# Internal energy. Internal energy of an ideal gas

The internal energy U of a thermodynamic system is the energy it contains. It can be due to the motion of its particles (in the form of kinetic energy) and/or to their interactions. It generally depends on the state variables of the thermodynamic system (if it is a gas, p, V, T).

As it is a property of thermodynamic systems, the internal energy is a state function: its variation depends only on the initial and final states of the thermodynamic system and not on the nature of the process it undergoes. The internal energy is an extensive quantity.

In the following figure, two different processes (one reversible in blue, and another irreversible in red) that take a thermodynamic system from an equilibrium state A (at temperature TA) to another equilibrium state B (at temperature TB) are represented in a PV diagram. Since internal energy is a state function, its variation will be the same for the two processes. The International System (SI) unit for the internal energy is the joule (J).

The variation of internal energy in a thermodynamic cycle is zero. This is true regardless of the type of process, whether it is reversible or irreversible and of the type of substance that undergoes it.

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Internal energy of an ideal gas

An ideal gas is a theoretical model of gas whose equation of state is deduced assuming that the particles that constitute it have no volume and that there are no interactions between them.

It is possible to demonstrate that the internal energy of a perfect gas depends solely on its temperature: it does not depend on the pressure or the volume occupied by the gas. This was demonstrated experimentally by Joule. Since the interactions between the constituent particles are neglected in the ideal gas model, the only thing they can do is move around and be subjected to elastic collisions. They do not lose energy because these collisions are elastic. Therefore, they only have kinetic energy. And temperature is precisely a measure of the average kinetic energy that a particle system possesses.

The variation of the internal energy for an ideal gas when it goes from an initial state A to a final state B is given by: Where CV is the molar heat capacity at constant volume of an ideal gas.

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To prove it, we will assume that n moles of an ideal gas undergo the isochoric process AB represented in the following figure. This figure also includes the isotherms of the ideal gas (hyperbolas) corresponding to the initial and final temperatures. The heat exchanged by the ideal gas when it goes from state A to state B is given by: The work done by the gas when it goes from state A to B is zero, since there is no variation in volume during the process.

By applying the first law of Thermodynamics we get: And since the internal energy is a state function, when the temperature of an ideal gas goes from TA to TB, the variation of the internal energy will be the same regardless of the process the ideal gas has undergone, whether it is reversible or irreversible. The variation of the internal energy for an ideal gas is the same for all the processes shown in the figure above.

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