Exponential Decay Functions
Learning Objectives
Students will be able to:
- Solve first-order differential equations for concentration decay
- Calculate half-life and time constants from decay data
- Graph exponential decay and interpret its features
- Apply logarithms to linearize exponential data
- Determine decay rates from experimental measurements
The Big Question
"After a pollution event ends, how long does it take for indoor air to clear? What mathematical function describes this process?"
The First-Order Decay Equation
When sources are removed, particle concentration decays following first-order kinetics:
dC/dt = -lambda * C
Solution: C(t) = C0 * e-lambda*t
Where lambda = lambdav + lambdad + lambdaf (total loss rate)
Key Parameters
Half-Life (t1/2)
t1/2 = ln(2)/lambda
Time for 50% reduction
Time Constant (tau)
tau = 1/lambda
Time for 63.2% reduction
e-Folding Time
After 1 tau: 37% remains
After 3 tau: 5% remains
After 5 tau: 0.7% remains
Linearization with Logarithms
Taking the natural log of both sides:
ln(C) = ln(C0) - lambda*t
This is a linear equation: y = b + mx, where:
- y = ln(C)
- b = ln(C0) (y-intercept)
- m = -lambda (slope)
- x = t
Key insight: Plotting ln(C) vs. t gives a straight line with slope = -lambda
Activity: Decay Analysis
Problem Set
- Basic decay: A room has lambda = 2 h-1 and initial C0 = 100 ug/m3. Calculate C at t = 30 min, 1 h, 2 h, and 3 h.
- Half-life: Calculate the half-life for this room. How many half-lives until C < 5 ug/m3?
- Determine lambda: Measurements show C dropped from 80 to 20 ug/m3 in 45 minutes. Calculate lambda.
- Graphing: Plot the decay curve from problem 1, then create a semi-log plot (ln(C) vs t) and verify the linearity.
Key Takeaway
Exponential decay is the fundamental pattern for particle concentration decrease after sources are removed. The decay rate (lambda) determines how quickly air clears, with higher rates meaning faster clearing. Understanding this mathematics is essential for designing ventilation systems and air cleaners that can achieve target air quality within acceptable time frames.