# Internal energy

**Internal energy** (symbol U, also called thermodynamic energy) is the total energy contained in a thermodynamic system by virtue of the motion of and interactions among its molecules, excluding the energy of the system's motion or position as a whole. The [International Union of Pure and Applied Chemistry](https://www.edgepedia.org/international-union-of-pure-and-applied-chemistry) defines it as the quantity whose change equals the sum of the heat, \( q \), brought to the system and the work, \( w \), performed on it, \( \Delta U = q + w \).<sup>[1](https://goldbook.iupac.org/terms/view/I03103)</sup> It is a state function: its value depends only on the current state of the system, not on the path by which the system reached that state.<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)</sup>

| Key fact | Detail |
|---|---|
| IUPAC definition | Quantity whose change equals heat \( q \) added plus work \( w \) done on the system, \( \Delta U = q + w \); synonym: thermodynamic energy<sup>[1](https://goldbook.iupac.org/terms/view/I03103)</sup> |
| Physics sign convention | \( \Delta E_{\text{int}} = Q - W \), with \( W \) positive when work is done by the system<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)</sup> |
| Path independence | \( Q \) and \( W \) each depend on the thermodynamic path, but their difference does not; internal energy is a state function<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)</sup> |
| Ideal monatomic gas | \( E_{\text{int}} = \tfrac{3}{2} nRT \), a function of temperature alone, independent of pressure and volume<sup>[3](https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy)</sup> |
| Measurement | At constant volume no expansion work is done, so \( \Delta U = q \); a bomb calorimeter measures \( \Delta U \) directly<sup>[4](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/10%3A_Calorimetry/10.02%3A_Introduction_to_calorimetry)</sup> |
| Relation to enthalpy | \( H = U + PV \); at constant pressure \( \Delta H = q_p \), and for reactions involving gases \( \Delta H = \Delta U + \Delta n_{\text{gas}} RT \)<sup>[4](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/10%3A_Calorimetry/10.02%3A_Introduction_to_calorimetry)</sup> |
| Absolute value | Only changes in \( U \) can be measured; the zero of the scale is arbitrary<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_%28Lower%29/14%253A_Thermochemistry/14.02%253A_The_First_Law_of_Thermodynamics)</sup> |

## Definition and the first law

The first law of thermodynamics, first explicitly formulated by [Rudolf Clausius](https://www.edgepedia.org/rudolf-clausius) in 1850, is a statement of energy conservation for systems that exchange energy as heat and as work.<sup>[6](https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node203.html)</sup> A system can exchange energy with its surroundings in these two ways, and the net energy exchanged equals the change in the system's internal energy; an isolated system therefore has constant internal energy.<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)</sup>

Two sign conventions are in use, and they describe the same physics. Chemistry writes \( \Delta U = q + w \), counting work as positive when performed on the system.<sup>[1](https://goldbook.iupac.org/terms/view/I03103)</sup> Physics writes \( \Delta E_{\text{int}} = Q - W \), counting \( W \) as positive when done by the system, so that heat added raises the internal energy and work done by the system (as in an expansion) lowers it.<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)</sup>

**Path independence.** Heat and work are energies in transit, not properties stored in a system, and each depends on the particular process connecting two equilibrium states. Their difference does not: along any path between the same two states, \( Q_1 - W_1 = Q_2 - W_2 \).<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)</sup> This is what makes internal energy easier to work with than either heat or work, and it underlies [Hess's law](https://www.edgepedia.org/hesss-law) in thermochemistry.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_%28Lower%29/14%253A_Thermochemistry/14.02%253A_The_First_Law_of_Thermodynamics)</sup>

## What internal energy contains

Microscopically, internal energy is the sum of the mechanical energies, kinetic and potential, of all the molecules in the system.<sup>[3](https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy)</sup> The kinetic part includes translational, rotational, and vibrational motion; the potential part arises from interactions between molecules, including chemical bonds and other intermolecular forces.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_%28Lower%29/14%253A_Thermochemistry/14.02%253A_The_First_Law_of_Thermodynamics)</sup> Engineering treatments list the stored modes as translational, rotational, vibrational, electronic, chemical, and nuclear energy.<sup>[7](https://archive.nptel.ac.in/content/storage2/courses/112103016/module1/lec3/4.html)</sup> The system's location and bulk motion are irrelevant: moving a system to the roof of a 100-story building or placing it on a moving train does not change its internal energy.<sup>[3](https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy)</sup>

