Thermodynamics is a branch of physics
which deals with the energy and work of a system.
Thermodynamics
deals only with the
large scale response
of a system which we can
observe and measure in experiments. In aerodynamics, we are most
interested in thermodynamics for the role it plays in
high speed flight
and
engine design.
High temperature
real gas effects
play a major role in
hypersonic aerodynamics.
As the flight
Mach number
increases above Mach = 3, some of the kinetic energy of the moving object
is converted into
heat.
Many fluid dynamics processes, like
isentropic flows
and
shock waves,
are altered by real gas effects.
On the figure we present some equations that relate the
specific heat capacities
of air at hypersonic conditions to the
specific heat capacities
at lower speeds.

For air at low speeds, the ratio of the specific heat capacities
is a numerical constant equal to 1.4.
If the specific heat capacity is a constant value, the gas is said
to be calorically perfect and if the specific heat capacity
changes with temperature, the gas is said to be calorically imperfect.
At subsonic and low supersonic
Mach numbers, air is calorically perfect.
But under low hypersonic conditions, air is
calorically imperfect. The specific heat capacity changes with the
temperature
of the flow because of excitation of the vibrational modes
of the diatomic nitrogen and oxygen of the atmosphere.
This computer simulation illustrates molecular vibration:

Click on the slider bar, hold the mouse button down and drag to the
right to increase the temperature. As the temperature increases, the
vibration increases and more energy is associated with the vibration.

The equations shown on the figure were developed using the
kinetic theory
of gases including a simple harmonic vibrator for the diatomic gases.
The details of the analysis were given by Eggars in
NACA Report 959.
A synopsis of the report is included in
NACA Report 1135.

The equation for the specific heat capacity at constant volume is:

where cv is the specific heat capacity at constant volume, (cv)perf
is the specific heat capacity for a calorically perfect gas, gamp is the
ratio of heat capacities for a perfect gas, theta is a thermal constant
equal to 5500 degrees Rankine, and T is the static temperature.

Similarly, for the specific heat capacity at constant pressure:

where cp is the specific heat capacity at constant pressure, and (cp)perf
is the specific heat capacity for a calorically perfect gas.
The ratio of specific heats is designated by gam, which is given by:

The variation of the value of the specific heat capacities with temperature
causes the values of many derived functions and processes to also become
functions of temperature in hypersonic flows.