Internal Energy (U) & Work

The First Law of Thermodynamics is one of the most fundamental principles in chemistry and physics: energy cannot be created or destroyed, only transferred or converted from one form to another. For a chemical system, this means that any change in the system's internal energy must be exactly accounted for by the heat exchanged with the surroundings and the work done on or by the system. This bookkeeping principle allows chemists to precisely track energy flow during reactions, phase changes, and physical processes. Internal energy (U) is the total energy stored within a chemical system — the sum of all kinetic energies (translational, rotational, vibrational motion of molecules) and potential energies (intermolecular forces, chemical bonds). While we cannot measure the absolute value of U for any real system, we can measure changes in internal energy (ΔU) by carefully monitoring heat and work. The equation ΔU = q + w captures this relationship precisely, where q represents heat and w represents work exchanged with the surroundings. Understanding sign conventions is critical when applying the First Law. Chemists use a system-centric perspective: energy flowing INTO the system is positive, and energy flowing OUT of the system is negative. This means heat absorbed by the system (endothermic) is positive q, while heat released (exothermic) is negative q. Similarly, work done ON the system (compression) is positive w, while work done BY the system (expansion) is negative w. These conventions reflect the International Union of Pure and Applied Chemistry (IUPAC) standard used in modern chemistry.

Key takeaways
  • What Is Internal Energy (U)?
  • Heat (q) and Its Sign Convention
  • Work (w) and Pressure–Volume Work
  • Applying the First Law: ΔU = q + w
  • State Functions vs. Path Functions

What Is Internal Energy (U)?

Internal energy U encompasses every form of energy stored within the molecular assembly of a system. This includes translational kinetic energy of molecules moving through space, rotational kinetic energy of molecules spinning, vibrational kinetic energy of bonds stretching and bending, electronic energy (arrangement of electrons in orbitals), and nuclear energy. In practice, ΔU is what matters: when a reaction occurs, the internal energy of the products minus that of the reactants gives ΔU. A negative ΔU means the system has lost energy to the surroundings; a positive ΔU means it has gained energy.

Heat (q) and Its Sign Convention

Heat (q) is the transfer of thermal energy between a system and its surroundings due to a temperature difference. When a system absorbs heat — as in endothermic reactions like dissolving ammonium nitrate (\ceNH4NO3(s)−>NH4+(aq)+NO3−(aq)\ce{NH4NO3(s) -> NH4^+(aq) + NO3^-(aq)}, ΔH = +25.7 kJ/mol) — q is positive because energy enters the system. When a system releases heat — as in exothermic combustion of methane (\ceCH4(g)+2O2(g)−>CO2(g)+2H2O(l)\ce{CH4(g) + 2O2(g) -> CO2(g) + 2H2O(l)}, ΔH = −890 kJ/mol) — q is negative. Measuring q in a calorimeter allows calculation of ΔU or ΔH.

Work (w) and Pressure–Volume Work

In chemistry, the most important type of work is pressure–volume (PV) work, also called expansion work or boundary work. It occurs when a gas expands or is compressed against an external pressure. The mathematical expression is w = −P_ext × ΔV, where P_ext is the external pressure and ΔV = V_final − V_initial. When a gas expands (ΔV > 0), the system does work on the surroundings, so w is negative (energy leaves the system). When a gas is compressed (ΔV < 0), the surroundings do work on the system, so w is positive (energy enters the system). For reactions at constant volume (ΔV = 0), w = 0 and ΔU = q_v.

Applying the First Law: ΔU = q + w

The First Law combines heat and work into a single energy balance. Consider a gas in a cylinder with a movable piston. If the gas absorbs 500 J of heat from a flame (q = +500 J) and simultaneously expands against a constant external pressure, doing 200 J of work on the surroundings (w = −200 J), then ΔU = 500 + (−200) = +300 J. The system's internal energy increased by 300 J. Note: some older textbooks write ΔU = q − w, where w is defined as work done BY the system. The IUPAC convention ΔU = q + w defines w as work done ON the system — always check the sign convention being used.

State Functions vs. Path Functions

A critical distinction: internal energy U is a state function — its value depends only on the current state of the system (temperature, pressure, composition), not on how the system arrived at that state. Therefore ΔU depends only on initial and final states, not the path taken. In contrast, heat (q) and work (w) are path functions — their individual values depend on the specific process. For example, a gas expanding isothermally (constant temperature) versus adiabatically (no heat exchange) will have different q and w values, but ΔU may differ because the final states differ. This is why thermodynamic cycles, like the Carnot cycle, can be analyzed systematically: after a complete cycle, ΔU = 0 because the system returns to its original state.

