Adiabatic Process Derivation: Formula, Examples & Equation

Adiabatic process can be defined as a thermodynamic process wherein no amount of heat energy is transmitted across a system’s boundary. However, it doesn’t mean that temperature continues to be constant, but that the heat is not transferred in or out of the system. Adiabatic process, simply, is where no heat is gained or lost by a given system. According to the first law of thermodynamics, it is known that Q = 0 indicates that all the change in internal energy is in the form of work done. As per the Adiabatic process, we can say that, ∆U = U₂ - U₁ and W = – ∆U = U₁ - U₂.

Key Terms: Adiabatic Process, Thermodynamics, Heat, Internal Energy, External Energy, Work, Energy, Adiabatic Index

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Adiabatic Process Formula

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Adiabatic process formula can be represented by:

Herein,

• P = Pressure of the system
• V = Volume of the system
• γ = Adiabatic index (ratio of heat capacity with a constant pressure of C_p to heat capacity at constant volume C_v)

Adiabatic Index

Adiabatic index, also called heat capacity ratio, can be expressed as the ratio of heat capacity at constant pressure C_p to heat capacity at a constant volume, i.e., C_v. Adiabatic Index is also referred to as Isentropic Expansion Factor. It is denoted by ɣ.

\(\begin{array}{l}\gamma =\frac{C_{p}}{C_{v}}=\frac{c_{p}}{c_{v}}\end{array}\)

Here,

• C = Heat capacity
• c = Specific heat capacity

The adiabatic index finds its application in reversible thermodynamic processes which involves ideal gases. The speed of sound also depends on the adiabatic index.

Read More: Wien’s Displacement Law

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Adiabatic Process Derivation

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Adiabatic process can be derived from the first law of thermodynamics which explains the change in internal energy dU to the amount of work W done by the system, alongside the heat dQ added to it.

Adiabatic Process Graph

Thus, it can be said, dU = dQ – dW

dQ = 0 (as known)

Therefore, 0 = dQ = dU + dW

\(\therefore\) Work done dW for volume V change by dV is expressed as PdV.

The first term here relates to specific heat (i.e., heat added change per unit temperature, per mole of a substance). The added heat increases the internal energy U such that it fulfills the definition of specific heat (constant volume). It can be shown by:

⇒\( \begin{array}{l}C_{v}=\frac{dU}{dT}\frac{1}{n}\end{array}\) (\(\because\) n = number of moles)

Thus, 0 = nC_vdT + PdV … (1)

Now, for ideal gas, we can claim:

⇒ nRT = PV … (2)

Hence, nRdT = PdV + VdP … (3)

Now, after combining both values (1) and (2),

⇒\(\begin{array}{l}-PdV=nC_{v}dT=\frac{C_{v}}{R}(PdV+VdP)\end{array}\)

⇒\(\begin{array}{l}0=(1+\frac{C_{v}}{R})PdV+\frac{C_{v}}{R}VdP\end{array} \)

⇒ \(\begin{array}{l}0=\frac{R+C_{v}}{C_{v}}(\frac{dV}{V})+\frac{dP}{P}\end{array}\)

Now, after heat is added with constant pressure C_p

⇒ C_p + C_v = R

Thus, \(\begin{array}{l}0=\gamma (\frac{dV}{V})+\frac{dP}{P}\end{array}\)

Herein, specific heat is expressed as:

\(\rightarrow\)\(\begin{array}{l}\gamma\equiv \frac{C_{p}}{C_{v}}\end{array}\)

Now, as per calculus, we can say, 

⇒ \(d(lnx)=\frac{dx}{x}\)

⇒ \(\begin{array}{l}0=\gamma d(lnV)+d(lnP)\end{array}\)

⇒ \(\begin{array}{l}0=d(\gamma lnV+lnP)=d(lnPV^{\gamma })\end{array}\)

⇒ \(PV^{\gamma }=constant\)

Also Check: Hydrostatic Paradox

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Adiabatic Process Examples

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There are several examples of Adiabatic Process. Some of them include:

• Turbine that uses heat to generate work
• Release of air from pneumatic tire
• Adiabatic Efficiency can be applied to nozzles, turbines and compressors
• Oscillating pendulum in a vertical plane

Read Also:

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Previous Year Questions

1. For a gas … 
2. If enthalpies of formation for … [NEET 1995]
3. The shown p- V diagram represents the thermodynamic cycle … [JEE Mains 2013]
4. Consider a spherical shell of radius … [JEE Mains 2015]
5. Consider an ideal gas confined in an isolated closed chamber … [JEE Mains 2015]
6. If the heat of 110J is added to a gaseous system … 
7. A perfect gas is found to obey the relation … 
8. A cyclic process is shown in the figure … 
9. A Carnot engine whose sink is at … [NEET 2006]
10. The molar specific heat at constant pressure of an ideal gas … [NEET 2006]
11. For a gas undergoing an adiabatic process … 
12. Heat is supplied to a diatomic gas at constant pressure …  
13. In an isothermal process, there is … 

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Things to Remember

• Adiabatic process can be explained as thermodynamic process where no amount of heat energy is conveyed across a system’s boundary.
• Adiabatic Process formula is PV^γ = Constant.
• The system, in an adiabatic process, should always be insulated from the surrounding.
• The process in adiabatic process should be quickly carried out such that there is a sufficient amount of time for the heat transfer to take place.
• One of the major examples of Adiabatic process is the release of air from pneumatic tire.