A Mass Balance in Chemical Kinetics
In Lecture 1, we encountered a fundamental relationship between reaction rates and concentration changes. For any species X involved in a reaction, its rate of concentration change is related to the overall reaction rate \(\nu\) by:
\[\nu = \frac{1}{n}\frac{\mathrm{d}[\mathrm{X}]}{\mathrm{d}t}\]
where \(n\) is the stoichiometric coefficient of X (positive for products, negative for reactants). This elegant relationship unifies the rates of change of all species in a reaction—but where does it come from?
The answer lies in a fundamental principle of chemistry: the conservation of mass. When we examine how reaction concentrations evolve over time, mass balance relationships provide the key to understanding how different species’ rates of change are connected. This appendix explores how mass conservation leads naturally to these rate relationships.
A.1 The Origins of Mass Balance
Consider a simple reaction:
\[\mathrm{A} \rightarrow \mathrm{B}\]
While concentrations of A and B change over time, the total mass of the system remains constant - this is the principle of conservation of mass. For this simple reaction, at any time \(t\):
\[[\mathrm{A}]_t + [\mathrm{B}]_t = [\mathrm{A}]_0 + [\mathrm{B}]_0\]
where \([\mathrm{A}]_0\) and \([\mathrm{B}]_0\) are the initial concentrations at \(t=0\).
# Plot showing [A] and [B] vs time with horizontal line showing constant totalIn cases where \([\mathrm{B}]_0 = 0\) (starting with pure reactant), this simplifies to:
\[[\mathrm{A}]_t + [\mathrm{B}]_t = [\mathrm{A}]_0\]
A.2 From Mass Balance to Rate Relationships
The mass balance equation provides a constraint that links the rates of change of A and B. Taking the derivative with respect to time:
\[\frac{\mathrm{d}}{\mathrm{d}t}([\mathrm{A}]_t + [\mathrm{B}]_t) = \frac{\mathrm{d}}{\mathrm{d}t}[\mathrm{A}]_0\]
Since \([\mathrm{A}]_0\) is constant, its derivative is zero:
\[\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} + \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = 0\]
Rearranging:
\[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = -\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t}\]
This tells us that for this simple reaction, the rate of formation of B equals the rate of consumption of A.
A.3 Mass Balance with Stoichiometry
Now consider a reaction where one molecule of A produces two molecules of B:
\[\mathrm{A} \rightarrow 2\mathrm{B}\]
# Diagram showing A → 2B transformationThe mass balance equation becomes:
\[2[\mathrm{A}]_t + [\mathrm{B}]_t = 2[\mathrm{A}]_0\]
The factor of 2 for [A] accounts for the fact that each A molecule contains the mass equivalent of two B molecules. Taking the derivative:
\[2\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} + \frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = 0\]
Rearranging:
\[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = -2\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t}\]
This shows that B is produced twice as fast as A is consumed, consistent with the stoichiometry.
A.4 Generalizing to Complex Reactions
For a general reaction:
\[\mathrm{A} + 2\mathrm{B} \rightarrow \mathrm{C} + 3\mathrm{D}\]
The reaction rate \(\nu\) can be expressed in terms of any species’ concentration change:
\[\nu = -\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} = -\frac{1}{2}\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{d}t} = \frac{1}{3}\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{d}t}\]
This leads to the general form:
\[\nu = \frac{1}{n}\frac{\mathrm{d}[\mathrm{X}]}{\mathrm{d}t}\]
where \(n\) is the stoichiometric coefficient of species X (negative for reactants, positive for products).
A.5 Equilibrium Reactions and Mass Balance
For an equilibrium reaction:
\[\mathrm{A} \underset{k_{-1}}{\stackrel{k_1}{\rightleftharpoons}} 2\mathrm{B}\]
# Diagram showing equilibrium with labeled rate constantsThe rate of change of [A] has contributions from both forward and reverse reactions:
\[\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} = -k_1[\mathrm{A}] + k_{-1}[\mathrm{B}]^2\]
The first term represents unimolecular decomposition of A, while the second term represents the bimolecular recombination of B.
Similarly for B:
\[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = +2k_1[\mathrm{A}] - 2k_{-1}[\mathrm{B}]^2\]
These equations satisfy mass balance:
\[\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{d}t} = -2\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t}\]
A.6 Key Principles
- Total mass is conserved in chemical reactions
- Mass balance provides constraints on concentration changes
- Stoichiometric coefficients determine the relative rates of concentration changes
- Mass balance applies to both irreversible and equilibrium reactions
# Diagram illustrating key concepts