Irreversible of first order

Consider initially a first-order reaction. From the expression of the rate (3.9), when a =l,b = 0 and n=l, together with Equation 1.17. Thus, for a constant T, we have  

For a tubular reactor (Equation 5.3), the expression of the rate is substituted, and after the integration, we obtain the following equation  

This solution is different from that of Equation 5.5 for a system at constant volume. The solution is the same when sa = 0. 

For a batch reactor, the expression change, since we start from the expression at variable volume (Equation 4.6), i.e.  

For gas-phase reactions, such as in a plug-flow batch reactor at constant pressure, the volume varies according  

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