Important Formulas to Memorize for APES
What You Need to Know
AP Environmental Science (APES) math is usually plug-and-chug + unit logic. The exam loves questions where you:
- Convert units cleanly (dimensional analysis)
- Compute rates, percent change, density
- Use a few core ecology/energy/chemistry equations correctly
- Interpret the result with correct units and reasonableness
If you memorize a small set of formulas and get fast at unit conversions, you can pick up a lot of “easy” points.
Critical reminder: In APES, the units are often half the problem. If your units don’t make sense, your setup is probably wrong.
Core idea (the “APES math mindset”)
Most APES calculations reduce to one of these structures:
- Rate: \text{rate} = \frac{\text{amount}}{\text{time}}
- Density/Concentration: \text{density or concentration} = \frac{\text{amount}}{\text{space}}
- Percent: \% = \frac{\text{part}}{\text{whole}}\times 100
Step-by-Step Breakdown
A) Dimensional analysis (unit conversion) — the highest-yield skill
- Write the given value with units.
- Multiply by conversion factors (fractions that equal 1) so units cancel.
- Cancel units top/bottom until only desired units remain.
- Do the arithmetic last (helps you avoid mistakes).
Mini-example (speed): Convert 72\ \text{km/hr} to \text{m/s}.
- Setup:
72\ \frac{\text{km}}{\text{hr}}\times \frac{1000\ \text{m}}{1\ \text{km}}\times \frac{1\ \text{hr}}{3600\ \text{s}} - Cancel \text{km} and \text{hr} → left with \text{m/s}.
- Compute: 72\times \frac{1000}{3600} = 20\ \text{m/s}.
B) Percent change (common in population, emissions, trends)
- Identify old value and new value.
- Compute difference: \text{new} - \text{old}
- Divide by old: \frac{\text{new} - \text{old}}{\text{old}}
- Multiply by 100.
Formula: \%\ \text{change} = \frac{\text{new} - \text{old}}{\text{old}}\times 100
C) Population growth + doubling time
- Decide if you’re being asked for growth rate, new population, or doubling time.
- Use the correct equation (see tables below).
- Sanity-check: if growth rate is positive, population should increase.
D) Electricity/energy cost problems
- Convert power to kilowatts if needed: \text{kW} = \frac{\text{W}}{1000}
- Compute energy used: E = P\times t (use \text{kWh})
- Cost: \text{cost} = (\text{kWh})\times (\text{price per kWh})
E) pH / acidity
- If given [H^+], use \text{pH} = -\log_{10}([H^+]).
- Remember: lower pH = more acidic.
- Each pH unit is a 10× change in [H^+].
Key Formulas, Rules & Facts
1) Population & demography
| Formula | When you use it | Notes / units |
|---|---|---|
| \text{Population density} = \frac{\text{population}}{\text{area}} | People/organisms per space | Units like \text{people/km}^2 |
| \Delta N = \text{births} + \text{immigration} - \text{deaths} - \text{emigration} | Net population change | Works for populations (humans/wildlife) |
| r = \frac{(b + i) - (d + e)}{N} | Per-capita growth rate | b,i,d,e are counts per time; N is population size |
| \text{Growth rate (\%)} = \frac{\text{CBR} - \text{CDR}}{10} | Human populations using rates per 1000 | CBR/CDR are births/deaths per 1000 people per year |
| t_d \approx \frac{70}{\text{growth rate (\%)}} | Doubling time estimate | “Rule of 70” (sometimes 69.3 is used; 70 is APES-friendly) |
| N(t)=N_0 e^{rt} | Exponential growth (continuous) | Usually conceptual, but can appear with given values |
| \frac{dN}{dt}=rN\left(\frac{K-N}{K}\right) | Logistic growth concept | K = carrying capacity (you rarely compute from scratch) |
Quick unit warning: If you use \frac{70}{\%}, the percent must be a number like 2, not 0.02.
