E represents the total energy stored in a system. There are two modes in which the energy may be stored in a system.

• Macroscopic energy mode

• Microscopic energy mode.

**4.3.1 Macroscopic Energy Mode**

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Kinetic energy and potential energy of a system are macroscopic energy.

Kinetic Energy = E_{k} = ½ m v^{2} where m is the mass of the system and V is the centre of mass velocity and,

Potential Energy = E_{p }= mgz where z is the elevation or height from the datum. This is the energy stored by virtue of position of the system.

**4.3.2 Microscopic Energy Mode**

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Molecular internal energy or simply internal energy (U) stored in molecules or atoms of a system are the microscopic energy. A system is composed of matter which contains molecules and atoms. Molecules of gases are in random motion with an average velocity. They collide with other molecules and walls of the container and due to collision, the molecules may gain rotational energy, and vibrational energy. They also translational, electronic, chemical, nuclear energy. If E represents the total energy of one molecule, then

If N is the total number of molecules in a system then total internal energy U is given by

U=Nε

For ideal gas, U is a function of T only

U= f(T)

Magnetic energy, electrical energy and surface tension energy can also be present in the system. In absence of these energies, the total stored energy can be written as

E= E_{K }+ E_{p}+ U

In absence of motion and gravity

i.e., E_{K} = 0 and E_{p} = 0,

E=U

So equation (4.4 (a)) becomes.

dQ = dE +dW

or dQ = dU + dW

In absence of shaft work, electrical work etc., we can write

dW = Displacement work only

= PdV

or from (4.7) dQ = dU + PdV. Integrating between state 1 and 2 we get

If we consider Kinetic Energy and potential energy equation (4.8) can be expanded to

Note that internal energy is an extensive property because it depends on the mass of the system. Similarly Kinetic Energy and Potential Energy are extensive properties of a system.

Specific internal energy (u) is defined as the internal energy per unit mass, so.

U = U/m.