## Internal Energy of an Ideal Gas

The **internal energy** is the total of all the energy associated with the motion of the atoms or molecules in the system. **Microscopic forms** of energy include those due to the **rotation**, **vibration**, **translation**, and** interactions** among the molecules of a substance.

## Monatomic Gas – Internal Energy

For a **monatomic ideal gas** (such as helium, neon, or argon), the only contribution to the energy comes from **translational kinetic energy**. The average translational kinetic energy of a single atom depends **only** on the **gas temperature** and is given by equation:

* K _{avg} = 3/2 kT.*

The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:

*E _{int} = 3/2 NkT = 3/2 nRT*

where n is the number of moles. **Each direction** (x, y, and z) contributes ** (1/2)nRT **to the

**internal energy**. This is where the

**equipartition of energy idea**comes in – any other contribution to the energy must also contribute

**. As can be seen, the internal energy of an ideal gas**

*(1/2)nRT***depends only on temperature**and the number of moles of gas.

## Diatomic Molecule – Internal Energy

If the gas molecules contain more than one atom, there are **three translation directions**, and **rotational kinetic energy** also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

**Diatomic ideal gas:**

*E _{int} = 5/2 NkT = 5/2 nRT*

This is only an approximation and applies at intermediate temperatures.** At low temperatures** only the **translational kinetic energy contributes**, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration. The **internal energy will be greater** at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.

The internal energy of real gases also depends mainly on temperature, but similarly as the **Ideal Gas Law**, the internal energy of real gases depends also somewhat on **pressure** and **volume**. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes **electrical potential energy** associated with the forces (or **chemical bonds**) between atoms and molecules.

## Specific Heat at Constant Volume and Constant Pressure

**Specific heat** is a property related to** internal energy** that is very important in thermodynamics. The **intensive properties c_{v}** and

*are defined for pure, simple compressible substances as partial derivatives of the*

**c**_{p}**internal energy**and

*u(T, v)***enthalpy**, respectively:

*h(T, p)*where the subscripts **v** and **p** denote the variables held fixed during differentiation. The properties **c _{v} **and

**c**are referred to as

_{p}**specific heats**(or

**heat capacities**) because under certain special conditions they relate the temperature change of a system to the amount of energy added by heat transfer. Their SI units are

**J/kg K**or

**J/mol K**. Two specific heats are defined for gases, one for

**constant volume (c**and one for

_{v})**constant pressure (c**.

_{p})According to the **first law of thermodynamics**, for constant volume process with a monatomic ideal gas the molar specific heat will be:

*C _{v} = 3/2R = 12.5 J/mol K*

because

*U = 3/2nRT*

It can be derived that the **molar specific heat** at constant pressure is:

**C _{p} = C_{v} + R = 5/2R = 20.8 J/mol K**

This ** C_{p}** is greater than the molar specific heat at constant volume

**, because energy must now be supplied**

*C*_{v}**not only**to

**raise the temperature**of the gas but also for the

**gas to do work**because in this case volume changes.

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