Reciprocal Relations Between Kinetic Curves

We study coupled irreversible processes. For linear or linearized kinetics with microreversibility, $\dot{x}=Kx$, the kinetic operator $K$ is symmetric in the entropic inner product. This form of Onsager's reciprocal relations implies that the shift in time, $\exp (Kt)$, is also a symmetric operator. This generates the reciprocity relations between the kinetic curves. For example, for the Master equation, if we start the process from the $i$th pure state and measure the probability $p_j(t)$ of the $j$th state ($j\neq i$), and, similarly, measure $p_i(t)$ for the process, which starts at the $j$th pure state, then the ratio of these two probabilities $p_j(t)/p_i(t)$ is constant in time and coincides with the ratio of the equilibrium probabilities. We study similar and more general reciprocal relations between the kinetic curves. The experimental evidence provided as an example is from the reversible water gas shift reaction over iron oxide catalyst. The experimental data are obtained using Temporal Analysis of Products (TAP) pulse-response studies. These offer excellent confirmation within the experimental error.