APES
Priority topic · Calculation FRQ favorite

Energy Calculations

APES rewards every line of unit cancellation. Q3 (Calculations) almost always includes energy or rate work. Show the formula, every conversion factor, and box the final answer with units.

Key conversions

Energy
1 kWh = 3.6 × 10⁶ J = 3.6 MJ 1 kcal (food Calorie) = 4184 J 1 BTU ≈ 1055 J
The food "Calorie" with capital C is 1 kcal. Most APES problems use kWh (electricity) or J (chemistry).
Power vs. energy
1 W = 1 J/s 1 kW = 1000 W E = P × t (kWh = kW × hr)
Power is energy per unit time. A 100-W bulb running 10 hours uses 1 kWh.
Cost
Cost = energy × rate = kWh × $/kWh
Most US electricity costs $0.10–$0.20 per kWh. APES often gives a rate; multiply at the end.
Efficiency
% Efficiency = (useful output / input) × 100
A 30%-efficient power plant produces 30 J of electricity per 100 J of fuel energy. The other 70 J is lost as waste heat.
Half-life
N = N₀ × (½)^(t/T) After n half-lives, fraction remaining = (½)ⁿ
T = half-life. After 3 half-lives, 1/8 remains. After 4, 1/16. Common APES isotopes: I-131 (8 days), Cs-137 (30 yr), U-235 (704 million yr), Pu-239 (24,000 yr).

Worked example #1: appliance cost

A 60-W lightbulb runs 5 hr/day for 30 days at $0.12/kWh. What is the cost?

Step 1 · Total energy
  = 60 W × 5 hr/day × 30 days × (1 kW / 1000 W)
  = 60 × 5 × 30 / 1000  kWh
  = 9 kWh

Step 2 · Cost
  = 9 kWh × $0.12/kWh
  = $1.08

Worked example #2: fuel energy yield

Coal yields 25 MJ/kg. A power plant burns 1000 kg of coal at 30% efficiency. How much electrical energy is produced?

Step 1 · Total chemical energy
  = 1000 kg × 25 MJ/kg = 25,000 MJ

Step 2 · Apply efficiency
  = 25,000 MJ × 0.30 = 7,500 MJ

Step 3 · Convert to kWh (optional)
  = 7,500 MJ × (1 kWh / 3.6 MJ) ≈ 2,083 kWh

Worked example #3: half-life

Iodine-131 has a half-life of 8 days. Starting with 80 g, how much remains after 32 days?

n = t / T = 32 / 8 = 4 half-lives

Remaining = 80 × (½)⁴
          = 80 / 16
          = 5 g

Example FRQs

Long FRQ A coal plant burns 500 short tons of coal per day. Coal yields 25 MJ/kg. The plant is 35% efficient.

(a) Convert 500 short tons to kg.

500 tons × (2000 lb / ton) × (1 kg / 2.2 lb)
   = 500 × 2000 / 2.2
   ≈ 4.55 × 10⁵ kg

(b) Calculate total electrical energy produced per day in MJ.

Total chemical energy = 4.55 × 10⁵ kg × 25 MJ/kg = 1.14 × 10⁷ MJ
Electrical = 1.14 × 10⁷ MJ × 0.35 ≈ 3.98 × 10⁶ MJ/day

(c) Convert to kWh.

3.98 × 10⁶ MJ × (1 kWh / 3.6 MJ) ≈ 1.1 × 10⁶ kWh/day
Long FRQ A household uses 900 kWh/month. A 250-W solar panel produces useful power 5 hr/day. (a) Energy from one panel per month? (b) How many panels are needed to cover the household demand?

Answer:

(a) Per panel per month:
  0.250 kW × 5 hr/day × 30 days = 37.5 kWh

(b) Panels needed:
  900 kWh / 37.5 kWh per panel = 24 panels
MCQ A radioactive isotope has a half-life of 50 yr. How long until 1/8 of the original amount remains? (A) 50 yr (B) 100 yr (C) 150 yr (D) 200 yr

Answer: C. 1/8 = (½)³, so 3 half-lives × 50 yr = 150 years.

MCQ A 1.5-kW microwave is used 12 min per day. Annual electricity in kWh? (A) 109.5 (B) 657 (C) 1.5 (D) 18

Answer: A. 1.5 kW × (12/60) hr/day × 365 day/yr = 109.5 kWh/yr.

Drill flashcards

energy-calculations 1 kWh = ? Tap / Space to flip
energy-calculations 1 kWh = 3,600,000 J = 3.6 MJ = 3.6 × 10⁶ J. The amount of energy a 1000-W appliance uses in 1 hour.
energy-calculations 1 calorie Tap / Space to flip
energy-calculations 1 cal = 4.184 J. 1 Calorie (food) = 1 kcal = 4184 J. Always check capitalization!
energy-calculations 1 BTU Tap / Space to flip
energy-calculations 1 BTU ≈ 1055 J. Energy to raise 1 lb of water by 1 °F. Used for fossil-fuel and heating-system ratings.
energy-calculations Power vs. energy Tap / Space to flip
energy-calculations Power (W) = energy per time. Energy (J or kWh) = power × time. A 100-W bulb running 10 h uses 1 kWh.
energy-calculations Half-life formula Tap / Space to flip
energy-calculations After n half-lives, fraction remaining = (1/2)ⁿ. Plutonium-239 has a half-life of ~24,000 years.
energy-calculations Dimensional analysis Tap / Space to flip
energy-calculations Show every unit conversion as a fraction. APES grading rewards setup; lose units anywhere = lose the answer point.
energy-calculations Efficiency calculation Tap / Space to flip
energy-calculations Efficiency (%) = (useful energy output / total energy input) × 100. A 30%-efficient power plant wastes 70% as heat.
energy-calculations Energy density (per kg) Tap / Space to flip
energy-calculations Approximate energy per kg: U-235 ≫ natural gas (~55 MJ/kg) > oil (~42) > coal (~24) > wood (~16) > batteries (~0.7).
energy-calculations Rate of consumption Tap / Space to flip
energy-calculations kWh/day × 365 = kWh/year. Multiply by people × $/kWh for total cost. Always carry units through.
energy-calculations CO₂ per kWh Tap / Space to flip
energy-calculations Coal ≈ 1.0 kg CO₂/kWh; natural gas ≈ 0.4 kg/kWh; nuclear, wind, solar PV near 0 (operational).

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