## Internal Energy – Thermal Energy

**Potential energy** and** kinetic energy**, that was discussed in previous chapters, are **macroscopic forms of energy**. They are dependent on macroscopic variables such as the position and the velocity of objects.

In thermodynamics,** internal energy** (also called the **thermal energy**) is defined as the energy associated with **microscopic forms of energy**. It is an extensive quantity, it depends on the size of the system, or on the amount of substance it contains. The SI unit of** internal energy** is the** joule (J)**. It is the energy contained within the system, excluding the kinetic energy of motion of the system as a whole and the potential energy of the system. **Microscopic forms of energy** include those due to the **rotation**, **vibration, translation**, and **interactions** among the molecules of a substance. None of these forms of energy can be measured or evaluated directly, but techniques have been developed to evaluate the change in the total sum of all these microscopic forms of energy.

In addition, energy is can be stored in the chemical bonds between the atoms that make up the molecules. This energy storage on the atomic level includes energy associated with electron orbital states, nuclear spin, and binding forces in the nucleus.

**Internal energy** is represented by the symbol **U**, and the change in internal energy in a process is U_{2} – U_{1}.

## Microscopic Energy

**Internal energy** involves energy on the **microscopic scale**. It may be divided into microscopic potential energy, *U*_{pot}, and microscopic kinetic energy, *U*_{kin}, components:

**U = U _{pot} + U_{kin}**

where the microscopic kinetic energy, U_{kin}, 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, **U _{pot}**, 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

**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 / N_{A} = [8.31 J/mol K] / [6.02 x 10^{23} 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

**K _{avg} = 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:

**E _{int} = 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:

**E _{int} = (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.

## Joule’s Second Law

For any gas whose equation of state is given exactly by **pV = nRT** (or pv = RT), the **specific internal energy** depends on temperature only. This rule was originally found in 1843 by Joule experimentally for real gases and is known as **Joule’s second law**:

*The internal energy of a fixed mass of an ideal gas depends only on its temperature (not pressure or volume).*

The specific enthalpy of a gas described by pV = nRT also depends on temperature only. Note that the enthalpy is the thermodynamic quantity equivalent to the total heat content of a system. It is equal to the internal energy of the system plus the product of pressure and volume. In intensive variables the **Joule’s second law **is therefore given by **h = h(T) = u(T) + pv = u(T) + RT**.

These three equations constitute the ideal gas model, summarized as follows:

**pv = RT**

**u = u(T)**

**h = h(T) = u(T) + RT**

## 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:**

**∆E**_{int}** = Q – W**

where **E _{int }**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 E

_{int}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 E_{int} is path independent.

**Differential form:**

**dE _{int} = dQ – dW**

The internal energy E_{int} 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.

## Distinguishing Temperature, Heat, and Internal Energy

Using the kinetic theory, a clear distinction between these three properties can be made.

**Temperature**is related to the**kinetic energies**of the molecules of a material. It is the average kinetic energy of individual molecules.**Internal energy**refers to the total energy of all the molecules within the object. It is an**extensive property**, therefore when two equal-mass hot ingots of steel may have the same temperature, but two of them have twice as much internal energy as one does.- Finally,
**heat**is the amount of energy flowing from one body to another spontaneously due to their temperature difference.

It must be added, when a **temperature difference** does exist heat flows spontaneously **from the warmer system to the colder system**. Thus, if a 5 kg cube of steel at 100°C is placed in contact with a 500 kg cube of steel at 20°C, heat flows from the cube at 300°C to the cube at 20°C even though the internal energy of the 20°C cube is much greater because there is so much more of it.

A particularly important concept is **thermodynamic equilibrium**. In general, when two objects are brought into** thermal contact**, **heat will flow** between them **until** they come into **equilibrium** with each other.

**Nuclear and Reactor Physics:**

- J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
- J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
- W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
- Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
- W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
- Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN 978-0471805533
- G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
- Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
- U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

**Advanced Reactor Physics:**

- K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
- K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
- D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
- E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.