Equations Giving the Molar Flow of Different Molecules Linked By a Reaction
In this article, we will explore the equations that give the molar flow of different molecules linked by a reaction. We will consider a steady-state CSTR, where the reaction:
aA + bB <--> cC
is taking place. Here, a, b, and c are the stoichiometric coefficients for the species A, B, and C, respectively. We will also use the following notations:
Fi is the molar flow of species i,
r is the rate of reaction,
V is the volume,
X is the conversion at the end of the reactor,
νi is the stoichiometric coefficient of species i, and
ξ is the extent of reaction.
Mass Balance Equation
To derive the equations giving the molar flow of species A, B, and C, we will start by writing the mass balance equation:
IN + PROD - CONS = OUT + VAR
where IN, PROD, and CONS are the molar flows of species A and B going into the reactor and the molar flow of species C produced by the reaction, respectively. OUT is the molar flow of all species leaving the reactor, and VAR is any accumulation in the reactor volume. From this equation, we can write:
FA,in + 0 - a r V = FA,out
FB,in + 0 - b r V = FB,out
FC,in + c r V - 0 = FC,out
where F[i,in] is the molar flow of species i entering the reactor, and F[i,out] is the molar flow of species i leaving the reactor.
Equations for Species A and B
Using the expression for the extent of reaction, ξ, we can write:
ξ = - r V = - X (FA,in/νA) = -X(FB,in/νB)
This gives us:
FA,out = FA,in - a r V = FA,in + a ξ = FA,in - a X (FA,in/νA) = (1 - X) FA,in
FB,out = FB,in - b r V = FB,in + b ξ = FB,in - b X (FA,in/νA) = (1 - X) FB,in
These are the equations for the molar flow of species A and B.
The Problem with Species C
Now, let’s derive the equation giving the molar flow of species C:
FC,out = FC,in + c r V = FC,in - c ξ = FC,in + c X (FC,in/νC) = (1 + X) FC,in
At first glance, this equation seems to be correct, but if we think about it, we realize that it is not. It states that if there is no C at the beginning, there is no C produced at all, which is obviously wrong. So, where did we go wrong?
The Correct Equation for Species C
The correct equation for the molar flow of species C can be derived by noting that:
Substituting this expression into the mass balance equation for species C, we get:
FC,out = FC,in + c r V = (FA,in/νA) (c/b) (1 - X)/(X) + c ξ = FC,in [1 + c/b (1 - X)/X] = FC,in(1 + X)/(1 - X b/a)
This is the correct equation for the molar flow of species C.
Conclusion
In conclusion, we have derived the equations giving the molar flow of different molecules linked by a reaction in a steady-state CSTR. We have shown that the equations for species A and B are given by (1 – X) Fi,in, while the equation for species C is given by FC,in(1 + X)/(1 – X b/a). It is important to note that the equation for species C is different from what we originally derived, and the correct equation takes into account the fact that species C can be produced even if there is no C at the beginning.
What are the Equations Giving the Molar Flow of Different Molecules Linked By a Reaction?
Equations Giving the Molar Flow of Different Molecules Linked By a Reaction
In this article, we will explore the equations that give the molar flow of different molecules linked by a reaction. We will consider a steady-state CSTR, where the reaction:
is taking place. Here,
a,b, andcare the stoichiometric coefficients for the speciesA,B, andC, respectively. We will also use the following notations:Fiis the molar flow of speciesi,ris the rate of reaction,Vis the volume,Xis the conversion at the end of the reactor,νiis the stoichiometric coefficient of speciesi, andξis the extent of reaction.Mass Balance Equation
To derive the equations giving the molar flow of species
A,B, andC, we will start by writing the mass balance equation:where
IN,PROD, andCONSare the molar flows of speciesAandBgoing into the reactor and the molar flow of speciesCproduced by the reaction, respectively.OUTis the molar flow of all species leaving the reactor, andVARis any accumulation in the reactor volume. From this equation, we can write:where
F[i,in]is the molar flow of speciesientering the reactor, andF[i,out]is the molar flow of speciesileaving the reactor.Equations for Species A and B
Using the expression for the extent of reaction,
ξ, we can write:This gives us:
These are the equations for the molar flow of species
AandB.The Problem with Species C
Now, let’s derive the equation giving the molar flow of species
C:At first glance, this equation seems to be correct, but if we think about it, we realize that it is not. It states that if there is no
Cat the beginning, there is noCproduced at all, which is obviously wrong. So, where did we go wrong?The Correct Equation for Species C
The correct equation for the molar flow of species
Ccan be derived by noting that:From this, we can express
F[C,in]as:Substituting this expression into the mass balance equation for species
C, we get:This is the correct equation for the molar flow of species
C.Conclusion
In conclusion, we have derived the equations giving the molar flow of different molecules linked by a reaction in a steady-state CSTR. We have shown that the equations for species
AandBare given by (1 – X) Fi,in, while the equation for speciesCis given by FC,in(1 + X)/(1 – X b/a). It is important to note that the equation for speciesCis different from what we originally derived, and the correct equation takes into account the fact that speciesCcan be produced even if there is noCat the beginning.