If we consider the single general stoichiometric equation: $$a\text{A} + b\text{B} \longrightarrow c\text{C} + d\text{D}$$ which can be rewitten as, $$\text{A} + \frac{b}{a}\text{B} \longrightarrow \frac{c}{a}\text{C} + \frac{d}{a}\text{D}$$ We can express the total molar flowrates from the reactor effluent for a PFR in terms of conversion ($X$) using a stoichiometric table as shown below,
| Species | Reactor Feed | Change | Reactor Effluent |
| A | $$n_{A0}$$ | $$-n_{A0}X$$ | $$n_A = n_{A0}(1-X)$$ |
| B | $$n_{B0}=(n_{B0}/n_{A0})n_{A0}=\theta_B n_{A0}$$ | $$-(b/a)n_{A0}X$$ | $$n_B=n_{A0}(\theta_B -(b/a)X)$$ |
| C | $$n_{C0}=(n_{C0}/n_{A0})n_{A0}=\theta_C n_{A0}$$ | $$(c/a)n_{A0}X$$ | $$n_C=n_{A0}(\theta_C +(c/a)X)$$ |
| D | $$n_{D0}=(n_{D0}/n_{A0})n_{A0}=\theta_D n_{A0}$$ | $$(d/a)n_{A0}X$$ | $$n_D=n_{A0}(\theta_D +(d/a)X)$$ |
| I | $$n_{I0}=(n_{I0}/n_{A0})n_{A0}=\theta_I n_{A0}$$ | $$0$$ | $$n_I=\theta_I n_{A0}$$ |
| Total | $$n_{T0}$$ | $$n_T=n_{T0}+\delta F_{A0}X$$ |
Note that the change in total molar flowrates (i.e. $n_T - n_{T0}$) can be calculated in terms of the conversion ($X$) and is equal to $\delta n_{A0}X$, where $\delta = (d/a + c/a - b/a - 1)$ is the change in moles per mole of A reacted. This quantity is especially important when working with gases in which case the change in total number of moles leads to a change in volumetric flowrate along the length of the PFR reactor. Thus in general for a single reaction,
$$n_T = n_{T0}(1+\delta y_{A0}X) = n_{T0}(1+\epsilon X)$$
where $y_{A0}=n_{A0}/n_{T0}$.
For a flowing gas or vapour, we can use the ideal gas law to relate the volumetric flowrate ($v$) to the total molar flowrate (assuming we can make the assumption the system is approximately ideal). In this case, we take the time derivative of the ideal gas law to give the relationship between the volumetric and total molar flowrates at a given distance along the PFR in terms of the pressure and temperature at the same location,
$$Pv=n_T RT$$
Similarly at the inlet to the PFR we have,
$$P_0v_0=n_{T0}RT_0$$
Taking the ratio of the local flowrate to the inlet flowrate above give,
$$v=v_0\left(\frac{n_T}{n_{T0}}\right)\left(\frac{P_0}{P}\right)\left(\frac{T}{T_0}\right)$$
and taking the ratio of our molar flow rates for a single reaction as above, gives,
$$v=v_0\left(1+\epsilon X\right)\left(\frac{P_0}{P}\right)\left(\frac{T}{T_0}\right)$$
This equation is especially significant because the concentrations of the components depend on the volumetric flowrate according to,
$$C_A=\frac{n_A}{v}$$
Thus, if we plug in the equation for $n_A$ from the stoichiometric table, we find that,
$$C_A=C_{A0}\frac{1-X}{1+\epsilon X}\left(\frac{P}{P_0}\right)\left(\frac{T_0}{T}\right)$$
Similar, the concentrations of the other species can be calculated, for example for C as,
$$C_C=C_{C0}\frac{\displaystyle\theta_C+\frac{c}{a}X}{\displaystyle 1+\epsilon X}\left(\frac{P}{P_0}\right)\left(\frac{T_0}{T}\right)$$
The graph below gives an example for the reaction,
$$\text{A} \longrightarrow b\text{B}$$
in the presence of an inert, I, with reaction rate,
$$-r_A=kC_A$$
The key parameters can be varied to see the comparison with the constant density case.