State the differential equation for adiabatic change
Answers
P is the pressure of the system. V is the volume of the system. γ is the adiabatic index and is defined as the ratio of heat capacity at constant pressure Cp to heat capacity at constant volume C.
Answer:
The thermodynamic process in which there is no exchange of heat from the system to its surrounding neither during expansion nor during compression.
The adiabatic process can be either reversible or irreversible. Following are the essential conditions for the adiabatic process to take place:
The system must be perfectly insulated from the surrounding.
The process must be carried out quickly so that there is a sufficient amount of time for heat transfer to take place.
Adiabatic process
For instance, the gas compression within an engine cylinder is expected to happen so fast that on the compression process timescale, a minimum amount of the energy of the system could be produced and sent out in the form of heat.
Despite the cylinders being not insulated and having a conductive nature, the process is deemed to be adiabatic. The same could be considered to be true for the enlargement process of such a system.
The adiabatic process can be derived from the first law of thermodynamics relating to the change in internal energy dU to the work dW done by the system and the heat dQ added to it.
dU=dQ-dW
dQ=0 by definition
Therefore, 0=dQ=dU+dW
The word done dW for the change in volume V by dV is given as PdV.
The first term is specific heat which is defined as the heat added per unit temperature change per mole of a substance. The heat that is added increases the internal energy U such that it justifies the definition of specific heat at constant volume is given as:
Cv=dUdT1n
Where,
n: number of moles
Therefore, 0=nCvdT+PdV (eq.1)
From the ideal gas law, we have
nRT=PV (eq.2)
Therefore, nRdT=PdV+VdP (eq.3)
By combining the equation 1. And equation 2, we get
−PdV=nCvdT=CvR(PdV+VdP) 0=(1+CvR)PdV+CvRVdP 0=R+CvCv(dVV)+dPP
When the heat is added at constant pressure Cp, we have
Cp=Cv+R 0=γ(dVV)+dPP
Where the specific heat ɣ is given as:
γ≡CpCv
From calculus, we have, d(lnx)=dxx 0=γd(lnV)+d(lnP) 0=d(γlnV+lnP)=d(lnPVγ) PVγ=constant
Hence, the equation is true for an adiabatic process in an ideal gas