Adiabatic Process Derivation: Formula, Examples & Equation

Edited By Vishal kumar | Updated on Jul 02, 2025 05:31 PM IST

Adiabatic Process Derivation - The first law of thermodynamics, which relates the change in internal energy due to the work (W) that the system does and the heat dQ introduced, can be used to construct the equation for an adiabatic process. PdV is the amount of work dW that was put in to modify volume V by dV.

This Story also Contains

Adiabatic Process
Different Applications Of The Adiabatic Assumption
Heating and Cooling - Adiabatic
Adiabatic Process Examples
Adiabatic Process Derivation
Adiabatic Index

Adiabatic Process Derivation: Formula, Examples & Equation

A thermodynamic process known as an adiabatic process occurs when no heat energy is transported across the system's boundaries. This doesn't imply a constant temperature, just that no heat is being moved into or out of the system. Learn about the adiabatic process, examples, derivation of the adiabatic process equation and the adiabatic index in this article.

The term "adiabatic" refers to a process in which there is no heat transmission into or out of a system, resulting in Q = 0. Such a system is referred to as being adiabatically isolated. It's common to make the simplistic assumption that a process is adiabatic. For instance, it is presumable that a gas is compressed so quickly inside an engine cylinder that very little of the system's energy can be released as heat to the environment during the compression process. The process is envisioned as adiabatic even though the cylinders are not insulated and quite conducive. The expansion phase of such a system follows a similar pattern.

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat energy is transported across the system's limits. This doesn't imply a constant temperature, just that no heat is being moved into or out of the system. Learn about the adiabatic process, examples, derivation of the adiabatic process equation and the adiabatic index in this article.

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A process in which no heat is transferred to or from a system results in Q = 0, and such a system is said to be adiabatically isolated. A common simplifying assumption is that a process is adiabatic. For example, the compression of a gas within an engine cylinder is assumed to occur so quickly that little of the system's energy can be transferred out as heat to the surroundings on the time scale of the compression process. Even though the cylinders are not insulated and are extremely conductive, the process is idealised to be adiabatic. The same can be said for the system's expansion process.

Different Applications Of The Adiabatic Assumption

For a closed system, the first law of thermodynamics can be written as: U = QW, where U denotes the change in internal energy of the system, Q is the amount of energy added to it as heat, and W is the work done by the system on its surroundings.

If the system's walls are so rigid that no work can be transferred in or out (W = 0), the walls are not adiabatic, energy is added in the form of heat (Q > 0), and there is no phase change, the system's temperature will rise.
If the system's walls are so rigid, pressure-volume work cannot be done. Still, the walls are adiabatic (Q = 0), and energy is added as isochoric (constant volume) performed in the form of friction or the stirring of the viscous fluid inside the system (W 0). There is no phase change; the system's temperature will rise.
The temperature of the system will increase if the system walls are adiabatic (Q = 0), but not stiff (W 0), and energy is provided to the system in the form of frictionless, non-viscous pressure-volume work (W 0) in a hypothetical idealised process. Such a process is referred to as "reversible" and is called an isentropic process. The energy could be completely recovered as work done by the system if the process was reversed. The system's entropy would appear to decrease if the system, which contains compressible gas, had a smaller volume. However, because the process is isentropic (S = 0), the system's temperature will increase. If the work is added in a way that causes the system to experience friction or viscous forces, the process will not be isentropic; additionally, if there is no phase change, the system's temperature will increase, the process is referred to as "irreversible," and the work added to the system is not entirely recoverable as work.

Heating and Cooling - Adiabatic

The temperature drops due to adiabatic expansion up against pressure or a spring. On the other hand, free expansion is an isothermal process for a perfect gas.

Adiabatic heating happens when a gas's pressure is raised by the work its surroundings do on it, such as when a piston compresses a gas inside of a cylinder, raising the temperature in a situation where, in many real-world circumstances, heat conduction through walls can be slower than the compression time. This is used in diesel engines, which depend on the fuel vapour temperature being raised enough to ignite due to the lack of heat dissipation during the compression stroke.

When the pressure on an adiabatically isolated system is reduced, allowing it to expand and conduct work on its all-around, Adiabatic cooling occurs. When a parcel of gas is subjected to less pressure, the gas is free to expand; as the volume grows, the temperature reduces due to the internal energy of the gas decreasing. With orographic lifting and lee waves, adiabatic cooling takes place in the Earth's atmosphere, which can result in the formation of piles or lenticular clouds.

