CD Nozzle And The Adiabatic Process

For adiabatic processes the realtion between pressure change and corresponding, temperature change is given by the equation where symbol gamma stands for Adiabatic Index of the fluid the second equation give you the realtion between Temperature and volume of the fluid

thanks so much slugger.

I spent several hours trying to figure this out, I assumed that I was missing a part of the equation, but even after going through doesn's of wikipedia pages, including a page descibing the adiabatic index I was still at a loss as I couldn't figure out the more advanced math formulas.

my undestanding of the adiabatic index is as follows:

cosider two cases of expanding gas inside a cylinder pushing a piston, expanding from 1 litre to 2 litres. both gases have the same pressure, 10 atmospheres , but have different temperatures 200 k (-173 Celsius) and 400 k (27 degrees celcius). the amount of work done will be equal since they arte both pushing a piston with the same amount of pressure. but if they where both to loose half their temperature, the hotter gas would loose more energy while doing the same amount of work which defies the "conservation of energy" principal.

(energy can change forms but will never apear or disapear. e.g. to total amount of energy inside any system will always remain the same although it may change form, in this case heat and pressure are coverted into mechanical work).

please correct me if i got it wrong

Well Adiabatic Index, going by the Textbook defintion, is the ratio of Specific heat of the gas at constant pressure to the Specific heat of the gas at constant volume
Defintion of Specific Heat
Amount of Heat required to raise the temperature of 1 kg of substance by 1 degree
For ex let us, for the sake of convienience, assume that we have 1Kg of Oxygen @ 253 K. Now we want to raise its temp to 254 K. If the amount of heat needed to be supplied for this is 12KJ, then 12KJ would become the specific heat of oxygen

Specific Heat at a constant volume
Amount of Heat required to raise the temperature of 1 kg of substance by 1 degree, keeping the volume constant
Now assume that the same Oxygen is stored in a sealed container at the same temp as above and is its temp is to be raised by same 1 degree. So the heat now supplied to the 1Kg oxygen will be the Specific Heat of oxygen at a constant volume. In this case, since the oxygen can not expand, all the heat supplied to the oxygen will be utilised in raising its temperature.

Specific Heat at a constant pressure
Amount of Heat required to raise the temperature of 1 kg of substance by 1 degree, keeping the pressure constant
Now assume that the same Oxygen is stored in a container, which has an adjustable lid that can be used to regulate pressure, at the same temp as above and is its temp is to be raised by same 1 degree. So the heat now supplied to the 1Kg oxygen will be the Specific Heat of oxygen at a constant pressure.

Since the pressure is to be mantained constant, by adding heat to the oxygen, it will expand, ie, it performs work. So, now the heat supplied to the oxygen will be utilised for to purposes - - perform work of expansion and raise temperature of oxygen, ie increase its internal energy. Therefore in a constant pressure process more heat will have to be supplied to obtain the same 1 degree rise in temp. Thus Specific Heat at a constant pressure is always more than Specific Heat at a constant volume

The ratio of this Specific Heat at a constant pressure to the Specific Heat at a constant volume is known as Adiabatic Index