# Internal Energy, Enthalpy, & Specific Heat of Ideal Gases

Instead of using tables to find a gases internal energy and enthalpy, the ideal gas equations could be used. However, you have to be sure that the ideal gas law can be used for that substance without major loss of accuracy, and you need to have an understanding of what specific heat is.

##### What is an Ideal Gas?

An ideal gas is a gas that can have its density, pressure, and temperature related through the ideal gas law. The ideal gas law is represented in equation 1.

(Eq 1) $ρ=\frac{P}{RT}$

It is important to realize that the ideal gas law provides an approximation of what the actual value for the gas could be. The ideal gas law works best for gases that have a low density state, is in a low pressure state, or has a high temperature state. The ideal gas law does not work well for water vapor or refrigerant coolants; thermodynamic tables should be used instead.

##### Specific Heat

The specific heat represents the amount of energy required to raise a substance by one degree. There are two specific heat constants that can be found in tables for different substance. They are the specific heat at constant volume (cv) and the specific heat at constant pressure (cp). The specific heat at constant volume is the amount of energy required to raise a substance by one degree while it remains at a constant volume, while the specific heat at constant pressure is the amount of energy required to raise the substance by one degree while it remains at a constant pressure. The value for cp is always greater than the value for cv. Refer to the tables below to see the specific heat for some substance when they are considered an ideal gas.

Specific Heat of Ideal Gases at 300 K

 Gas cp (kJ/kg)*K cv (kJ/kg)*K k Air 1.005 0.718 1.400 Carbon Dioxide 0.846 0.657 1.289 Helium 5.1926 3.1156 1.667 Hydrogen 14.307 10.183 1.405 Nitrogen 1.039 0.743 1.400 Oxygen 0.918 0.658 1.395

In some cases you may hear someone talking about specific heat ratios (k). The specific heat ratio is a ratio of cp and cv. Refer to the equation below.

(Eq 2) $k=\frac{c_p}{c_v}$

##### Internal Energy & Enthalpy of an Ideal Gas

For an ideal gas the change in internal energy and enthalpy can be calculated for a temperature change of that gas. This was discovered when Joule conducted an experimented where one tank contained air at a high pressure, and another tank was evacuated. These tanks were connected to each other by a valve and then placed in a water bath until thermal equilibrium was obtained. After thermal equilibrium was obtained he opened the valve letting air from the higher pressure tank pass into the other tank until the pressures had equalized. After this was done Joule observed that there was no temperature change of the water bath, and concluded that internal energy is a function of temperature. Since internal energy is a function of temperature, for an ideal gas equation 3 can be used to calculate the change in internal energy for a change in temperature.

(Eq 3) $Δu = c_{v_{avg}}\left(T_2-T_1\right)$

$c_v$ = Specific Heat at Constant Volume

$T$ = Temperature

For an ideal gas, change in enthalpy is also a function of temperature. This can be shown by combining the ideal gas law and the definition of enthalpy. Refer to equation 4.

(Eq 4)  $h=u+RT,~\cases{h=u+Pν \cr Pν=RT}$

$h$ = enthalpy

$u$ = internal energy per unit mass

$P$ = Pressure

$v$ = specific volume

$R$ = Ideal Gas Constant

Finally the change in enthalpy for temperature difference of an ideal gas can be calculated using equation 5.

(Eq 5)  $Δh = c_{p_{avg}}\left(T_2-T1\right)$

$c_p$ = Specific Heat at Constant Pressure