First Law of Thermodynamics — Energy Conservation Explained

Energy accounting for the universe

Every thermal process is an energy transaction, and the first law is the accountant. It says something profound yet simple: energy cannot be created or destroyed — it can only change form or move between a system and its surroundings.

For a gas in a cylinder, the energy ledger has three entries: heat flowing in, work done by expansion, and the system's own stored energy — its internal energy $U$.

The equation

In the physics convention:

What internal energy actually is

Internal energy is everything happening at the molecular scale: the kinetic energy of jiggling molecules (which sets the temperature), plus the potential energy stored in molecular bonds and attractions. For an ideal gas, attractions are negligible, so internal energy depends only on temperature — heat a sealed rigid tank and every joule becomes molecular motion.

Change the internal energy and you change the state of the system: hotter, higher pressure, or a different phase.

Working the signs

Worked example. A gas absorbs 800 J of heat while expanding against a piston, doing 500 J of work. Find ΔU.

$$\Delta U = Q - W = 800 - 500 = +300\ \mathrm{J}$$

The gas keeps 300 J as internal energy — it ends hotter than it started. Now reverse it: compress a gas with 400 J of work while it loses 150 J of heat. Then $Q = -150$, $W = -400$ (work done ON the gas), and:

$$\Delta U = (-150) - (-400) = +250\ \mathrm{J}$$

Sign discipline is everything. Write each sign explicitly before computing.

Four special cases, one law

Four textbook processes are just the first law with terms zeroed out:

• Isochoric (constant volume): $W = 0$, so $\Delta U = Q$ — all heat becomes internal energy.
• Isothermal (constant temperature, ideal gas): $\Delta U = 0$, so $Q = W$ — heat in equals work out.
• Adiabatic (no heat exchange): $Q = 0$, so $\Delta U = -W$ — expansion cools the gas.
• Isobaric (constant pressure): $W = P\Delta V$, the full law applies.

Real engines cycle through these continuously — which is exactly what the second law then constrains.

Variables and units

Common mistakes

Real-world applications

• Steam power plants: chemical energy becomes heat, heat becomes work — the first law balances every joule in the plant's energy accounts.

• Refrigerators: electrical work drives heat from the cold interior to the warm kitchen, with energy conserved at every step.

• Human body: food energy (Q in, chemically) becomes body heat plus the mechanical work of muscles — dietitians and physiologists both use this balance.

• Internal combustion engines: only a fraction of fuel heat becomes shaft work; the first law tells you exactly where the rest went (exhaust and coolant).