The First Law of Thermodynamics — Energy, Heat & Work
The First Law of Thermodynamics is conservation of energy applied to heat and work. Its mathematical statement fits on one line: ΔU = Q − W. But behind this compact equation lies one of the most important insights in 19th-century physics — that heat and work are not fundamentally different things. They are both simply ways of transferring energy across a system’s boundary. Understanding this equivalence changed how we build engines, understand the human body, and predict the behaviour of every gas on Earth.
1. Internal Energy — What Is U?
The internal energy U of a system is the total kinetic and potential energy of all its molecules. For an ideal monatomic gas, internal energy is purely kinetic — the random translational motion of its atoms. For more complex substances, internal energy also includes rotational energy of molecules, vibrational energy of bonds, and the potential energy stored between molecules.
Internal energy is a state function: its value depends only on the current state of the system (temperature, pressure, volume, composition) — not on how the system arrived at that state. This is a crucial property. Whether you heated a gas quickly or slowly, compressed it first and then heated it, or any other path — if it ends at the same temperature and pressure, it has the same internal energy.
Internal energy U is a state function — defined by the current state, not how you got there.
Heat Q and Work W are path functions — they depend on the process taken, not just the start and end states. Two processes connecting the same initial and final states can involve completely different amounts of heat and work — as long as their difference (Q − W) is the same.
2. The First Law — Statement and Meaning
The change in internal energy of a system equals the heat added to the system minus the work done by the system.
This is simply conservation of energy. The internal energy of a system changes only when energy crosses its boundary — either as heat (Q) or as work (W). Add heat without letting the system do work, and all the heat increases internal energy. Let the system expand and do work without adding heat, and it loses internal energy.
Sign Convention Matters: Some textbooks write ΔU = Q + W where W is work done on the system (opposite sign convention). Always check which convention is being used. In this article: Q > 0 means heat flows INTO the system; W > 0 means the system does work ON its surroundings (like expanding gas pushing a piston outward).
3. Heat — What It Is and What It Isn’t
Heat (Q) is energy in transit driven by a temperature difference. It flows spontaneously from regions of higher temperature to lower temperature and stops when temperatures equalise. Heat is not something a system “contains” — it is energy crossing a boundary.
This distinction matters enormously. A hot object has high internal energy — not “lots of heat.” Heat is the process of energy transfer, not a stored quantity. Once energy has been transferred by heat and absorbed by the system, it becomes internal energy — not heat.
| Sign of Q | Meaning | Effect on System | Example |
|---|---|---|---|
| Q > 0 | Heat flows INTO system | Internal energy increases | Heating water on a stove |
| Q < 0 | Heat flows OUT of system | Internal energy decreases | Cooling a hot metal in cold water |
| Q = 0 | Adiabatic process — no heat exchange | All energy change is via work | Rapid compression of gas in insulated cylinder |
4. Work Done by a Gas
For a gas in a cylinder with a movable piston, work is done when the gas expands or is compressed. The work done by the gas against a constant external pressure p over a volume change ΔV is:
| Sign of W | Meaning | Effect on System | Example |
|---|---|---|---|
| W > 0 | System does work on surroundings | Internal energy decreases | Gas expanding, pushing piston out |
| W < 0 | Surroundings do work on system | Internal energy increases | Gas compressed by external pressure |
| W = 0 | Isochoric (constant volume) process | All energy change is via heat | Heating gas in rigid sealed container |
5. The Four Thermodynamic Processes
| Process | Condition | Simplification | Example |
|---|---|---|---|
| Isothermal | Constant temperature (ΔT = 0) | For ideal gas: ΔU = 0, so Q = W | Slow compression in contact with heat reservoir |
| Adiabatic | No heat exchange (Q = 0) | ΔU = −W (work comes entirely from internal energy) | Rapid compression, insulated cylinder |
| Isobaric | Constant pressure | W = pΔV, ΔU = Q − pΔV | Gas heated in piston at atmospheric pressure |
| Isochoric | Constant volume (W = 0) | ΔU = Q (all heat → internal energy) | Gas heated in sealed rigid container |
6. Worked Examples
Problem: 850 J of heat is added to a gas. The gas expands, doing 320 J of work on the piston. What is the change in internal energy?
Problem: A piston compresses a gas, doing 700 J of work on it. Simultaneously, 250 J of heat escapes to the surroundings. What is ΔU?
Problem: A gas undergoes adiabatic compression — no heat is exchanged. The surroundings do 1,200 J of work on the gas. What happens to internal energy?
7. Real-World Applications
Steam Engine
Heat (Q) is added from burning fuel; the expanding steam does work (W) on the pistons. ΔU = Q − W governs every stroke of the engine.
Refrigerator
Work is done on the refrigerant gas (compression). Heat is extracted from inside the fridge and expelled outside. Energy is moved, not destroyed.
Human Lungs
Breathing involves work done by chest muscles (W) and heat exchange with inhaled air (Q). The first law governs respiratory thermodynamics.
Diesel Engine
Rapid adiabatic compression (Q = 0) raises air temperature above diesel’s ignition point — no spark plug needed. Pure first law thermodynamics.
8. Common Misconceptions
“Heat and temperature are the same thing.” Temperature is a measure of average molecular kinetic energy. Heat is the flow of energy due to a temperature difference. A swimming pool at 20°C contains far more thermal energy than a cup of boiling water at 100°C — even though the water has a higher temperature — because the pool has vastly more mass and therefore more total molecular kinetic energy.
“The first law says energy is conserved, so perpetual motion machines are theoretically possible.” The first law prohibits machines that create energy (perpetual motion of the first kind). The second law goes further — it also prohibits machines that convert heat to work with 100% efficiency. Both laws together rule out all perpetual motion machines, in principle and in practice.
9. Frequently Asked Questions
Conclusion
The First Law of Thermodynamics, ΔU = Q − W, is one of the most powerful equations in all of science. It tells us that internal energy changes only through heat transfer or work — and that these two mechanisms, though physically different, are equivalent ways of transferring energy. Heat and work are interconvertible; energy is conserved.
The four special processes — isothermal, adiabatic, isobaric, isochoric — each simplify the first law in different ways, and together they describe the full range of thermodynamic behaviour of gases. Master the sign conventions and the physical meaning of each term, and the rest of thermodynamics becomes dramatically more accessible.