Efficiency

This lesson covers efficiency and how to calculate it, including the use of Sankey diagrams, as required by the Edexcel GCSE Combined Science specification (1SC0). Efficiency is a key concept that links energy stores, transfers and dissipation.

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What Is Efficiency?

Efficiency is a measure of how much of the input energy (or power) is usefully transferred. It is expressed as a decimal or a percentage.

$\text{Efficiency} = \frac{\text{useful output energy transfer}}{\text{total input energy transfer}}$Efficiency=total input energy transferuseful output energy transfer​

$\text{Efficiency (\%)} = \frac{\text{useful output energy transfer}}{\text{total input energy transfer}} \times 100\%$Efficiency (%)=total input energy transferuseful output energy transfer​×100%

The same equations work with power:

$\text{Efficiency} = \frac{\text{useful output power}}{\text{total input power}}$Efficiency=total input poweruseful output power​

Quantity	Unit
Efficiency (decimal)	No unit (between 0 and 1)
Efficiency (percentage)	% (between 0% and 100%)

Exam Tip: Efficiency can never be greater than 1 (or 100%). If your calculation gives a value above 100%, you have made an error — check which is the input and which is the output.

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Worked Examples

Worked Example 1

A light bulb is supplied with 100 J of energy. It transfers 10 J as light and 90 J as heat. Calculate the efficiency.

1. $\text{Efficiency} = \frac{\text{useful output}}{\text{total input}} = \frac{10}{100} = 0.10$Efficiency=total inputuseful output​=10010​=0.10
2. As a percentage: $0.10 \times 100\% = 10\%$0.10×100%=10%
3. Answer: Efficiency = 10% (or 0.10)

Worked Example 2

An LED lamp has an efficiency of 0.60. If the input power is 8 W, what is the useful output power?

1. Rearrange: $\text{useful output power} = \text{efficiency} \times \text{total input power}$useful output power=efficiency×total input power
2. $\text{useful output} = 0.60 \times 8 = 4.8 \text{ W}$useful output=0.60×8=4.8 W
3. Answer: Useful output power = 4.8 W

Worked Example 3

A motor has a useful output of 1500 J and wastes 500 J. What is its efficiency?

1. Total input = useful output + wasted = 1500 + 500 = 2000 J
2. $\text{Efficiency} = \frac{1500}{2000} = 0.75$Efficiency=20001500​=0.75
3. As a percentage: 75%
4. Answer: Efficiency = 75%

Exam Tip: If the question gives you the useful output and the wasted energy, you must add them together to find the total input. A common error is to divide useful output by wasted energy — this is wrong.

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Sankey Diagrams

A Sankey diagram is a visual representation of energy transfers. The width of each arrow is proportional to the amount of energy.

flowchart LR
    A["Input energy\n100 J"] -->|"Useful output\n40 J"| B["Useful store"]
    A -->|"Wasted\n60 J"| C["Thermal store\n(surroundings)"]

Reading a Sankey Diagram

Feature	What it tells you
Width of input arrow	Total energy input
Width of useful output arrow	Useful energy transferred
Width of wasted arrow(s)	Energy dissipated (wasted)
Relative widths	Show efficiency visually

Rules for Sankey Diagrams

1. The input arrow enters from the left.
2. The useful output arrow continues to the right.
3. Wasted energy arrows branch off downwards.
4. The total width of all output arrows = the width of the input arrow (conservation of energy).

Worked Example 4

A Sankey diagram for a petrol engine shows: input = 1000 J, kinetic energy output = 250 J, thermal waste = 750 J. What is the efficiency?

1. $\text{Efficiency} = \frac{250}{1000} = 0.25$Efficiency=1000250​=0.25
2. Answer: Efficiency = 25%

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Improving Efficiency

No device is 100% efficient — some energy is always wasted. However, we can improve efficiency by reducing the wasted energy.

Device	Method to improve efficiency
Engine	Lubricate moving parts to reduce friction
Building	Add insulation to reduce heat loss
Light bulb	Use LED instead of filament (less heat wasted)
Motor	Use better bearings, reduce air resistance
Power station	Use waste heat for district heating (combined heat and power)

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Comparing Efficiencies

Device	Typical efficiency
Filament light bulb	5%
Energy-saving (CFL) bulb	15%
LED bulb	30–60%
Electric motor	80–95%
Petrol engine	25%
Diesel engine	35%
Gas boiler	90%
Electric heater	~100% (all energy becomes heat)
Solar cell	15–25%

Exam Tip: An electric heater has nearly 100% efficiency because all the electrical energy is converted to heat — and heat is the desired output. This does not mean it is the cheapest or best way to heat a room.

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Efficiency with Power

Worked Example 5

A motor has an input power of 500 W and an efficiency of 0.80. What is the useful output power?

1. $\text{useful output power} = \text{efficiency} \times \text{input power}$useful output power=efficiency×input power
2. $\text{useful output} = 0.80 \times 500 = 400 \text{ W}$useful output=0.80×500=400 W
3. Answer: 400 W

Worked Example 6

A machine has a useful output power of 120 W and a total input power of 160 W. What is its efficiency?

1. $\text{Efficiency} = \frac{120}{160} = 0.75$Efficiency=160120​=0.75
2. Answer: Efficiency = 75%

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Summary

• Efficiency = useful output energy (or power) ÷ total input energy (or power).
• Efficiency is expressed as a decimal (0 to 1) or percentage (0% to 100%).
• Efficiency can never exceed 100% — energy is always conserved.
• Sankey diagrams represent energy transfers visually; arrow widths are proportional to energy.
• Efficiency can be improved by reducing friction (lubrication), heat loss (insulation) or using more efficient technology (LEDs).
• When given useful output and wasted energy, add them to find the total input.

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Extended Worked Examples

Worked Example 7 — Car Engine

A car engine is supplied with 60 MJ of fuel energy and produces 18 MJ of useful kinetic energy in the car.

1. $\text{Efficiency} = \frac{18}{60} = 0.30 = 30\%$Efficiency=6018​=0.30=30%
2. Wasted: $60 - 18 = 42 \text{ MJ}$60−18=42 MJ — transferred by heating to the thermal store of the engine block, exhaust and surroundings.

Worked Example 8 — Gas Boiler with Heat Loss

A domestic gas boiler is rated 24 kW with an efficiency of 92%. Over 30 minutes it consumes gas with energy content 43.2 MJ. Calculate useful output and waste.