First Law of Thermodynamics

A gas turbine takes in natural gas with 500 MW of chemical energy. It produces 200 MW of electricity. Where did the other 300 MW go? The first law says: nowhere. It is all accounted for, rejected as waste heat. Energy in always equals energy out.

The first law of thermodynamics is conservation of energy for thermal systems: ΔU = Q - W, where ΔU is the change in internal energy, Q is heat added, and W is work done by the system. For a power plant running at steady state, this simplifies to Q_in = W_out + Q_rejected. Every joule of fuel energy either becomes useful work (electricity) or waste heat. No exceptions.

This is the accounting identity behind every power plant. A coal plant at 38% efficiency burning 1,000 MW of fuel produces 380 MW of electricity and 620 MW of waste heat. A combined-cycle gas plant at 60% efficiency: 600 MW electricity, 400 MW waste heat. The first law lets you calculate any unknown if you know the other two quantities.

Worked Example

A combined-cycle plant burns natural gas providing 800 MW of thermal input and produces 480 MW of electricity.

• Calculate waste heat. Q_rejected = Q_in - W_out = 800 - 480 = 320 MW.
• Calculate efficiency. η = W_out / Q_in = 480 / 800 = 60%.

A salesperson claims a new boiler produces 110% efficiency. What does the first law tell you?

The claim is impossible. Energy out cannot exceed energy in. The salesperson is either measuring wrong, defining efficiency misleadingly (some boiler ratings use lower heating value, which can give numbers above 100% relative to a different baseline), or lying.

The first law does not tell you which conversions are possible, only that the books must balance. For directionality, you need the second law.

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