AP Environmental Science — Quantitative Relationships
6 sections · 35 entries · print it and keep it beside your practice sets
AP Environmental Science does not publish an equation sheet, and a calculator is allowed on both sections — so about a quarter of the points come from arithmetic you set up yourself. These are the relationships the exam reuses year after year. Show every unit conversion: setup earns credit even when the final number is wrong.
Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.
Human population
Doubling time from a percentage growth rate
A country growing at 2%/yr doubles in about 35 years. Works for any exponentially growing quantity, including energy demand.
Natural increase rate from crude birth and death rates (per 1,000 people)
Including migration
Total fertility rate — average children per woman
Replacement-level fertility is about 2.1 in developed countries; higher where child mortality is high.
People per unit area
Environmental impact model
Impact = Population × Affluence × Technology.
Percent change, rates & efficiency
Change relative to the starting value
Always divide by the ORIGINAL value. Note the difference between a change of 5 percentage points and a 5% change.
Any per-unit-time quantity
Useful output over total input
The 10% rule for energy transfer between trophic levels
Average time a molecule spends in a reservoir
Energy & power
Energy from power and time
1 kW running for 1 hour = 1 kWh. This single line is behind most APES energy math.
Kilowatt-hour in joules
Energy returned on energy invested
Dimensionless. Greater than 1 means net energy gain; conventional oil is high, tar sands and corn ethanol are low.
How much of nameplate capacity a plant actually delivers
Typical setup
Pollution, decay & toxicity
Amount remaining after time t
After n half-lives, the fraction left is 1/2ⁿ: 3 half-lives leaves 12.5%.
Concentration in water
Mass of pollutant delivered by a flow
Dose lethal to 50% of a test population
A LOWER LD50 means a MORE toxic substance.
Biochemical oxygen demand — oxygen consumed by decomposers
High BOD drives dissolved oxygen down, which is what kills fish in a eutrophic lake.
Global warming potential, relative to CO₂ over 100 years
CO₂ = 1 by definition; methane ≈ 25–30; nitrous oxide ≈ 265–300. Multiply mass by GWP to compare gases.
Water & land
Stream discharge from cross-sectional area and velocity
Units: m² × m/s = m³/s. Doubling the width of a channel at the same depth and speed doubles the discharge.
Useful conversions
Typical setup
Mass lost per area per year
Biodiversity
Diversity index; n = individuals of one species, N = total individuals
Higher D means greater diversity. D = 0 is a monoculture; the maximum approaches 1.
Simple count of species present
Richness ignores evenness — two communities with the same richness can have very different Simpson values.
Share of the community made up by one species
Lincoln index estimate of population size
# Constants & conversions
Approximate current atmospheric carbon dioxide concentration
Pre-industrial was about 280 ppm.
Composition of dry air by volume
Distribution of water
Incoming solar radiation at the top of the atmosphere
Fraction of sunlight reflected