The First Law — Statement and Meaning

The First Law of Thermodynamics is a statement of energy conservation applied to thermodynamic systems. In its most common form:

ΔU = Q − W
Change in internal energy = Heat added to system − Work done by the system

Every term in this equation requires careful definition, because casual use of the words "heat," "energy," and "work" leads to sign errors and conceptual confusion. Let's take each one seriously.

ΔU (change in internal energy) is the difference between the total microscopic energy of the system after and before the process. It includes all kinetic energy of the molecules (thermal motion), potential energy of molecular interactions, vibrational and rotational energy of molecules, and electronic energy. It does not include the bulk kinetic or potential energy of the system as a whole — only the internal, microscopic energy.

Q (heat) is the energy transferred across the system boundary due to a temperature difference between the system and its surroundings. It is not a property of the system — it's a process. You cannot say a gas "contains heat"; you can only say heat flowed in or out of it.

W (work) is the energy transferred by any other means — most commonly by the system expanding or being compressed, doing work on its surroundings or having work done on it.

Sign Convention (the standard physics convention)

Q positive: heat flows INTO the system (system gains energy). Q negative: heat flows OUT. W positive: system does work on surroundings (system loses energy). W negative: surroundings do work on system (system gains energy). Chemistry textbooks sometimes use ΔU = Q + W (where W is work done on the system) — be aware of which convention a source uses.

Internal Energy: What It Is and Isn't

Internal energy U is a state function — it depends only on the current state of the system (temperature, pressure, volume, composition), not on how the system got there. This is a profound property. It means that if a gas starts at state A and ends at state B, the change in internal energy ΔU = U_B − U_A is the same regardless of the path taken — whether the process was slow, fast, reversible, or involved some bizarre series of compressions and expansions.

Heat and work, by contrast, are path-dependent — they depend on how the process was carried out. You might get from A to B by adding lots of heat and doing lots of work, or by adding a small amount of heat with no work. The path is different; the ΔU is identical. This path-independence of ΔU is what makes the First Law a conservation law.

For an ideal monatomic gas (like helium or argon), the internal energy is purely kinetic: U = (3/2)nRT, where n is the number of moles and R = 8.314 J/mol·K. For real gases and other materials, additional molecular interactions contribute, but the essential point holds: U is a function of state.

Work in Thermodynamics

The most common form of thermodynamic work is expansion work — work done by a gas pushing against an external pressure. For a quasi-static (very slow, reversible) process at constant pressure P:

W = PΔV = P(V_f − V_i)
Work done by a gas expanding against constant external pressure. W > 0 when gas expands (V_f > V_i).

For variable pressure, we integrate: W = ∫P dV. This integral — the area under a P-V diagram — represents the work done. Understanding P-V diagrams is essential for analysing thermodynamic cycles like the Carnot cycle and Otto cycle. See the full Thermodynamics guide for P-V diagram analysis.

Note that work requires displacement — if a gas heats up inside a rigid container (ΔV = 0), no PΔV work is done. All the added heat goes directly into raising the internal energy.

The Four Thermodynamic Processes

Isothermal

Constant temperature. ΔT = 0, so for an ideal gas, ΔU = 0. All heat added equals work done.

Q = W  (ΔU = 0)

Adiabatic

No heat exchange. Q = 0. Any work done changes internal energy only. Gas cools when it expands.

ΔU = −W  (Q = 0)

Isochoric

Constant volume. ΔV = 0, so W = 0. All heat added goes to internal energy.

ΔU = Q  (W = 0)

Isobaric

Constant pressure. W = PΔV. Heat added goes to both internal energy and expansion work.

Q = ΔU + PΔV

These four idealised processes are building blocks for real thermodynamic cycles. A car engine's Otto cycle approximates two adiabatic and two isochoric processes. Understanding each process on a P-V diagram — what the curve looks like, how much area (work) it encloses — is the key to analysing any heat engine.

Physical Intuition for Adiabatic Processes

When a gas expands adiabatically (no heat exchange), it does work on the surroundings by pushing against them. That work energy must come from inside the gas — so the gas cools. This is why a bicycle pump gets warm when you compress air quickly (approximately adiabatic — the compression is faster than heat can escape), and why the temperature drops when you open a pressurised gas cylinder rapidly.

Heat Capacity: How Much Energy Does It Take to Heat Things Up?

The heat capacity of a substance is the amount of heat required to raise its temperature by 1 kelvin. But there are two different heat capacities, depending on conditions:

Heat CapacityConditionDefinitionWhy It's Larger/Smaller
C_VConstant volumeQ = nC_VΔTAll heat goes to raising internal energy — no work done.
C_PConstant pressureQ = nC_PΔTHeat goes to internal energy AND expansion work — must add more heat for same ΔT. C_P > C_V always.

The relationship between them: C_P − C_V = R (for an ideal gas), where R = 8.314 J/mol·K. For a monatomic ideal gas: C_V = (3/2)R, C_P = (5/2)R. The ratio γ = C_P/C_V = 5/3 ≈ 1.67 appears in the adiabatic process equations.

Worked Examples

Example 1 — Basic

Apply ΔU = Q − W directly

GivenQ = 450 J added to a gas  |  Gas expands, doing W = 180 J of work. Find: ΔU.
01

Direct application: ΔU = Q − W = 450 − 180 = +270 J

02

Interpretation: the gas gained 450 J from heat, used 180 J to push outward (do work), and kept 270 J as increased internal energy — its temperature rose.

