# What is Internal Energy Formula – Definition

Internal Energy Formula. The internal energy is the total of all the energy associated with the motion of the atoms or molecules in the system. Thermal Engineering

## Microscopic Energy

Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, Upot, and microscopic kinetic energy, Ukin, components:

U = Upot + Ukin

where the microscopic kinetic energy, Ukin, involves the motions of all the system’s particles with respect to the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of energy will transfer to the molecule of lower kinetic energy. However, for polyatomic gases there is rotational and vibrational kinetic energy as well.

The microscopic potential energy, Upot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system (e.g. electric or magnetic fields).

In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.

## Specific Internal Energy

The specific internal energy (u) of a substance is its internal energy per unit mass. It is an intensive property. It equals to the total internal energy (U) divided by the total mass (m).

u = U/m

where:

u = specific internal energy (J/kg)

U = internal energy (J)

m = mass (kg)

## Internal Energy of an Ideal Gas

What is Ideal Gas
An ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces. An ideal gas can be visualized as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other. In reality, no real gases are like an ideal gas and therefore no real gases follow the ideal gas law or equation completely.

At temperatures near a gases boiling point, increases in pressure will cause condensation to take place and drastic decreases in volume. At very high pressures, the intermolecular forces of a gas are significant.

However, most gases are in approximate agreement at pressures and temperatures above their boiling point. The ideal gas law is utilized by engineers working with gases because it is simple to use and approximates real gas behavior.

In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume.

pV = nRT

where:

p is the absolute pressure of the gas

n is the amount of substance

T is the absolute temperature

V is the volume

R  is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant,

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R=8.31 J/mol K. We can rewrite the previous equation in an alternative form, in terms of a constant called the Boltzmann constant k, which is defined as:

k = R / NA = [8.31 J/mol K] / [6.02 x 1023 mol-1] = 1.38 x 10-23 J/K

The internal energy is the total of all the energy associated with the motion of the atoms or molecules in the system. Microscopic forms of energy include those due to the rotation, vibration, translation, and interactions among the molecules of a substance.

## Monatomic Gas

For a monatomic ideal gas (such as helium, neon, or argon), the only contribution to the energy comes from translational kinetic energy. The average translational kinetic energy of a single atom depends only on the gas temperature and is given by equation

Kavg = 3/2 kT.

The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:

Eint = 3/2 NkT = 3/2 nRT

where n is the number of moles. Each direction (x, y, and z) contributes (1/2)nRT to the internal energy. This is where the equipartition of energy idea comes in – any other contribution to the energy must also contribute (1/2)nRT. As can be seen, the internal energy of an ideal gas depends only on temperature and the number of moles of gas.

## Diatomic Molecule

If the gas molecules contain more than one atom, there are three translation directions, and rotational kinetic energy also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

Diatomic ideal gas:

Eint = (5/2)NkT = (5/2)nRT

This is only an approximation and applies at intermediate temperatures. At low temperatures only the translational kinetic energy contributes, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration.

The internal energy will be greater at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.

The internal energy of real gases also depends mainly on temperature, but similarly as the Ideal Gas Law, the internal energy of real gases depends also somewhat on pressure and volume. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes electrical potential energy associated with the forces (or “chemical” bonds) between atoms and molecules.

## Internal Energy and the First Law of Thermodynamics

In thermodynamics the concept of energy is broadened to account for other observed changes, and the principle of conservation of energy is extended to include a wide variety of ways in which systems interact with their surroundings. The only ways the energy of a closed system can be changed are through transfer of energy by work or by heat. Further, based on the experiments of Joule and others, a fundamental aspect of the energy concept is that energy is conserved. This principle is known as the first law of thermodynamics. The first law of thermodynamics can be written in various forms:

In words:

Equation form:

∆Eint = Q – W

where Eint represents the internal energy of the material, which depends only on the material’s state (temperature, pressure, and volume). Q is the net heat added to the system and W is the net work done by the system. We must be careful and consistent in following the sign conventions for Q and W. Because W in the equation is the work done by the system, then if work is done on the system, W will be negative and Eint will increase.

Similarly, Q is positive for heat added to the system, so if heat leaves the system, Q is negative. This tells us the following: The internal energy of a system tends to increase if heat is absorbed by the system or if positive work is done on the system. Conversely, the internal energy tends to decrease if heat is lost by the system or if negative work is done on the system. It must be added Q and W are path dependent, while Eint is path independent.

Differential form:

dEint = dQ  – dW

The internal energy Eint of a system tends to increase if energy is added as heat Q and tends to decrease if energy is lost as work W done by the system.

References:
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