Chapter 1: Mole Balances

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Topics

  1. Chemical Identity
  2. Reaction Rate
  3. General Mole Balance Equation
  4. Mole Balance on Different Reactor Types
  5. Self Test Exercises

Chemical Identity

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A chemical species is said to have reacted when it has lost its chemical identity. The identity of a chemical species is determined by the kind, number, and configuration of that species' atoms.

Reaction YouTube Video: Carbon Dioxide and Magnesium

Three ways a chemical species can lose its chemical identity:

  1. Decomposition

    \(\text{CH}_3\text{CH}_3 \rightarrow \text{H}_2 + \text{H}_2\text{C} = \text{CH}_2\)

  2. Combination

    \(\mathrm{N_2 + O_2 \rightarrow 2NO}\)

  3. Isomerization

    \(\mathrm{C_2H_5CH=CH_2 \rightarrow CH_2=C(CH_3)_2}\)

Reaction Rate

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The reaction rate is the rate at which a species looses its chemical identity per unit volume. The rate of a reaction can be expressed as the rate of disappearance of a reactant or as the rate of appearance of a product. Consider species A:

A → B

rA = the rate of formation of species A per unit volume
-rA = the rate of a disappearance of species A per unit volume
rB = the rate of formation of species B per unit volume

Example: A → B

If B is being created at a rate of 0.2 moles per decimeter cubed per second (i.e. the rate of formation of B is rB = 0.2 mole/dm3/s),
then A is disappearing at the same rate (-rA = 0.2 mole/dm3/s). This also means that the rate of formation of A is rA = -0.2 mole/dm3/s.

For a catalytic reaction, we refer to -rA', which is the rate of disappearance of species A on a per mass of catalyst basis.

Note: dCA/dt is not the rate of reaction

Example Is sodium hydroxide reacting?

Consider species j:

  • rj is the rate of formation of species j per unit volume [e.g. mol/dm3*s]
  • rj is a function of concentration, temperature, pressure, and the type of catalyst (if any)
  • rj is independent of the type of reaction system (batch, plug flow, etc.)
  • rj is an algebraic equation, not a differential equation.

We use an algebraic equation to relate the rate of reaction, -rA, to the concentration of reacting species (e.g., CA) and to the temperature (T) at which the reaction occurs [e.g. -rA = k(T)CA2].



Example Would you like to see some other rates of reaction?
Self Test The Convention for Rates of Reaction

General Mole Balance Equation

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A diagram illustrating the general mole balance equation in a chemical reactor from the book ‘Elements of Chemical Reaction Engineering’. The diagram shows a reactor with an inlet stream labeled F_A0 (entering molar flow rate of A) and an outlet stream labeled F_A (exiting molar flow rate of A). Inside the reactor, there is a volume V and a number of moles N_A (number of moles of A inside the system volume V). The rate of generation of species A, G_A, is represented as the integral of the reaction rate r_A (rate of generation of A per unit volume) over the volume V. The diagram includes the equation G_A = ∫_V r_A dV, showing the relationship between the rate of generation and the reaction rate over the reactor volume.

FA0= Entering molar flow rate of A (mol/time)
FA= Exiting molar flow rate of A (mol/time)
GA= Rate of generation(formation) of A (mol/time)
V = Volume (vol e.g. m3)
rA= rate of generation(formation) of A (mole/time•vol)
NA= number of moles of A inside the system Volume V (mol)
t = time (e.g. s)

IN - OUT + GENERATION = ACCUMULATION

\(\mathrm{F_{A0} - F_{A} + \int_{0}^{V} r_{A} dV = \frac{dN_{A}}{dt}}\)

Self Test How about testing what you've learned?

Mole Balance on Different Reactor Types

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The General Mole Balance Equation (GMBE) applied to the four major reactor types (and the general reaction, A → B):


Reactor Differential Algebraic Integral
Batch

\[\frac{dN_A}{dt} = r_A V\]

\[ t = \int_{N_{A0}}^{N_A} \frac{dN_A}{r_A V} \]

A graph showing the exponential decay of N_A over time t. The y-axis represents the quantity N_A and the x-axis represents time t. The curve starts at a high value on the y-axis and decreases asymptotically towards zero as time progresses. Derive
CSTR

\[ V = \frac{F_{A0} - F_A}{-r_A} \]

Derive
PFR

\[ \frac{dF_A}{dV} = r_A \]

\[ V = \int_{F_{A0}}^{F_A} \frac{dF_A}{r_A} \]

A graph showing the decay of F_A over the volume V . The y-axis represents the molar flow rate F_A and the x-axis represents the volume V . The curve starts at a high value on the y-axis and decreases asymptotically towards zero as the volume increases. Derive
PBR

\[ \frac{dF_A}{dV} = r'_A \]

\[ W = \int_{F_{A0}}^{F_A} \frac{dF_A}{r'_A} \]

A graph showing the decay of F_A over the weight W . The y-axis represents the molar flow rate F_A and the x-axis represents the weight W . The curve starts at a high value on the y-axis and decreases asymptotically towards zero as the weight increases Derive

Self Test Exercises

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Self Test Fourteen Multiple Choice Questions
Self Test Batch Reactor Time
Self Test What's Wrong With This Solution?
Link Smog in Los Angeles Basin Web Module
Link Getting Unstuck on a Problem and the Problem Solving Algorithm
Link Visual Encyclopedia of Chemical Reactors
PolymathPolymath Tutorials
Assess Objective Assessment of Chapter 1

The following humorous video is set to Randy Newman's song "You've got a friend in me" was made by Professor Lane's 2008 Chemical Reaction Engineering class at the University of Alabama, Tuscaloosa.

Link Reaction YouTube Video: Fogler Zone (You've Got a Friend in Fogler)

The following animation is an excerpt taken from the lecture slides that Professor Fogler uses in his Reactions course, illustrating the importance of "Keeping Up"

Link Keeping Up