✓ ΔU = +270 J (internal energy increased)
Example 2 — Isochoric

Gas heated at constant volume

Given2 moles of ideal monatomic gas  |  Temperature rises from 300 K to 400 K  |  Constant volume  |  C_V = (3/2)R
01

Work done: W = PΔV = 0 (constant volume, no expansion)

02

Heat added: Q = nC_VΔT = 2 × (3/2)(8.314) × 100 = 2 × 12.471 × 100 = 2,494 J

03

First Law: ΔU = Q − W = 2494 − 0 = 2,494 J

✓ Q = ΔU = 2,494 J. All heat went to raising internal energy — no work done.
Example 3 — Isobaric

Gas heated at constant pressure

Given2 moles of ideal monatomic gas  |  T: 300 K → 400 K  |  P = 1.0 × 10⁵ Pa (constant)  |  C_P = (5/2)R
01

Heat added: Q = nC_PΔT = 2 × (5/2)(8.314) × 100 = 4,157 J

02

Internal energy change: ΔU = nC_VΔT = 2,494 J (same temperature rise, same ΔU as Example 2)

03

Work done by gas: W = Q − ΔU = 4157 − 2494 = 1,663 J

04

Verify: W = nRΔT = 2 × 8.314 × 100 = 1,663 J ✓ (confirms C_P − C_V = R relation)

✓ Q = 4,157 J  |  ΔU = 2,494 J  |  W = 1,663 J (extra heat went to expansion work)
Example 4 — Real World

Adiabatic compression in a diesel engine

GivenAir compressed adiabatically  |  Initial T = 300 K, V_i = 500 cm³  |  Compression ratio: V_i/V_f = 20 (typical diesel)  |  γ = 1.4 (diatomic air)
01

For adiabatic: TV^(γ-1) = constant. So T_f = T_i(V_i/V_f)^(γ-1) = 300 × (20)^0.4

02

20^0.4 = e^(0.4 × ln20) = e^(0.4 × 2.996) = e^1.198 = 3.31

03

T_f = 300 × 3.31 = 993 K = 720°C

04

This is above diesel fuel's autoignition temperature (~250°C) — which is exactly why diesel engines don't need spark plugs. The adiabatic compression alone heats the air enough to ignite the fuel.

✓ Compressed air reaches ~993 K — hot enough to spontaneously ignite diesel fuel.

Why the First Law Rules Out Perpetual Motion Machines

A perpetual motion machine of the first kind (PMM1) is a device that produces more energy than it consumes — effectively creating energy from nothing. The First Law makes this impossible by definition: ΔU = Q − W means that the work output W can never exceed the heat input Q without depleting internal energy. A machine cannot have W > Q while ΔU = 0 — that would require Q > W and W > Q simultaneously, which is a contradiction.

Throughout history, inventors have proposed thousands of perpetual motion designs. Some were ingenious; all failed. The First Law isn't a guideline or an approximation — it's a fundamental consequence of the nature of energy itself. Any device that appears to violate it contains a hidden energy source, a measurement error, or outright fraud.

The PMM Test

Whenever you see a claim that a device "outputs more energy than it takes in" — whether in engineering, finance, or social media — the First Law provides an immediate, unambiguous answer: impossible. No exceptions. No new discoveries will change this, because conservation of energy follows from the structure of time itself (Noether's theorem).

Frequently Asked Questions

Internal energy (U) is a state property — it describes how much energy is stored inside the system at any given moment. Heat (Q) is an energy transfer process — energy moving from a hotter region to a cooler one. You can say a gas has an internal energy of 5,000 J. You cannot say it "contains heat" — heat is the process of transferring that energy, not the energy itself.
When a gas expands against external pressure, it does work (W > 0). In an adiabatic process, Q = 0, so the First Law gives ΔU = −W < 0. The internal energy decreases — meaning the average kinetic energy of the molecules drops — and temperature falls. The expansion literally extracts energy from the molecules' thermal motion to push the surroundings. This is why pressurised aerosol cans get cold when you spray them.
A state function depends only on the current state of the system, not on the history of how it got there. Like altitude — it doesn't matter whether you hiked straight up or took a zigzag path; your height above sea level is the same. For internal energy: if a gas starts at T=300K and ends at T=400K, ΔU is the same regardless of whether you heated it at constant volume, constant pressure, or via some complicated multi-step process. This path-independence is what makes energy accounting tractable.
At constant pressure, when you add heat, the gas does two things: its internal energy rises (ΔU), and it expands, doing work on the surroundings (PΔV). At constant volume, none of the heat goes to expansion — all of it raises internal energy. So to achieve the same temperature increase ΔT, you need more heat at constant pressure than at constant volume. Hence C_P > C_V, with C_P − C_V = R for ideal gases.
The First Law tells you how much energy is conserved. The Second Law tells you which direction processes go — it introduces entropy and explains why not all of the heat input can be converted to work. Together they're the complete foundation of thermodynamics: the First Law says you can't win (create energy), and the Second Law says you can't even break even (recover all energy as useful work). See our full Thermodynamics guide for the Second Law.

The Bigger Picture: All Four Laws

The First Law is just one of four laws governing thermodynamics. The full topic guide covers entropy, Carnot efficiency, absolute zero, and kinetic theory.

Full Thermodynamics Guide → What Is Energy?

Sources & Further Reading

  1. Atkins, P., & de Paula, J. (2014). Physical Chemistry (10th ed.). Oxford University Press. Chapter 2: The First Law.
  2. Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). Wiley. Chapter 1: The Problem and the Postulates.
  3. Fermi, E. (1956). Thermodynamics. Dover Publications. Chapter 1: Fundamental Concepts.
  4. Zemansky, M. W., & Dittman, R. H. (1997). Heat and Thermodynamics (7th ed.). McGraw-Hill. Chapters 5–6.
  5. Halliday, D., Resnick, R., & Krane, K. S. (2002). Physics (5th ed., Vol. 2). Wiley. Chapter 21: Thermodynamics.