2) Energy, power, efficiency, and electricity bills
| Formula | When you use it | Notes / units |
|---|---|---|
| P = \frac{E}{t} | Relate power, energy, time | P in W, E in J, t in s |
| E = P\times t | Energy used by device | On bills use \text{kWh} |
| \text{kWh} = \text{kW}\times \text{hr} | Utility energy unit | 1\ \text{kWh} = 3.6\times 10^6\ \text{J} |
| P = IV | Electric power | I in A, V in V, P in W |
| \%\ \text{efficiency} = \frac{\text{useful output}}{\text{total input}}\times 100 | Engines, power plants, appliances | Waste heat explains losses |
| \text{Energy at next trophic level} \approx 0.10\times \text{previous level} | 10% rule (food webs) | Ecological rule of thumb; not exact |
3) Ecosystem productivity (carbon/energy flow)
| Formula | When you use it | Notes |
|---|---|---|
| \text{NPP} = \text{GPP} - R | Net primary productivity | R = respiration by producers |
| \text{Biomass change} \approx \text{NPP} - \text{losses} | Conceptual biomass questions | Losses include consumption/decomposition |
4) Chemistry: concentration, dilution, pH
| Formula | When you use it | Notes / units |
|---|---|---|
| \text{Concentration} = \frac{\text{mass of solute}}{\text{volume of solution}} | Pollution in water, nutrients, etc. | Commonly \text{mg/L} |
| \text{ppm} \approx \frac{\text{mg}}{\text{L}} | Water (dilute solutions) | Because 1\ \text{L water} \approx 1\ \text{kg} |
| C_1V_1 = C_2V_2 | Dilution problems | Assumes solute amount conserved |
| \text{pH} = -\log_{10}([H^+]) | Acidity | Lower pH = higher [H^+] |
| \text{pH} + \text{pOH} = 14 | Acid-base relation (at 25°C) | Useful if hydroxide is given |
5) Atmosphere & emissions (common “environmental math” setups)
| Formula | When you use it | Notes |
|---|---|---|
| \text{Emissions} = \text{Activity}\times \text{Emission factor} | Carbon footprint style | Units must match (e.g., \text{kg CO}_2/\text{kWh}) |
| \text{Mass} = \text{Concentration}\times \text{Volume} | Total pollutant mass | Great for “how much total pollutant” questions |
| \text{ppb} = \text{ppm}\times 1000 | Unit changes | Also 1\ \text{ppm} = 1000\ \text{ppb} |
6) Radioactivity / half-life (nuclear power, contamination)
| Formula | When you use it | Notes |
|---|---|---|
| N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}} | Remaining after time t | t_{1/2} is half-life |
| t = n\cdot t_{1/2} with N = \frac{N_0}{2^n} | “How many half-lives?” style | Fast mental math when time is a multiple of half-life |
7) Heat & water (occasionally tested, especially in labs)
| Formula | When you use it | Notes |
|---|---|---|
| Q = mc\Delta T | Heating/cooling water, climate/thermal pollution | c for water is high; units must be consistent |
8) Biodiversity indices (sometimes used in APES classes)
These appear more often in labs/classwork than on the national exam, but if your course emphasizes them, know the setup.
| Formula | When you use it | Notes |
|---|---|---|
| D = 1 - \sum\left(\frac{n}{N}\right)^2 | Simpson’s Index of Diversity | Higher D = higher diversity (with this common form) |
| H = -\sum p_i\ln(p_i) | Shannon index | p_i = \frac{n_i}{N}; higher H = higher diversity |
Examples & Applications
1) Doubling time (Rule of 70)
A country’s population grows at 2.5% per year. Estimate doubling time.
t_d \approx \frac{70}{2.5} = 28\ \text{years}
Exam angle: If they ask whether that’s “fast,” compare to typical developed-country growth rates (often <1%).
2) Electricity cost (kWh)
A 1500\ \text{W} space heater runs for 4\ \text{hr/day}. Electricity costs 0.20\ \text{USD/kWh}. What’s the daily cost?
1) Convert to kW: 1500\ \text{W}\times \frac{1\ \text{kW}}{1000\ \text{W}} = 1.5\ \text{kW}
2) Energy used: E = 1.5\ \text{kW}\times 4\ \text{hr} = 6\ \text{kWh}
3) Cost: 6\ \text{kWh}\times 0.20\ \text{USD/kWh} = 1.2\ \text{USD}
Exam angle: They may extend this to monthly cost by multiplying by 30.
3) Water pollution concentration (ppm and mg/L)
A stream sample has 8\ \text{mg} nitrate in 2\ \text{L} of water. Find concentration in \text{mg/L} and approximate ppm.
