Adiabatic Process Derivation

An adiabatic process is a thermodynamic process such that there is no heat transfer in or out of the system and is generally obtained by using a strong insulating material surrounding the entire system.

Adiabatic process examples

  • The vertical flow of air in the atmosphere
  • When the interstellar gas cloud expands or contracts.
  • Turbine is an example of the adiabatic process as it uses the heat a source to produce work.

Adiabatic process derivation

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:

\(C_{v}=\frac{dU}{dT}\frac{1}{n}\)

Where,

n: number of moles

Therefore, \(0=nC_{v}dT+PdV\) (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

\(-PdV=nC_{v}dT=\frac{C_{v}}{R}(PdV+VdP)\) \(0=(1+\frac{C_{v}}{R})PdV+\frac{C_{v}}{R}VdP\) \(0=\frac{R+C_{v}}{C_{v}}(\frac{dV}{V})+\frac{dP}{P}\)

When the heat is added at constant pressure Cp, we have

\(C_{p}=C_{v}+R\) \(0=\gamma (\frac{dV}{V})+\frac{dP}{P}\)

Where the specific heat ɣ is given as:

\(\gamma\equiv \frac{C_{p}}{C_{v}}\)

From calculus we have, \(d(lnx)=\frac{dx}{x}\) \(0=\gamma d(lnV)+d(lnP)\) \(0=d(\gamma lnV+lnP)=d(lnPV^{\gamma })\) \(PV^{\gamma }=constant\)

Hence, the equation is true for an adiabatic process in an ideal gas.

Adiabatic index

The adiabatic index is also known as heat capacity ratio and is defined as the ratio of heat capacity at constant pressure Cp to heat capacity at constant volume Cv. It is also known as the isentropic expansion factor and is denoted by ɣ.

\(\gamma =\frac{C_{p}}{C_{v}}=\frac{c_{p}}{c_{v}}\)

Where,

C: heat capacity

c: specific heat capacity

Adiabatic index finds application reversible thermodynamic process involving ideal gases and speed of sound is also dependent on the adiabatic index.

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