First Law of Thermodynamics
ΔU = Q − W — energy conservation applied to heat, work, and internal energy.
Statement and Formula
The first law of thermodynamics is the statement of energy conservation for thermodynamic systems:
ΔU = Q − W
where ΔU is the change in the system's internal energy, Q is the heat added to the system, and W is the work done by the system on its surroundings.
The first law has three equivalent formulations: (1) energy is conserved; (2) it is impossible to build a perpetual motion machine of the first kind (one that creates energy); (3) the internal energy of an isolated system is constant.
For infinitesimal processes: dU = δQ − δW, where the δ notation (inexact differentials) reminds us that Q and W are path-dependent, while U is a state function.
Variable Table
| Symbol | Quantity | Sign Convention |
|---|---|---|
| ΔU | Change in internal energy (J) | +: energy stored in system increases |
| Q | Heat (J) | +: heat flows INTO system |
| W | Work done BY system (J) | +: system expands (does work on surroundings) |
Thermodynamic Processes
Isothermal Process (T = constant)
For an ideal gas at constant temperature, internal energy is unchanged: ΔU = 0. Therefore Q = W — all heat absorbed goes into work done. For an ideal gas: W = nRT ln(V_f/V_i).
Adiabatic Process (Q = 0)
No heat exchange with surroundings. First law: ΔU = −W. Work done by the gas comes entirely from its internal energy: expanding gas cools; compressed gas heats. For an ideal gas: TV^(γ−1) = constant, where γ = Cp/Cv.
Isochoric Process (V = constant)
No volume change means no PdV work: W = 0. Therefore ΔU = Q — all heat goes into internal energy. This is the process in a rigid sealed container.
Isobaric Process (P = constant)
At constant pressure, W = PΔV. The heat at constant pressure is Q = nCpΔT, and ΔU = nCvΔT. The difference Cp − Cv = R for an ideal gas (Mayer's relation).
Worked Examples
Example 1 — Finding ΔU
A system absorbs 500 J of heat and does 200 J of work on its surroundings. Find ΔU.
Solution: ΔU = Q − W = 500 − 200 = 300 J. The internal energy increases by 300 J.
Example 2 — Adiabatic compression
An ideal gas is compressed adiabatically. The surroundings do 800 J of work on the gas. Find the change in internal energy.
Solution: Q = 0. Work done BY the gas W = −800 J (work is done ON it, so W is negative). ΔU = Q − W = 0 − (−800) = +800 J. The gas warms up.
Example 3 — Isothermal expansion
An ideal gas expands isothermally and does 350 J of work. How much heat did it absorb?
Solution: Isothermal → ΔU = 0. Q = ΔU + W = 0 + 350 = 350 J absorbed from surroundings.
Common Mistakes
- Sign convention errors. Physics uses ΔU = Q − W (W = work done BY system). Engineering often uses ΔU = Q + W (W = work done ON system). Always check which convention your textbook uses.
- Treating Q and W as state functions. Q and W are path-dependent; only ΔU = Q − W is path-independent (depends only on initial and final states).
- Forgetting that "adiabatic" means Q = 0, not ΔT = 0. Temperature can change dramatically in an adiabatic process.
- Assuming isothermal means Q = 0. Isothermal means ΔT = 0, which implies ΔU = 0 for an ideal gas, but Q ≠ 0 — heat is exchanged with the surroundings to maintain constant temperature.
Applications
- Heat engines: The first law governs the energy input (heat from hot reservoir) and output (work plus waste heat).
- Refrigerators and heat pumps: Work input drives heat from cold to hot reservoir — the first law sets the energy budget.
- Atmospheric science: Adiabatic lapse rate — rising air cools adiabatically, dry air at ~9.8°C/km.
- Combustion engines: The fuel-air mixture expands against a piston, converting internal energy (chemical) to mechanical work.
- Biology: Cellular respiration is analysed thermodynamically — energy from glucose appears as ATP synthesis and heat.
Relation to the Zeroth and Second Laws
The first law establishes that internal energy is a valid state function — it has a definite value at each thermodynamic state, independent of how that state was reached. This is what makes the first law non-trivial: it asserts that Q and W, though path-dependent individually, always combine to give the same path-independent ΔU.
The zeroth law defines temperature (transitivity of thermal equilibrium). The first law defines internal energy as a conserved quantity. The second law then restricts the direction of energy flow, introducing entropy. The third law concerns the behaviour of entropy at absolute zero. Together, these four laws constitute the complete foundation of classical thermodynamics.
For open systems (matter can flow in or out), the first law is extended to include enthalpy H = U + PV: at constant pressure, ΔH = Q_p. Enthalpy is the natural thermodynamic potential for chemical reactions at constant pressure, which is why it appears throughout chemistry and biochemistry.
Related Topics
Frequently Asked Questions
What does the first law state?
Energy is conserved: ΔU = Q − W. The internal energy change equals heat in minus work out.
What is internal energy?
The microscopic kinetic and potential energy of molecules. For an ideal gas: U = (f/2)nRT, depending only on temperature.
Sign convention for Q and W?
Q > 0: heat INTO system. W > 0: work done BY system (expansion). ΔU = Q − W.
What is an adiabatic process?
Q = 0. All work comes from internal energy. Expanding adiabatically cools the gas; compressing it heats it.
References
- Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). Wiley. Chapter 1.
- Zemansky, M. W., & Dittman, R. H. (1981). Heat and Thermodynamics (7th ed.). McGraw-Hill. Chapter 4.
- Schroeder, D. V. (2000). An Introduction to Thermal Physics. Addison-Wesley. Chapter 1.
- Fermi, E. (1956). Thermodynamics. Dover Publications. Chapter 1.