\text{Concentration} = \frac{8\ \text{mg}}{2\ \text{L}} = 4\ \text{mg/L}
For dilute water solutions: 4\ \text{mg/L} \approx 4\ \text{ppm}
Exam angle: They might flip it: given ppm and volume, find total mass.
4) Half-life (radioactive decay)
You start with 80\ \text{g} of a radionuclide with t_{1/2} = 10\ \text{years}. How much remains after 30\ \text{years}?
30\ \text{years} = 3\ \text{half-lives} \Rightarrow N = \frac{80}{2^3} = 10\ \text{g}
Exam angle: They may ask for percent remaining: \frac{10}{80}\times 100 = 12.5\%.
Common Mistakes & Traps
Mixing up “percent” and “decimal” growth rates
- Wrong: using 0.02 in \frac{70}{\text{growth rate}}.
- Right: use 2 if growth is 2%.
Forgetting to convert watts to kilowatts on electricity problems
- Wrong: treating 1500\ \text{W} as 1500\ \text{kW}.
- Fix: always do \text{kW} = \frac{\text{W}}{1000} before computing kWh.
Using the wrong “old value” in percent change
- Wrong denominator: dividing by new.
- Correct: \%\ \text{change} = \frac{\text{new} - \text{old}}{\text{old}}\times 100.
Not carrying units through the work
- Why it hurts: you miss cancellation mistakes (like hours not converted to seconds).
- Fix: write units on every number until the end.
Confusing ppm relationships across media
- Trap: \text{ppm} \approx \text{mg/L} is a water approximation (dilute, density near 1).
- Fix: Only use \text{ppm} \approx \text{mg/L} when it’s clearly dilute aqueous solution.
pH direction mistakes
- Wrong: thinking higher pH means more acidic.
- Correct: higher pH = more basic, and each +1 pH means 10× less [H^+].
Half-life exponent errors
- Wrong: plugging t_{1/2} into the exponent incorrectly.
- Correct: N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}} and check: if t = t_{1/2}, you must get \frac{1}{2}N_0.
Efficiency used backward
- Wrong: \%\ \text{efficiency} = \frac{\text{input}}{\text{output}}\times 100.
- Correct: \%\ \text{efficiency} = \frac{\text{useful output}}{\text{total input}}\times 100 (should usually be <100%).
Memory Aids & Quick Tricks
| Trick / mnemonic | What it helps you remember | When to use it |
|---|---|---|
| “Keep-Flip-Change” | Dividing by a fraction during conversions | Any dimensional analysis step |
| Rule of 70: t_d \approx \frac{70}{\%} | Fast doubling time | Population/economic growth estimate |
| pH “down = acidic” | Lower pH means more [H^+] | Acid rain, ocean acidification |
| kWh = kW × hr | Utility billing unit | Electricity cost problems |
| “Half-life ladder”: N_0\to \frac{N_0}{2}\to \frac{N_0}{4}\to \frac{N_0}{8} | Quick decay without exponents | When time is an integer multiple of half-life |
| ppm in water: \text{ppm} \approx \text{mg/L} | Fast concentration interpretation | Water pollution questions (dilute) |
| 10% rule | Energy drops hard up trophic levels | Food web/biomass questions |
Quick Review Checklist
- [ ] You can do dimensional analysis and cancel units cleanly.
- [ ] You know the “big three” structures: \text{rate} = \frac{\text{amount}}{\text{time}}, \text{density} = \frac{\text{amount}}{\text{space}}, \% = \frac{\text{part}}{\text{whole}}\times 100.
- [ ] You can compute percent change: \frac{\text{new} - \text{old}}{\text{old}}\times 100.
- [ ] You can estimate doubling time with \frac{70}{\%} (percent as a whole number).
- [ ] You can compute electricity use/cost with \text{kWh} = \text{kW}\times \text{hr} and convert \text{W}\to \text{kW}.
- [ ] You know \text{NPP} = \text{GPP} - R.
- [ ] You can work concentration problems and remember \text{ppm} \approx \text{mg/L} for dilute water.
- [ ] You can do pH: \text{pH} = -\log_{10}([H^+]) and interpret direction correctly.
- [ ] You can do half-life: N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}.
You’re aiming for clean setups, correct units, and fast interpretation—do that and APES math becomes free points.