For an ideal monatomic gas the molecules are single atoms with no rotation, vibration, or interatomic interactions, so the internal energy is purely translational kinetic energy and equals \( \tfrac{3}{2} nRT \) for \( n \) moles at temperature \( T \). It depends on temperature alone and is completely independent of pressure and volume.<sup>[3](https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy)</sup> For other systems no such simple expression exists, although an increase in internal energy can often be associated with an increase in temperature.<sup>[3](https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy)</sup>

## Measurement

Because absolute internal energy cannot be measured, thermodynamics works with changes \( \Delta U \), with the zero of energy assigned to an arbitrary reference state.<sup>[5](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_%28Lower%29/14%253A_Thermochemistry/14.02%253A_The_First_Law_of_Thermodynamics)</sup> [Calorimetry](https://www.edgepedia.org/calorimetry) supplies the practical route. In a rigid, sealed vessel such as a [bomb calorimeter](https://www.edgepedia.org/bomb-calorimeter), the volume cannot change, no expansion work is done, and all energy transfer occurs as heat, so \( \Delta U = q \); the temperature change of the calorimeter, combined with its known heat capacity, gives \( \Delta U \).<sup>[4](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/10%3A_Calorimetry/10.02%3A_Introduction_to_calorimetry)</sup> Bomb calorimeters are used, for example, to determine the energy content of food.<sup>[8](https://openstax.org/books/chemistry/pages/5-summary)</sup>

At constant pressure, the more common condition for laboratory chemistry, heat no longer equals \( \Delta U \) because some energy goes into expansion work. Defining the enthalpy as \( H = U + PV \) makes \( \Delta H = q_p \) at constant pressure, so \( \Delta H \) can be found by measuring heat alone. For reactions involving gases the two quantities are related by \( \Delta H = \Delta U + \Delta n_{\text{gas}} RT \), where \( \Delta n_{\text{gas}} \) is the change in moles of gas between reactants and products.<sup>[4](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/10%3A_Calorimetry/10.02%3A_Introduction_to_calorimetry)</sup>

## Role in property formulations

Internal energy is one of the four fundamental thermodynamic potentials. With entropy \( S \) and volume \( V \) as its natural variables, \( dU = TdS - PdV \) when only pressure-volume work is performed, and the enthalpy, Helmholtz energy, and Gibbs energy follow from it by Legendre transforms.<sup>[9](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Chemical_Thermodynamics_%28Supplement_to_Shepherd_et_al.%29/15%3A_Extension_11_-_Legendre_Transforms/15.01%3A_Differential_Forms_of_Fundamental_Equations)</sup> Reference-property tables compiled by the [National Institute of Standards and Technology](https://www.edgepedia.org/national-institute-of-standards-and-technology) list molar internal energy \( U \) in kJ/mol alongside enthalpy, entropy, and the other state functions for pure fluids.<sup>[10](https://trc.nist.gov/properties.htm)</sup>

## References

1. [Internal energy (I03103), IUPAC Compendium of Chemical Terminology (Gold Book), 5th ed.](https://goldbook.iupac.org/terms/view/I03103)
2. [3.3 First Law of Thermodynamics, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/3-3-first-law-of-thermodynamics)
3. [3.2 Work, Heat, and Internal Energy, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/3-2-work-heat-and-internal-energy)
4. [10.2 Introduction to calorimetry, Chemistry LibreTexts (Duke University)](https://chem.libretexts.org/Courses/Duke_University/Duke_Physical_Chemistry_Laboratory/03%3A_Physical_Chemistry_Lab_Manual/10%3A_Calorimetry/10.02%3A_Introduction_to_calorimetry)
5. [14.2 The First Law of Thermodynamics, Chemistry LibreTexts (Chem1)](https://chem.libretexts.org/Bookshelves/General_Chemistry/Chem1_%28Lower%29/14%253A_Thermochemistry/14.02%253A_The_First_Law_of_Thermodynamics)
6. [First Law of Thermodynamics, Farside Physics, University of Texas at Austin](https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node203.html)
7. [Module 1, Lecture 3: Macroscopic and microscopic energy, NPTEL](https://archive.nptel.ac.in/content/storage2/courses/112103016/module1/lec3/4.html)
8. [Ch. 5 Summary, Chemistry 2e, OpenStax](https://openstax.org/books/chemistry/pages/5-summary)
9. [15.1 Differential Forms of Fundamental Equations, Chemical Thermodynamics, Chemistry LibreTexts](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Chemical_Thermodynamics_%28Supplement_to_Shepherd_et_al.%29/15%3A_Extension_11_-_Legendre_Transforms/15.01%3A_Differential_Forms_of_Fundamental_Equations)
10. [NIST Thermophysical Properties code designations and units](https://trc.nist.gov/properties.htm)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Thermodynamics › Laws, states, and potentials › Thermodynamic potentials and free energy › Internal energy*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license