Equation of State.....
Relationships between P,V,T.....
Isothermal Process..... Internal Energy..... Constant Volume Process..... Constant Pressure Process.....
This page provides a limited notes on thermodynamic relationships useful to mechanical engineers.
Thermodynamic Process RelationshipsGeneral Polytropic Process
The majority of frictionless processes for ideal gases are called polytropic processes and are in accordance with the following relationship
PV n = constant
Equation of state for and Ideal Gas
PV = mRT
Thermodynamic Relationships between P,V & T
Consider a piston in a frictionless cylinder
The work done on/by the gas in moving the piston δx = (PA)δx = P δ V = δ W
The gas is assumed to be expanding in balanced resisted reversible process.
For a perfect gas - The relationship between Temperature , Pressure and Volume over a cycle
For an adiabatic process with no transfer of heat across the system boundary.(Q = 0 )
Consider a fixed mass of gas in a cylinder which is expanding in a reversible manner...
For an adiabatic process there is no heat transfer.
Therefore the increase in internal energy = - External work done by gas
It is shown below that cp - cv = R = cv ( cp / cv -1) and therefore
γ = 1.4 for Air, H 2, O 2, CO, NO, Hcl
γ = 1.3 for CO 2, SO 2, H 2O, H 2S, N 2O, NH 3, CL 2, CH 4, C 2H 2, C 2H 4
In a isothermal process the temperature = constant and therefore
PV = c and P = c / V
Internal Energy, Cp and Cv
Although it is not possible to determine the absolute value of the internal energy
of a substance. The internal energy change between the initial and final
equilibrium states of any process is definite and determinable.
If a definite mass of gas (m) at constant volume is a closed system is heated from initial conditions P1, V, T1, U1 to P2, V , T2,U2. As the volume is fixed then no work has been done. Then in accordance with the First Law of Thermodynamics (δQ = δU + δW ).
mCv (T2 - T1) = (U2 - U1) + 0
Heating at constant pressure....
If a definite mass of gas (m) at constant volume is a closed system is heated from initial conditions P, V1, T1, U1 to P, V1 , T2,U2. As the volume is fixed then no work has been done. Then in accordance with the First Law of Thermodynamics (δQ = δU + δW ).
mcp (T2 - T1)
mcv (T2 - T1) = U2 - U1 therefore mcp (T2 - T1) = mcv (T2 - T1) + mR (T2 - T1) therefore
cp = cv + R... and ..