Nothing Stays
The world is not static. It flows.
Concentrations rise and fall. Bonds form and break. Electrons jump between orbitals. Molecules vibrate, rotate, translate. Reactions proceed. Systems equilibrate. Everything changes.
CHANGE is the primitive that captures this flux. Before any formalism—before derivatives, before differential equations—there is the perception that something is becoming something else.
CHANGE answers: What is different?
And its refinement: How fast? (That is RATE, the next primitive.)
The Cognitive Basis of CHANGE
1.1 Perception of Transformation
Motion perception is hardwired.
The visual system contains specialized neurons for detecting movement—direction-selective cells in the visual cortex that fire when something in the visual field changes position. This is not learned. Newborns track moving objects within hours of birth.
Beyond motion, humans perceive transformation in:
- Quantity: More or less than before
- Quality: Different color, shape, texture, sound
- State: Solid to liquid, asleep to awake
- Identity: Caterpillar to butterfly, reactant to product
- Relationship: Together to apart, aligned to opposed
1.2 Change Requires Comparison
CHANGE is inherently relational. It requires:
A. A state space: The set of possible configurations.
B. Two states: A before-state and an after-state.
C. A difference: The states must be distinguishable.
D. An ordering: Which is "before" and which is "after."
1.3 Kinds of Change
| Type | Description | Example |
|---|---|---|
| Continuous | Smooth variation through intermediates | Temperature rising |
| Discrete | Jumps without intermediates | Electron transitions |
| Cyclic | Returns to starting point | Oscillation, rotation |
| Irreversible | Cannot return | Combustion, decay |
Quantifying Change
2.1 The Difference
2.2 The Difference Quotient
Change occurs over some interval. To compare changes, normalize by interval size:
2.3 The Problem of Instantaneous Change
The average rate depends on the interval. A reaction may be fast initially, then slow.
What is the rate right now?
This requires the interval to shrink to zero. But Δf/Δt with Δt = 0 is undefined (0/0). The resolution is PROXIMITY: take the limit.
The Derivative
3.1 Definition
The derivative is the limit of the difference quotient:
The derivative is the limit of the secant slope as the points converge.
3.2 Geometric Interpretation
The derivative is the slope of the tangent line.
- f'(t) > 0: function increasing (tangent slopes upward)
- f'(t) < 0: function decreasing (tangent slopes downward)
- f'(t) = 0: horizontal tangent (local extremum or inflection)
3.3 Physical Interpretation
The derivative measures sensitivity: how much does output change per unit change in input?
- Position f(t) → Velocity f'(t)
- Velocity v(t) → Acceleration v'(t)
- Concentration [A](t) → Reaction rate d[A]/dt
Differentiation Rules
4.1 Basic Rules
| Rule | Formula |
|---|---|
| Constant | d/dx [c] = 0 |
| Power | d/dx [xⁿ] = nxⁿ⁻¹ |
| Sum | d/dx [f + g] = f' + g' |
| Constant multiple | d/dx [cf] = c·f' |
4.2 Product and Chain Rules
4.3 Special Functions
| f(x) | f'(x) | Chemistry context |
|---|---|---|
| eˣ | eˣ | First-order kinetics |
| e^(kx) | ke^(kx) | Arrhenius, decay |
| ln(x) | 1/x | Entropy, equilibrium |
| 1/x | -1/x² | Coulomb potential |
| sin(ωt) | ω cos(ωt) | Spectroscopy |
The derivative at each point equals the slope of the tangent.
Higher Derivatives
5.1 The Second Derivative
5.2 Physical Interpretation
- f(t) position → f'(t) velocity → f''(t) acceleration
- f''(t) indicates whether the rate is speeding up or slowing down
5.3 Concavity
The second derivative determines concavity:
- f''(x) > 0: concave up (curve bends upward)
- f''(x) < 0: concave down (curve bends downward)
- f''(x) = 0: possible inflection point
Partial Derivatives
6.1 Functions of Multiple Variables
Many chemical quantities depend on several variables:
- Pressure P(V, T, n)
- Gibbs energy G(T, P)
- Rate k(T, [A], [B])
6.2 Definition
The partial derivative of f(x, y) with respect to x:
6.3 The Total Differential
6.4 Chemical Applications
For ideal gas PV = nRT:
- (∂P/∂V)_{T,n} = -nRT/V²
- (∂P/∂T)_{V,n} = nR/V
CHANGE in Chemical Systems
7.1 Reaction Progress
A chemical reaction transforms reactants into products: A → B
The rate of change of concentration:
- d[A]/dt < 0 (A is consumed)
- d[B]/dt > 0 (B is produced)
The derivative gives the instantaneous rate of concentration change.
7.2 Rate Laws
| Order | Rate Law | Integrated Form |
|---|---|---|
| Zero | r = k | [A] = [A]₀ - kt |
| First | r = k[A] | [A] = [A]₀ e^(-kt) |
| Second | r = k[A]² | 1/[A] = 1/[A]₀ + kt |
7.3 Potential Energy Surfaces
Reactions proceed through stationary points on the potential energy surface.
CHANGE and Other Primitives
8.1 CHANGE and PROXIMITY
The derivative requires the limit concept from PROXIMITY.
- CHANGE asks: What is different?
- PROXIMITY enables: What happens as Δt → 0?
8.2 CHANGE and SAMENESS
SAMENESS is the negation of CHANGE.
- Where SAMENESS holds, the derivative is zero
- Equilibrium: d[A]/dt = 0, d[B]/dt = 0
8.3 CHANGE and DIRECTION
In multivariate functions, CHANGE has DIRECTION.
The gradient ∇f points in the direction of steepest increase.
The gradient points in the direction of fastest increase.
Summary
9.1 What CHANGE Is
CHANGE is the perception of becoming—the recognition that the current state differs from the previous state.
The perception is primary. Infants perceive motion. Organisms respond to environmental change. The awareness of transformation precedes any formal quantification.
9.2 Key Concepts
Core Concepts
9.3 Forward
CHANGE is formalized through the derivative. The next primitive, RATE, applies this to temporal evolution.
- Lecture 14: Instantaneous Rate — the derivative definition
- Lecture 15: Rules of Change — differentiation techniques
- Lecture 16: The Kinetics Problem — RATE applied to reactions