Thermodynamics is an offshoot of physics which looks after the mechanism of energy and function of a system. Thermodynamics is specialized in taking care of the large scale response of a system that we can notice and gauge in experiments.
Here, we draw some calculations that have relations with the heat capacity of a gas to the gas constant used in the equation of state. We have determined that we will use specific values of the state variables. According to a scientist, a “specific” state variable means the value of the variable divided by the weight of the substance. This lets us draw relations between variables irrespective of the value of the substance that we have. All we can do is to the specific variable by the amount of the substance at any time to derive the correct value of the flow variable. On studying heat transfer thoroughly, we understand that the quantity of heat transferred between two objects is equivalent to the temperature variation between the objects and the heat capacity of the objects. The heating ability of the object is consistent and as a result of which we are aware of the amount of heat added per unit temperature rise. The value of the constant is different for different materials and depends on the process. Heat capacity is not a state variable.
If we are working with a gas, it is most conducive to apply types of the thermodynamics equations based on the enthalpy of the gas. From the definition of enthalpy:
h = e + p * v
Where h in the specific enthalpy, p is the pressure, v is the specific volume, and e is the specific internal energy. During a process, the values of these variables change. Let’s denote the change by the Greek letter delta which looks like a triangle. So “delta h” means the change of “h” from state 1 to state 2 during a process. Then, for a constant pressure process the enthalpy equation becomes:
delta h = delta e + p * delta v
The enthalpy, internal energy, and volume are all changed, but the pressure remains the same. From our derivation of the enthalpy equation, the change of specific enthalpy is equal to the heat transfer for a constant pressure process:
delta h = cp * delta T
where delta T is the change of temperature of the gas during the process,and c is the specific heat capacity. We have added a subscript “p” to the specific heat capacity to remind us that this value only applies to a constant pressure process.
The equation of state of a gas relates the temperature, pressure, and volume through a gas constant R . The gas constant used by aerodynamicists is derived from the universal gas constant, but has a unique value for every gas.
p * v = R * T
If we have a constant pressure process, then:
p * delta v = R * delta T
Now let us imagine that we have a constant volume process with our gas that produces exactly the same temperature change as the constant pressure process that we have been discussing. Then the first law of thermodynamics tells us:
delta e = delta q – delta w
where q is the specific heat transfer and w is the work done by the gas. For a constant volume process, the work is equal to zero. And we can express the heat transfer as a constant times the change in temperature. This gives:
delta e = cv * delta T
where delta T is the change of temperature of the gas during the process, and c is the specific heat capacity. We have added a subscript “v” to the specific heat capacity to remind us that this value only applies to a constant volume process. Even though the temperature change is the same for this process and the constant pressure process, the value of the specific heat capacity is different.
Because we have selected the constant volume process to give the same change in temperature as our constant pressure process, we can substitute the expression given above for “delta e” into the enthalpy equation. In general, you can’t make this substitution because a constant pressure process and a constant volume process produce different changes in temperature If we substitute the expressions for “delta e”, “p * delta v”, and “delta h” into the enthalpy equation we obtain:
cp * delta T = cv * delta T + R * delta T
dividing by “delta T” gives the relation:
cp = cv + R
The specific heat constants for constant pressure and constant volume processes are related to the gas constant for a given gas. This rather remarkable result has been derived from thermodynamic relations, which are based on observations of physical systems and processes. Using the kinetic theory of gases, this same result can be derived from considerations of the conservation of energy at a molecular level.
We can define an additional variable called the specific heat ratio, which is given the Greek symbol “gamma”, which is equal to cp divided by cv:
gamma = cp / cv
“Gamma” is just a number whose value depends on the state of the gas. For air, gamma = 1.4 for standard day conditions. “Gamma” appears in many fluids equations including the equation relating pressure, temperature, and volume during a simple compression or expansion process, the equation for the speed of sound, and all the equations for isentropic flows, and shock waves. Because the value of “gamma” just depends on the state of the gas, there are tables of these values for given gases. You can use the tables to solve gas dynamics problems.