In some regions of the Sahara desert, snowfall is infrequent because of adiabatic cooling in hilly regions.

Adiabatic Process Examples

Air moves vertically in the atmosphere
when the gas cloud between stars grows or shrinks.
As it employs heat to generate work, the turbine illustrates the adiabatic process.

Adiabatic Process Derivation

The first law of thermodynamics, which relates the change in internal energy dU to the work W that the system does and the heat dQ that is introduced to it, can be used to construct the equation for an adiabatic process.

dU=dQ-dW

dQ=0 by definition,

Therefore, 0=dQ=dU+dW.

Here, work done by dW for the change in volume V by dV is given as PdV.

Cv=dUdT1n

Where,

n: number of moles,

Therefore,

0=nCvdT+PdV… (eq.1)

From the ideal gas, we have

nRT=PV (eq.2)

Therefore, nRdT=PdV+VdP (eq.3)

By combining 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)

PV=constant

Since an adiabatic process in an ideal gas follows the equation, it is valid.

Adiabatic Index

The first phase has to do with specific heat, the heat added for every degree a substance's temperature changes per mole. The extra heat raises internal energy U to the point where it supports the definition of specific heat at constant volume as

γ=CpCv=cpcv

Where

C: heat capacity

c: specific heat capacity

The ratio of the heat capacity at constant volume Cv to the heat capacity at constant pressure Cp is the adiabatic index, which is sometimes referred to as the heat capacity ratio. The symbol represents it, also referred to as the isentropic expansion factor.

In reversible thermodynamic processes involving ideal gases, the adiabatic index is used.

Frequently Asked Questions (FAQs)

1. Is it true that the heat or mass is not transferred in the adiabatic process?

The statement is True

2. What do you mean by the adiabatic index?

The heat capacity ratio at the specific constant pressure Cp to heat capacity at constant volume Cv.

γ=CpCv=cpcv

3. Describe one example of an adiabatic process.

The turbine uses heat to produce work is a perfect example of an adiabatic process.

4. What is the expression for the adiabatic process in an ideal gas?

The expression for the adiabatic process in an ideal gas is,

PV=constant

5. An adiabatic index is also known as an?

As we see above in an adiabatic index, heat capacity and specific heat capacity are important factors. For adiabatic index is also known as the heat capacity ratio and is defined as the ratio of heat capacity at a constant temperature to heat capacity at a constant volume.

6. What is the adiabatic equation, and what does it represent?

The adiabatic equation is PVᵞ = constant, where P is pressure, V is volume, and γ (gamma) is the heat capacity ratio. This equation describes the relationship between pressure and volume during an adiabatic process.

7. What is the significance of the heat capacity ratio (γ) in adiabatic processes?

The heat capacity ratio (γ) is the ratio of specific heat at constant pressure to specific heat at constant volume. It determines how much the temperature changes during an adiabatic process and affects the relationship between pressure and volume changes.

8. What is the relationship between pressure and volume in an adiabatic process?

In an adiabatic process, pressure and volume are inversely related, but not linearly. Their relationship is described by the equation PVᵞ = constant, where γ is greater than 1, making the curve steeper than an isothermal process.

9. How does the specific heat ratio (γ) affect the temperature change in an adiabatic process?

A higher specific heat ratio (γ) results in a greater temperature change for a given volume change in an adiabatic process. This is because γ represents the ratio of energy stored in translational motion to that stored in rotational and vibrational modes.

10. How does the ideal gas law apply to adiabatic processes?

The ideal gas law (PV = nRT) still applies to adiabatic processes, but it must be used in conjunction with the adiabatic equation (PVᵞ = constant) to fully describe the behavior of the gas.

11. What is an adiabatic process?

An adiabatic process is a thermodynamic process in which no heat is transferred between the system and its surroundings. The system is thermally insulated, so any changes in temperature or pressure are solely due to work done on or by the system.

12. How does an adiabatic process differ from an isothermal process?

In an adiabatic process, no heat is exchanged with the surroundings, and the temperature of the system changes. In contrast, an isothermal process maintains a constant temperature throughout, allowing heat exchange with the surroundings.

13. What is the difference between an adiabatic and an isentropic process?

An adiabatic process involves no heat transfer, while an isentropic process is both adiabatic and reversible. All isentropic processes are adiabatic, but not all adiabatic processes are isentropic, as some may involve irreversibilities.

