The mass balance for the irreversible reaction A → 2B whose rate is \(n^{th}\) order in \(C_A.\)

\[\frac{dF_A}{dV} = -kC_A^n\]

\[\frac{dF_B}{dV} = 2kC_A^n\]

where \(F_A\) and \(F_B\) are molar flow rates (mol/s) of A and B, respectiviely, \(V\) is the cummulative volume of the reactor (L), \(k\) is the rate constant, and \(C_A\) is the molar concentration of A (mol/L). An example of initial conditions for these differential equations are at \(V\) = 0 (reactor inlet), then \(F_A = F_{A_{0}}\) and \(F_B = F_{B_{0}}\). These equations are integrated until \(V = V_T\) (the total reactor volume). 

These equations use the relations \(F_A = vC_A\) where \(v\) is the volumetric flow rate (L/s). For a liquid phase system, \(v\) can be assumed to be constant (constant density). For a gas-phase system in which the gases can be considered ideal.

\[v = \frac{F_TRT}{P}\]

where \(F_T = F_A +F_B\) = total molar flow rate (L/s), \(R\) is the ideal gas constant, \(T\) is the absolute temperature (K), and \(P\) is the pressure (bar). Thus, the volumetric flow rate can increase or decrease (as a function of distance down the reactor) for a gas-phase reaction.

For a tubular reactor with constant cross section section, the mass balances can also be written in terms of the distance down the reactor (\(z\)):

\[\frac{dF_A}{dz} = -kA_xC_A^n\]

\[\frac{dF_B}{dz} = 2kA_xC_A^n\]

where \(A_x\) is the cross-sectional area of a tubular reactor, and the mass balances are integrated from \(z\) = 0 to \(z\) = L (length of reactor).

Space velocity (\(SV\)) is often used as a variable for a plug flow reactor:

\[SV = \frac{v_0}{V_T}\]

where \(v_0\) is the feed volumetric flow rate to the reactor.

Fraction conversion for component a: 

\[X_A = \frac{F_{A_{0}} – F_A}{F_{A_{0}}}\]

Percent conversion = \(100X_A\)