The equilibrium constant \(K_e\) (dimensionless) at temperature \(T\) (K) is:   \[K_e\,=\,K_mexp{\left[\frac{\Delta H}{R}\left(\frac{1}{T_m}-\frac{1}{T}\right)\right]}\]where \(K_m\) is the equilibrium constant at \(T_m\),
\(\Delta H\) is the heat of reaction (J/mol)
\(R\) is the ideal gas constant \((\frac{J}{mol\,K})\)

The fractional conversion \(X\), which is referred to as conversion, for a chemical reaction \(A\rightleftharpoons B\); \[X\,=\,\frac{F_{A0}-F_A}{F_{A0}}\] where \(F_{A0}\) is the molar flow rate of A (mol/s) into the reactor and \(F_A\) is the molar flow rate of A out of the reactor.

For the reaction \(A \rightleftharpoons B\), the equilibrium conversion \(X_e\) is: \[X_e\,=\,\frac{K_e}{1+K_e}\] where \(K_e\) is a function of temperature as shown above.

For an adiabatic reactor, the energy balance is; \[H_{in}^{total}\,=\,H_{out}^{total}\] This energy balance yields the conversion (\(X_{energy}\)) at temperature \(T\) for the reaction \(A \rightleftharpoons B\) in an adiabatic reactor in which the heat capacities \(C_{PA},C_{PB}\) are equal and independent of temperature: \[X_{energy}\,=\,\frac{(C_{PA}+\alpha C_{PI})(T-T_{feed})}{-\Delta H_{rxn}}\] where \(\alpha\) is the ratio of inert flow rate to feed flow rate of \(A\), \(T_{feed}\) is the feed temperature into the reactor, and \(\Delta H_{rxn}\) is the heat of reaction (J/mol), which is independent of temperature.

Note that for constant \(C_P\)’s and constant heat of reaction, the conversion versus temperature is a straight line.

The energy balance for an isothermal reactor is; \[XF_{A0}\Delta H_{rxn}\,=\,\dot Q\] where \(\dot Q\) is the heat transfer rate (J/s).