14. What is the difference between quasi-static and non-quasi-static adiabatic processes?

A quasi-static adiabatic process occurs slowly enough for the system to remain in thermodynamic equilibrium throughout, while a non-quasi-static process happens too rapidly for equilibrium to be maintained. Quasi-static processes are reversible, while non-quasi-static ones are irreversible.

15. What is an adiabatic calorimeter, and how does it work?

An adiabatic calorimeter is a device designed to measure heat changes in chemical or physical processes under conditions where no heat is exchanged with the surroundings. It uses insulation and temperature control to maintain adiabatic conditions, allowing precise measurement of heat effects.

16. How do you derive the adiabatic equation?

The adiabatic equation is derived from the first law of thermodynamics, the ideal gas law, and the relationship between specific heats. By combining these principles and considering that no heat is exchanged, we can arrive at the equation PVᵞ = constant.

17. What is the formula for work done in an adiabatic process?

The work done in an adiabatic process is given by W = (P₂V₂ - P₁V₁) / (1 - γ), where P₁, V₁ are initial pressure and volume, P₂, V₂ are final pressure and volume, and γ is the heat capacity ratio.

18. What is the first law of thermodynamics, and how does it relate to adiabatic processes?

The first law of thermodynamics states that the change in internal energy of a system equals the heat added to the system minus the work done by the system. In an adiabatic process, since no heat is exchanged (Q = 0), the change in internal energy is solely due to work done.

19. How does the work done in an adiabatic process compare to that in an isothermal process?

The work done in an adiabatic process is generally less than in an isothermal process for the same initial and final volumes. This is because the pressure changes more rapidly in an adiabatic process, resulting in a smaller area under the P-V curve.

20. What is the significance of entropy in an adiabatic process?

In a reversible adiabatic process, the entropy of the system remains constant. This is because entropy changes are associated with heat transfer, and in an adiabatic process, no heat is exchanged with the surroundings.

21. How do adiabatic processes occur in nature?

Adiabatic processes occur in nature when changes happen quickly enough that heat transfer is negligible. Examples include the rising and falling of air parcels in the atmosphere, sound wave propagation, and rapid compression or expansion in engines.

22. How does the adiabatic lapse rate in the atmosphere relate to adiabatic processes?

The adiabatic lapse rate describes the rate at which the temperature of a parcel of air changes as it rises or falls in the atmosphere without exchanging heat with its surroundings. This is an example of an adiabatic process in nature.

23. What happens to the temperature of a gas during an adiabatic compression?

During an adiabatic compression, the temperature of the gas increases. This is because work is done on the gas, increasing its internal energy, and since no heat can escape, this energy is converted entirely into an increase in temperature.

24. How does an adiabatic expansion affect the temperature of a gas?

In an adiabatic expansion, the gas does work on its surroundings, decreasing its internal energy. Since no heat can enter the system, this results in a decrease in the gas's temperature.

25. Can you explain the concept of adiabatic efficiency?

Adiabatic efficiency is a measure of how close a real process is to an ideal, reversible adiabatic process. It compares the actual work done to the theoretical maximum work that could be done in an ideal adiabatic process.

26. What role do adiabatic processes play in heat engines?

Adiabatic processes are crucial in heat engines, such as internal combustion engines and gas turbines. The compression and expansion strokes in these engines are often approximated as adiabatic processes to maximize efficiency.

27. What is the adiabatic bulk modulus, and how does it relate to the isothermal bulk modulus?

The adiabatic bulk modulus describes a material's resistance to compression under adiabatic conditions. It's always larger than the isothermal bulk modulus because temperature changes during adiabatic compression provide additional resistance to volume change.

28. What is the significance of the critical pressure ratio in adiabatic flow through nozzles?

The critical pressure ratio in adiabatic flow through nozzles is the pressure ratio at which the flow reaches sonic velocity. It's important in designing nozzles for applications like rocket engines and determines whether the flow will be subsonic or supersonic.

29. How do adiabatic processes relate to the concept of enthalpy?

In an adiabatic process, the change in enthalpy (H) equals the work done (W) since no heat is exchanged. This relationship, ΔH = W, is useful in analyzing many thermodynamic systems, especially in engineering applications.

30. What is the adiabatic flame temperature, and why is it important in combustion processes?

The adiabatic flame temperature is the maximum temperature achieved in a combustion process under ideal, adiabatic conditions. It's important in understanding combustion efficiency, pollutant formation, and material selection for combustion chambers.

