This cheat sheet covers the most important equations and cycles used in AP Environmental Science. Students need these formulas to solve population, energy, pollution, and resource problems quickly and accurately. It also connects calculations to the major environmental systems that move matter and energy through ecosystems.
A clear reference helps students review for quizzes, labs, FRQs, and the AP exam.
The core equations include population growth, percent change, doubling time, half-life, energy conversions, and trophic efficiency. The major cycles include the carbon, nitrogen, phosphorus, and water cycles, with attention to reservoirs, fluxes, and human impacts. Energy flows one way through ecosystems, while matter cycles between living and nonliving parts of Earth.
Strong AP answers often combine a correct calculation with a clear explanation of the environmental process involved.
Key Facts
- Population growth rate can be estimated with r = birth rate - death rate when immigration and emigration are not included.
- Population change can be calculated with population change = (births + immigration) - (deaths + emigration).
- Percent change is calculated with percent change = ((new value - old value) / old value) x 100.
- Doubling time can be estimated with the Rule of 70: doubling time = 70 / percent growth rate.
- Half-life decay can be modeled by remaining amount = original amount x (1/2)^number of half-lives.
- Energy conversion commonly used in AP Environmental Science is 1 kilowatt-hour = 3,600,000 joules.
- Trophic efficiency is calculated with trophic efficiency = (energy transferred to next level / energy at previous level) x 100.
- Only about 10% of energy is typically transferred from one trophic level to the next, while most energy is lost as heat and metabolism.
Vocabulary
- Carrying capacity
- The maximum population size that an environment can support over time with available resources.
- Half-life
- The time required for half of a radioactive or unstable substance to decay.
- Trophic level
- A feeding position in a food chain or food web, such as producer, primary consumer, or secondary consumer.
- Biogeochemical cycle
- The movement of matter through living organisms, the atmosphere, water, rocks, and soil.
- Reservoir
- A place where a substance such as carbon, nitrogen, phosphorus, or water is stored for a period of time.
- Flux
- The rate at which matter moves from one reservoir to another in an environmental cycle.
Common Mistakes to Avoid
- Using the Rule of 70 with a decimal growth rate is wrong because the formula uses percent growth rate; use 2 instead of 0.02 for a 2% growth rate.
- Forgetting immigration and emigration in population change is wrong because migration can increase or decrease population size even when births and deaths stay constant.
- Confusing energy flow with nutrient cycling is wrong because energy moves one way through ecosystems, while matter is reused through biogeochemical cycles.
- Assuming the phosphorus cycle has a large atmospheric phase is wrong because phosphorus mainly moves through rocks, soil, water, and organisms.
- Treating trophic efficiency as always exactly 10% is wrong because 10% is a useful average, but actual transfer varies by ecosystem and organism.
Practice Questions
- 1 A town grows from 50,000 people to 57,500 people in 10 years. What is the percent change in population?
- 2 A radioactive pollutant has a half-life of 8 years. If 160 grams are released, how many grams remain after 24 years?
- 3 A producer level contains 20,000 kcal of energy, and the primary consumers receive 2,400 kcal. What is the trophic efficiency?
- 4 Explain why deforestation can affect both the carbon cycle and the water cycle without doing a calculation.
Understanding AP Environmental Science Equations and Cycles
Environmental calculations are useful only when their assumptions match the situation. A simple population growth rate ignores migration, so it works best for a closed population or a short time period when movement is small. Real populations rarely stay under ideal conditions.
Food supply, disease, predators, housing, and government policies can change birth and death rates. Growth that seems small each year can become large over decades because each new individual may add to later growth.
When using doubling time, check that the growth rate is a percent per year and that it stays roughly constant. A declining population has no meaningful doubling time under this method.
Percent change problems often test careful reading more than difficult arithmetic. The original value belongs in the denominator because it is the reference point. A change from ten to twenty is an increase of one hundred percent, while a change from twenty to ten is a decrease of fifty percent.
Students should label units at every step. A value in kilograms cannot be combined directly with a value in grams. Energy conversion problems require the same care.
Electricity use may be reported in kilowatt-hours, while ecosystem energy is often given in joules or calories. Convert before comparing values, then round only at the end.
Half-life describes a repeating proportional decrease, not the loss of one fixed amount each period. After each half-life, half of what remains disappears. This makes radioactive waste persist longer than many people expect.
A sample can become much less radioactive without ever reaching zero. Environmental scientists use this idea when discussing radioactive isotopes, radioactive contamination, and some dating methods. Do not confuse half-life with the time needed for a chemical to biodegrade.
Some pollutants break down through biological or chemical processes that do not follow a half-life pattern. In any decay problem, identify the starting amount, the length of one half-life, and the total elapsed time before calculating.
Trophic efficiency explains why food chains usually have few levels. Producers capture energy from sunlight and store some of it as chemical energy. Herbivores use much of the energy they eat for movement, body maintenance, and waste.
Predators do the same. Only the energy stored as new biomass is available to the next level. This limits the number of large predators an ecosystem can support.
Energy pyramids are usually upright, but biomass pyramids can look unusual in fast-growing aquatic systems. Tiny phytoplankton may have low standing biomass while replacing themselves quickly enough to support more consumer biomass.
The biogeochemical cycles show where matter can be stored and how it moves. Carbon can remain in forests, soils, oceans, fossil fuels, and carbonate rock. Burning fuels transfers long-stored carbon to the atmosphere quickly.
Nitrogen becomes usable to plants only after fixation and other microbial transformations. Fertilizer adds reactive nitrogen, which can wash into water and help cause algal blooms. Phosphorus has no major atmospheric phase, so it commonly moves slowly through rock, soil, water, and organisms.
The water cycle links all of these processes through evaporation, precipitation, runoff, infiltration, and groundwater flow. In written responses, name the reservoir, name the process moving material, then state one ecological effect of a human change.