The First Law of Thermodynamics: Energy, Heat, and Work Explained

Every engine ever built, every metabolic reaction in your body, every star burning in the sky — all operate under one inviolable constraint: the first law of thermodynamics. It is, at its core, the law of conservation of energy applied to thermodynamic systems, and it governs the relationship between heat, work, and internal energy with mathematical precision.

The Statement of the First Law

The first law of thermodynamics states: The change in internal energy of a system equals the heat added to the system minus the work done by the system.

Here, ΔU is the change in internal energy (the total kinetic and potential energy of all the molecules inside the system), Q is the heat transferred into the system (positive when heat flows in, negative when it flows out), and W is the work done by the system on its surroundings (positive when the system expands, negative when it's compressed).

This equation is a bookkeeping law. Energy cannot appear from nothing and cannot disappear — every joule of energy is accounted for. If you add heat to a gas and the gas doesn't do any work (constant volume), all the heat goes into increasing the internal energy — the gas gets hotter. If the gas expands and does work but no heat flows in, the internal energy decreases — the gas cools down. The first law tells you exactly how these quantities balance.

Internal Energy

Internal energy (U) is the sum of all microscopic energies in a system — the translational kinetic energy of molecules zooming around, rotational kinetic energy of molecules tumbling, vibrational energy of atoms within molecules, and the potential energy of intermolecular forces. For an ideal gas (a useful simplification), internal energy depends only on temperature:

where n is the number of moles, C_v is the molar heat capacity at constant volume, and T is absolute temperature in kelvin. This is why temperature is so fundamental in thermodynamics — it directly measures the internal energy of an ideal gas. Raising the temperature always means increasing internal energy.

Heat and Work: Two Ways to Change Internal Energy

There are exactly two ways to change the internal energy of a system: transfer heat across its boundary, or let it do (or have done on it) mechanical work. Heat is energy transfer driven by a temperature difference — it flows spontaneously from hotter to colder. Work is energy transfer through macroscopic mechanical means — a piston compressing a gas, for example.

Crucially, heat and work are not properties of a system — they are processes of energy transfer. You can't say a gas "contains" a certain amount of heat; you can only say heat was transferred to or from it. Internal energy, by contrast, is a property of the system's state. This distinction is one of the conceptual pillars of thermodynamics.

Thermodynamic Processes

The first law takes different simplified forms in special processes:

Isothermal process (constant temperature): For an ideal gas, ΔU = 0 (since U depends only on T). Therefore Q = W — all heat input goes directly into work output. This is why isothermal processes appear in ideal engine cycles.

Adiabatic process (no heat transfer, Q = 0): ΔU = −W. The system's internal energy changes only through work. When a gas expands adiabatically, it does positive work and its internal energy (and temperature) decreases — this is why air cools as it rapidly expands, and why diesel engines ignite fuel without spark plugs.

Isochoric process (constant volume, W = 0): ΔU = Q. All heat goes into changing internal energy. No work is done because there's no volume change. This is the scenario in a rigid sealed container.

Isobaric process (constant pressure): Both Q and W are non-zero, and the general first-law equation applies. Cooking at atmospheric pressure approximates this condition.

Connection to the Physics of Engines

The first law explains why a perfect heat engine — one that converts 100% of heat into work — is impossible. To run a cycle (return to the same state), the change in internal energy over a complete cycle is zero (ΔU = 0). Therefore Q = W for the cycle as a whole: you can only get out as much work as net heat flows in. The second law of thermodynamics then adds a further constraint — some heat must always be exhausted to the environment. The interplay of these two laws defines the maximum possible efficiency of any heat engine.

Everyday Applications

The first law is everywhere. Your body is a thermodynamic system: you consume food (chemical energy), your metabolism converts it to internal energy and heat, you do work (exercise), and you radiate heat to stay at constant temperature. Every calorie you count is a measure of the internal energy stored in food, governed by the same equation that governs steam engines and stars. The first law of thermodynamics connects the physics of energy to every process in the physical and biological world.