# How to calculate internal energy

### Definition

Internal energy is the energy possessed by a system. Internal energy can be defined in two ways:

• microscopic (atomic and molecular view)
• macroscopic

From the microscopic point of view, which examines the system on the atomic and molecular scale, the internal energy U [J] of a system is the sum of the kinetic and potential energies of its atoms and molecules. Since the sum of kinetic energy and potential energy is the mechanical energy, the internal energy of a system is the sum of atomic and molecular mechanical energy.

In other words, internal energy is all the energy of a system that is associated with its microscopic components (consisting of atoms and molecules), when viewed from a reference frame at rest with respect to the object. Internal energy includes kinetic energy of translation, rotation, and vibration of molecules, potential energy within molecules, and potential energy between molecules.

In physics, a more common way to view the internal energy of a system is in terms of its macroscopic characteristics, which are very similar to atomic and molecular average values.

From the macroscopic point of view, the change in internal energy ΔU [J] is defined as the difference between the energy received Q [J] (as heat) and energy lost W [J] (as work).

Image: Internal energy

In thermodynamics, we are not interested in the absolute value of the internal energy of a body (system) but on the variation (change) of its internal energy. This is because the internal energy variation of a system represents the energy exchange of the system with its surroundings (environment).

ΔU = Q – W
(1)

where:

• ΔU [J] – change in internal energy
• Q [J] – net heat transfer
• W [J] – total work done on and by the system

Image: Internal energy – heat transfer

Q [J] is the sum of all heat transfers into and out of the system. If Q [J] is positive when heat is transferred into the system (heat received), and is negative when heat is taken out from the system (heat lost).

Q = Qin – Qout
(2)

W [J] is positive when more work is done by the system on the environment than on it.

W = Wout – Win
(3)

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### Formula

The formula (equation) to calculate internal energy variation is [1]:

ΔU = Q – W
(4)

Heat Q [J] and work W [J] are calculated with the equations (2) and (3) above.

The unit of measurement of internal energy variation is joule [J].

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### Example

At one point in time a system receives 50 J of heat and performs 30 J of work. Later, the system losses 25 J of heat and 17 J of work is performed on it. Calculate the net change of internal energy for the system.

Step 1. Calculate the net heat transfer of the system using equation (2):

Q = Qin – Qout = 50 – 25 = 25 J

Step 2. Calculate the net work done on the system using equation (3):

W = Wout – Qin = 30 – 17 = 13 J

Step 3. Calculate the change in internal energy using equation (1):

ΔU = Q – W = 25 – 13 = 12 J

The system increased its internal energy by 12 J due to increase of its temperature (more heat received).

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### Calculator

The internal energy calculator allows you to calculate the change in internal energy of a system with a given input and output heat and input and output work.

The default unit of measurement for energy is Joule. If you want the result displayed in another unit, use the drop down list to choose and click the CALCULATE button again.

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### References

[1] David Halliday, Robert Resnick, Jearl Walker, Fundamentals of Physics, 7th edition, John Wiley & Sons, 2004.
[2] Benjamin Crowell, Light and Matter – Physics, 2007.
[3] Raymond A. Serway and John W. Jr. Jewett, Physics for Scientists and Engineers, 6th edition, Brooks/Cole Publishing Co.,2004
[4] Jiansong Li, Jiyun Zhao, and Xiaochun Zhang, A Novel Energy Recovery System Integrating Flywheel and Flow Regeneration for a Hydraulic Excavator Boom System, Energies 2020.
[5] Leo H. Holthuijsen, Waves in oceanic and coastal waters, Cambridge University Press, 2007.
[6] Kira Grogg, Harvesting the Wind: The Physics of Wind Turbines, Carleton College, 2005.