The Key Equation

ΔU=q+w=q+(−PextΔV)\Delta U = q + w = q + (-P_{\text{ext}} \Delta V)
  • \Delta U — Change in Internal Energy (joules (J) or kilojoules (kJ)): The difference between the internal energy of the final state and the initial state (U_final − U_initial). Negative means the system lost energy; positive means the system gained energy.
  • q — Heat (joules (J)): Thermal energy transferred between system and surroundings. Positive when heat flows into the system (endothermic); negative when heat flows out (exothermic).
  • w — Work (joules (J)): Mechanical energy transferred between system and surroundings. Using IUPAC convention: positive when work is done ON the system (compression); negative when the system does work on surroundings (expansion).
  • P_{\text{ext}} — External Pressure (pascals (Pa) or atmospheres (atm)): The constant pressure exerted by the surroundings on the system. For an irreversible process against constant external pressure, this is used directly in w = −P_ext ΔV.
  • \Delta V — Change in Volume (cubic meters (m³) or liters (L)): V_final − V_initial. Positive for expansion (gas does work on surroundings), negative for compression (surroundings do work on gas).

Worked Example

Problem. A sample of gas in a piston absorbs 850 J of heat from a hot plate. The gas then expands against a constant external pressure of 1.50 atm, and its volume increases from 2.00 L to 5.00 L. Calculate (a) the work done, w, and (b) the change in internal energy, ΔU. (Use the conversion: 1 L·atm = 101.325 J)

w=−PextΔV=−(1.50 atm)(3.00 L)=−456 J;ΔU=q+w=850+(−456)=+394 Jw = -P_{\text{ext}}\Delta V = -(1.50\text{ atm})(3.00\text{ L}) = -456\text{ J}; \quad \Delta U = q + w = 850 + (-456) = +394\text{ J}

Answer. Work done on the system w = −456 J (system did expansion work); Change in internal energy ΔU = +394 J (internal energy increased).

Figure
Diagram
First Law: Sign Conventions for Heat and WorkEXPANSION (w < 0)PistonGas (system)expandΔV > 0w = −P·ΔV < 0System loses energyCOMPRESSION (w > 0)PistonGas (system)compressΔV < 0w = −P·ΔV > 0System gains energyΔU = q + wq > 0: heat flows IN (endothermic) | q < 0: heat flows OUT (exothermic)w > 0: compression (work IN) | w < 0: expansion (work OUT)IUPAC convention: positive = energy enters the systemq absorbed
Origins · 1 minRudolf Clausius and how this was discovered
RC
Rudolf Clausius
1822–1888 · German

Rudolf Clausius was one of the primary founders of thermodynamics. In 1850, he published a landmark paper that mathematically formulated the First and Second Laws of Thermodynamics, introducing the concept that heat and work are both forms of energy transfer. He was the first to clearly state that the internal energy of an isolated system is constant — the essence of the First Law — and later introduced the concept of entropy (S) for the Second Law. His work unified the mechanical theory of heat with experimental observations of steam engines, revolutionizing both chemistry and physics. Clausius coined the word 'entropy' in 1865, deliberately choosing a Greek-rooted term that would parallel the word 'energy,' so that the two most fundamental thermodynamic quantities would sound similar in all languages.

  • Enthalpy (H) and Constant-Pressure Processes — Enthalpy H = U + PV is derived directly from internal energy. At constant pressure, ΔH = q_p, making it more practical than ΔU for most laboratory reactions open to the atmosphere.
  • Calorimetry: Measuring Heat Transfer — Calorimeters experimentally measure q, which is required to calculate ΔU using the First Law. Bomb calorimeters measure ΔU directly at constant volume (w = 0).
  • Hess's Law and State Functions — Because internal energy U is a state function, ΔU for any reaction can be calculated by combining thermochemical equations — the foundation of Hess's Law.
  • Second Law of Thermodynamics and Entropy — The Second Law builds on the First Law by explaining the direction of energy flow, introducing entropy (S) to determine which processes are spontaneous.
  • Bond Enthalpy and Bond Energy — Bond energies contribute directly to the internal energy of molecules. Breaking bonds requires energy input (positive ΔU), while forming bonds releases energy, linking molecular structure to thermodynamic quantities.