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Internal Energy

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Every system is maid up of tiny particles, known as molecules. The molecules of a substance possess kinetic energy due to the translatory motion or temperature. Translational kinetic energy of molecules is proportional to the absolute temperature. The molecules also possess potential energy due to their position or inter molecular force of attraction. 

Internal energy is the energy of a system that is related with its microscopic constituents that is atoms and molecules when viewed from a frame of reference at rest with respect to the center of mass of the system.  
The sum of kinetic energy and potential energy of all molecules of a system is called the internal energy or intrinsic energy of the system. That is the internal energy of the system maid up of kinetic and potential energy of the constituent particles.
The kinetic energy is the function of temperature. Internal energy changes when heat is transferred from system to surrounding or vice-versa. 

Internal Energy Equation

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According to first law of thermodynamics, the internal energy denoted as $U$, for the isolated system is always constant. First law dose not provide any way to measure absolute value of internal energy, however, it only provides ways to measure the change of internal energy ($\Delta U$), that is the difference between internal energies of initial and final states.

$\Delta U$ = $U_{final}-U_{initial}$

Where
$\Delta U$ - Change in internal energy
$U_{initial}$ - Internal energy of the initial state
$U_{final}$ - Internal energy of the final state

Change In Internal Energy

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Internal energy of a system depends upon state of the system and not upon how the system attains the state. The internal energy is, therefore a state function.

The internal energy of a system can be changed in two ways:
  1. Either by allowing the heat to flow into the system or out if the system
  2. By work done on the system or by the system.

Internal Energy of Ideal Gas

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From the definition we know that the internal energy is the sum of kinetic energy and potential energy of all molecules of a system.

$U$ = $K.E + P.E$

In the case of ideal gases there is no molecular attraction between the molecules. Hence, they have no potential energy. Thus, the internal energy of an ideal gas is only the kinetic energy of its molecules.