# Internal Energy of an Ideal gas: Several Facts and Detailed Explanations:

The “Internal energy of an ideal gas” is not depending upon the path of a system which is closed but the internal energy of an ideal gas depends on the initial state and final state of the system.

From the law of thermodynamics we get a crystal clear concept about the internal energy of an ideal gas. The internal energy of an ideal gas can be explain as, the total amount of energy is amalgamated with the motion that could be vibration motion, rotation motion or translation motion of the molecules or atoms of a matter in the system.

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## What is internal energy of an ideal gas?

For an ideal gas the amount of internal energy for a system only depend upon temperature. But for the real gas the amount of internal energy for a system depend upon temperature, volume, pressure.

The internal energy of an ideal gas is a property of extensive and the amount of energy of a gaseous matter cannot determine directly. The internal energy of an ideal gas is in a system the molecules of a gaseous matter, the amount of internal energy transferring in the form of thermodynamic work and heat. Image – The carnot heat engine of internal energy of an ideal gas; Image Credit – Wikipedia

For an ideal gas the total amount of internal energy is directly proportional to the temperature and also the total number of the molecules of mole of a substance which is present in the gaseous state.

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So mathematically the internal energy of an ideal gas can be express as,

dU = nCvdT…… eqn (1)

Or, U = CvnT…………. eqn (2)

From the equation (1) term nCvT  is used from the kinetic energy of an ideal gas.

Where,

U = The amount of internal energy of a gas

Cv = At constant volume the amount of heat capacity of a gaseous substance

n = The total number of moles of a gaseous substance

T = Temperature of the system

## Internal energy of an ideal gas formula:

In the thermodynamics the change of total amount of internal energy which is expressed as ΔU can determine but for an ideal gas the amount of absolute internal energy can estimate.

Internal energy of an ideal gas formula is,

Where,

U = Internal energy of an ideal gas

cv = Heat capacity of the specific isochoric

m = Mass of an ideal gas

T = Temperature

To calculate the amount Internal energy of an ideal gas at first we need imagine a gas substance is blockaded to a cylinder that time the volume of the ideal gas should be in constant state and the ideal gas should to cool down and reaches at absolute zero temperature.

In this particular state all particles of the ideal gas at rest position and there is no internal energy is present. The total amount of heat is expressed as Q is transferred at the constant state of volume until the ideal gas temperature is reaches to T. Now in this state the total amount of heat which is necessary for the internal energy is reaches at U.

## Internal energy of an ideal gas derivation:

In a system of thermodynamic the amount of internal energy can be converted into potential energy or kinetic energy. For the system of the thermodynamics three types of energy such as internal energy, potential energy and kinetic energy can contained.

### Derivation internal energy for an ideal gas:-

For an ideal gas substance the internal energy depend upon the kinetic energy and potential energy.

We know that,

M/m = Na

Kavg = 1/2 3RT/Na

Kavg = 3/2 kT because  k = R/Na

## How does the internal energy of an ideal gas differ from that of real gas?

The ideal gas explain as, the gaseous substance which are obeys the law of gases at any condition of temperature and pressure. The real gas explain as, the gaseous substance which are not obeys the law of gases.

The difference between the internal energy of an ideal gas and real gas is discuss below,

## Specific internal energy of an ideal gas:

The specific internal energy of an ideal gas which is expressed as u explain as, the amount of internal energy of an ideal gas matter in per unit mass of the particular ideal gas matter.

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The formula of the specific internal energy of an ideal gas is,

u = U/m

Where,

u = Specific internal energy of an ideal gas in joule per kilogram

U = Internal energy of an ideal gas in joule

m = Mass of an ideal gas in kilogram

The S.I. unit of the specific internal energy of an ideal gas is joule per kilogram. The dimension of specific internal energy of an ideal gas is L2T-2.

## Change in internal energy of an ideal gas:

From the laws of kinetic energy it’s clearly shown that kinetic energy of a particle has a directly relation with temperature from that change in internal energy of an ideal gas directly connected.

Change in internal energy of an ideal gas only depends on the temperature it is not depend on the other physical parameters like volume, pressure. If initial temperature, final temperature is known for the system then change in internal energy of an ideal gas easy to determined.

Whether the system can follow any process like isentropic, isobaric or isochoric or any other method the change in internal energy of an ideal gas is irrelevant. In one word we can say change in internal energy of an ideal gas only ruled by the state of the gaseous matter not ruled by the process of the gaseous matter. If the temperature is differ in the system only for that case internal energy can be differ for an ideal gaseous substance. The change in internal energy of an ideal gas can be zero in the process of isothermal. Image – Isothermal Process; Image Credit – Wikimedia Commons

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By the process of the thermodynamic the clear relation between change in internal energy of an ideal gas and temperature easily can investigate of gaseous matter.  In the process of isochoric on the gas no workdone is happened. In the process of isochoric on the gas heat is input for this reason change in internal energy of an ideal gas is increases.

## What is the change of internal energy?

In a system of thermodynamic the change in internal energy is derive in this way the sum of the internal energy changes for the gaseous matter is equal to  the net workdone of a thermodynamic system and the total amount of heat is deposal to the system and the surrounding of the system. Image – Internal energy and Temperature; Image Credit – Wikimedia Commons

The formula for the change in internal energy of an ideal gas is,

Δ U = Q + W

Where,

ΔU = The total amount of change in internal energy of an ideal gas in a system

Q = The amount of heat transfer between the system and the system’s surroundings

W = Work done by a system

In some process there is no change in internal energy. The processes are cyclic process, isothermal and free expansion. In these processes the amount of internal energy is same because the temperature of the system remains unchanged. Image – Free expansion in gas; Image Credit – Wikipedia

## How to calculate change in internal energy of an ideal gas?

From the 1st law of the thermodynamic we can a concept about change in internal energy of an ideal gas.The amount of internal energy of an ideal gas is equal to the heat flow and PV workdone by the system.

The quantity of the internal energy that could be change for a gaseous matter that always should be equal to the workdone of the system and the amount of input heat and amount of output heat.

Formula for calculate change in internal energy of an ideal gas:-

Q =ΔU = W…….eqn (1)

Q = ΔU + PV

Because we know that, the amount of heat is added or removed is always equal to the total sum of the internal energy which is changed and the workdone of PV.

From the eqn (1) after arranging we get,

ΔU = Q – PV……. eqn (2)

## Question: – Is all-time the values of the internal energy of a substance remain positive or it can be negative?

Solution: – No, all-time the values of the internal energy of a substance cannot remain positive.

Some time the value of the internal energy can be negative. We can calculate the value of internal energy from the sum of workdone and heat. Negative value of internal energy of an ideal gas means the value of final energy is low than the value of initial energy.

## Question: – Give some examples of internal energy.

Solution: – Some examples of internal energy listed below,

1. Vapor of a liquid substance
2. Shaking of a liquid substance
3. Batteries
4. Compressed gasses
5. Increasing the temperature of a substance

Indrani Banerjee

Hi..I am Indrani Banerjee. I completed my bachelor's degree in mechanical engineering. I am an enthusiastic person and I am a person who is positive about every aspect of life. I like to read Books and listen to music. Let's connect through LinkedIn-https://www.linkedin.com/in/indrani-banerjee-2487b4214