To understand and perform any sort of thermodynamic calculation, we have to be very clear about the fundamental laws and concepts of thermodynamics. For example, work and heat are dependent terms with interrelated concepts. Heat is the transfer of thermal energy between two objects which are at different temperatures, and also not equal to thermal energy. Work is the force used to transfer energy between the system and its surroundings. It is needed to create heat and transfer internal energy. In this article, we will see the concept of the internal energy formula with examples. Let us learn things!

The relationship between heat and work, the two concepts can be analyzed through the topic of Thermodynamics, which is the scientific study of the interaction of heat and other types of energy.

**Table of content**

## Internal Energy

In chemistry and physics, the internal energy is the microscopic energy contained in a substance is given by the random, disorganized kinetic energy of the molecules. It also comprises the potential energy among these molecules as well as the nuclear energy inherent in their atoms.

In other words, **internal energy** is the entire energy of a closed system of molecules or the sum of a substance’s molecular kinetic and potential energy. Internal energy is represented by the symbol U, and the unit of measurement is the joules (J).

According to the First Law of Thermodynamics, the internal energy of a system can be modified by doing work on it, adding/removing heat from it, or a mixture of the two. When a system is isolated, it is not allowed to interact with its surroundings, which means that the internal energy cannot change.

Internal energy is also known as thermal energy because the temperature is essentially a measure of a system’s internal energy – it’s defined as the average kinetic energy of the system’s molecules.

**Internal Energy Formula**

**Concept of Internal Energy**

To understand the relationship between work and heat, we need to understand the factor of linking factors. This is the change in internal energy. We cannot create nor destroy energy but we can convert or transfer it. Internal energy refers to the energy within a given system, which includes the kinetic energy of molecules and the energy stored in all of the chemical bonds between the molecules.

With the interactions of heat, work and internal energy, there is a transfer of energy and conversions every time. But, no net energy is created or lost during these transfers. This is the main theme of the first law of thermodynamics.

According to the First Law of Thermodynamics, energy can be converted from one form to another with the interaction of heat, work, and internal energy. But no one can create nor destroy it, under any circumstances.

**Get the huge list of Physics Formulas here**

**Internal Energy Formal Definition**

The energy accumulated within the system is associated with random motions of the particles along with the potential energies of the molecules due to their orientation.

The energy due to random motion includes many forms as translational, rotational, and vibrational energy. We represent it as U. Therefore, we can say that internal energy is a state function and in all the processes in internal energy from one state to another state will be the same.

**The Formula of Internal Energy**

Mathematically, we can represent it,

\( \Delta U=q+w \)

Where,

\( \Delta U \) | total change in internal energy of a system, |

q | heat exchanged between a system and its surroundings |

w | work done by or on the system |

## Internal Energy Explanation

Internal energy, U, of a system or body with well-defined boundaries is the sum of kinetic energy owing to molecular motion and potential energy due to vibrational motion and electric energy of atoms within molecules. Internal energy contains the energy contained in all chemical bonds. Internal energy increases when the temperature rises and states or phases transition from solid to liquid and liquid to gas.

Internal energy is an extensive quantity that is a state function of a system. The joule is the SI unit of energy (J). The specific internal energy is the internal energy related to mass with the unit J/kg.

## Internal Energy Change Equations

One of the most important equations when dealing with internal energy is the first law of thermodynamics, which states that the change in internal energy of a system equals the heat added to the system minus the work done by the system (or, plus the work done on the system).

The first law of thermodynamics -> **ΔU = q+w**

where, q is heat and w is work

Internal energy indicators –

- The energy entering the system is POSITIVE (+), indicating that heat is absorbed, and q>0. Thus, work is done on the system, w>0.
- The energy exiting the system is NEGATIVE (-), indicating that the system emits heat, q<0, and performs work, w<0.

## Internal Energy of an Ideal Gas

An ideal gas’s internal energy is a good representation of a real-world system. In such a system, the particles in an ideal gas are seen as point objects that collide in entirely elastic ways. This model accurately describes the behaviour of monatomic gases (such as helium and argon).

Internal energy in an ideal gas is proportional to the number of particles per mole and the temperature:

**U = ncT**

In this equation,

U = internal energy

c = heat capacity at constant volume

n = number of moles

T = temperature

## Internal Energy Change

Every substance has a definite amount of energy that is determined by its chemical constitution and state of existence. This is referred to as intrinsic energy. Every substance has a specific amount of internal energy that is equal to the sum of the energies of its constituents, which are atoms, ions, or molecules.

The distinction between the internal energies of the two states can be used to calculate the change in the internal energy of a reaction.

Let E_{A} and E_{B} represent the starting energy in states A and B. The difference in starting energy in the two states will then be

**ΔU = E _{B} – E_{A}**

The internal energy differential has a set value and is independent of the path followed between two states A and B. The change in the internal energy of a chemical reaction can be defined as the difference between the internal energies of the products and the reactants.

**ΔU = E _{products} – E_{reactants}**

**Solved Examples**

Q.1: A system has constant volume and the heat around the system increases by 45 J. Then,

(i) What will be the sign for heat (q) for the system?

(ii) What will be \( \delta U\) equal to?

(iii) Find out the value of the internal energy of the system in Joules?

Solution:

Since the system has constant volume, so \( \delta V=0. \)

i.e. -P \( \delta V=0. \)

So, Work is equal to zero.

Thus, in the equation

\( \delta U=q+w \)

Put w=0, then

\( \delta U=q. \)

Therefore, the internal energy is equal to the heat of the system. The surrounding heat increases, and then the heat of the system decreases because heat is not created nor destroyed. Therefore, heat is taken away from the system making it exothermic and hence negative. So, the value of Internal Energy will be the negative value of the

(i) Negative (q<0)

(ii) \( \delta U=q + w \)

= q+ 0

= q

(iii) \( \delta U = -45 J \)