Page 66 - The engineering of chemical reactions

50 Reaction Rates, the Batch Reactor, and the Real World
For the reversible reaction
A+BZCC, Y = k&ACB -k&c

among ideal gases, if CAM = CB~ and Cco = 0, the substitution of CjS in terms of X yields

which again can be integrated for X(t) and substituted back to find CA, Cs, and Cc.
Example 2-5 Find the conversion versus time for the reaction

A -+ 3B, r=kCi
n a constant-pressure batch reactor assuming A and B are ideal gases starting with pure A
it CAo.
The number of moles of A and B are found by setting up the mole table

Species Moles initially Moles
1 NAO NAO(~ --xl
3 0 ~NAoX
otal moles NAO NAO(~ +2x)
volume VO Vdl +2x)

Therefore, the concentrations are

and
3 x
NB
CB = - = cAo------
V 1 + 2 x
In a constant-pressure batch reactor the mass balance on A becomes
dNA = -kCiV
~ _ -hTAog = -rV
dt
V,(l +2X)

so that
dX
- =kCA,(;;ZX:’
dt
This equation can be separated to yield
X
1 1 + 2 x
- ___
t = kc,.,,, s (1 - X)2 dX
0
which must be integrated by partial fractions to give
3 x
- + 2 ln(1 - X)
t= 1-x
This equation cannot be inverted to find X(t) explicitly, and so we can only find
CA (t ) numerically.