applied in this case. However you can get the magnitude of the energy involved by treating the problem as a reversible adiabatic expansion. If you want to accurately calculate the final state of the system (supposing we deal with a perfect gas) for the first law you have $\Delta U = w_{irr}$. If the transition takes place from ($P_1,T_1$) to ($P_2,T_2$) we have

$w_{irr} = P_{2} \Delta V$

where $P_2 = P_{ext}$; if you use the equation of state of a perfect gas you can substitute $\Delta V$ and get

$w_{irr} = −P_2(\frac{nRT2}{P2} − \frac{nRT1}{P1})$

Since $\Delta U = n \cdot C_{V,m} (T_2-T_1) = w_{irr}$, and considering $C_{V,m}$ independent of temperature:

$C_{V,m}(T_2 − T_1) = −RP_2\left(\frac{T_2}{P_2} − \frac{T1}{P_1}\right)$

you now solve for the particular variable you're interested in using the fact that $C_{V,m} - C_{P,m} =R$.

For example if you want to calculate the final temperature knowing ($P_1,T_1$) and $P_2$ the former yields to

$T_2 = T_1 \left[\frac{C_{V,m} + R(P_2/P_1)} {C_{P,m}}\right]$

It may also interest you: https://gioretikto.github.io/chemistry/thermodynamics/adiabatic_expansion.html