31. How does the adiabatic process contribute to the formation of clouds?

As air parcels rise in the atmosphere, they expand and cool adiabatically. If the air cools enough to reach its dew point, water vapor condenses, forming clouds. This process is crucial in understanding atmospheric dynamics and weather patterns.

32. What is the adiabatic flame temperature?

The adiabatic flame temperature is the maximum temperature that can be achieved in a combustion process under adiabatic conditions, where no heat is lost to the surroundings. It represents the theoretical limit of flame temperature.

33. How do real-world processes deviate from ideal adiabatic conditions?

Real-world processes often deviate from ideal adiabatic conditions due to heat transfer with the surroundings, friction, and other irreversibilities. These factors can lead to lower efficiency and different outcomes compared to ideal adiabatic processes.

34. What is the difference between dry and moist adiabatic lapse rates?

The dry adiabatic lapse rate applies to unsaturated air and is about 9.8°C per kilometer of altitude change. The moist adiabatic lapse rate applies to saturated air and is less steep due to the release of latent heat during condensation.

35. How do adiabatic processes affect the efficiency of refrigeration cycles?

In refrigeration cycles, the compression of the refrigerant is ideally an adiabatic process. The efficiency of this compression affects the overall coefficient of performance of the refrigeration system. Real compressors deviate from ideal adiabatic behavior, impacting efficiency.

36. What is the significance of the Poisson relation in adiabatic processes?

The Poisson relation, T₂/T₁ = (V₁/V₂)ᵞ⁻¹, relates the initial and final temperatures to the initial and final volumes in an adiabatic process. It's derived from the adiabatic equation and ideal gas law, providing a way to calculate temperature changes.

37. How does the adiabatic index relate to the degrees of freedom of a gas molecule?

The adiabatic index (γ) is related to the degrees of freedom (f) of a gas molecule by the equation γ = (f + 2) / f. For monatomic gases with 3 degrees of freedom, γ = 5/3, while for diatomic gases with 5 degrees of freedom at room temperature, γ ≈ 7/5.

38. How does the concept of adiabatic processes apply to sound waves?

Sound waves propagate through a medium as adiabatic compressions and rarefactions. The rapid nature of these pressure changes means there's insufficient time for heat transfer, making the process effectively adiabatic. This affects the speed of sound in different media.

39. How do adiabatic processes contribute to the formation of Föhn winds?

Föhn winds form when air moves over a mountain range. As it rises, it cools adiabatically and may lose moisture through precipitation. When descending on the other side, it warms adiabatically but is now drier, resulting in warm, dry winds on the leeward side of mountains.

40. How does the adiabatic process relate to the concept of internal energy in thermodynamics?

In an adiabatic process, the change in internal energy of a system is solely due to work done on or by the system, as no heat is exchanged. This relationship is expressed as ΔU = -W, where ΔU is the change in internal energy and W is the work done.

41. What is the significance of the adiabatic exponent in gas dynamics?

The adiabatic exponent, also known as the heat capacity ratio (γ), plays a crucial role in gas dynamics. It affects the speed of sound in a gas, the behavior of shock waves, and the efficiency of thermodynamic cycles, influencing the design of aircraft, engines, and other fluid systems.

42. How do adiabatic processes affect the temperature of stars?

In stars, adiabatic processes play a key role in energy transport. As plasma rises in a star's convection zone, it expands and cools adiabatically. When it sinks, it compresses and heats adiabatically. This process helps distribute energy within the star.

43. What is the adiabatic invariant in plasma physics?

In plasma physics, an adiabatic invariant is a quantity that remains nearly constant when parameters of the system change slowly compared to the characteristic frequencies of motion. Examples include the magnetic moment of a charged particle in a slowly varying magnetic field.

44. How does the concept of adiabatic processes apply to the expansion of the universe?

The expansion of the universe is often treated as an adiabatic process on large scales. As the universe expands, it cools without exchanging heat with any external system. This adiabatic cooling has important implications for the evolution of matter and radiation in cosmology.

45. What is the role of adiabatic processes in the working of a Stirling engine?

In a Stirling engine, the compression and expansion phases are ideally adiabatic processes. The working fluid (usually air or helium) is compressed adiabatically, heated at constant volume, expanded adiabatically, and then cooled at constant volume, completing the cycle.

46. How do adiabatic processes contribute to the formation of temperature inversions in the atmosphere?

Temperature inversions can form when a layer of warm air descends and undergoes adiabatic compression, heating it further. This warm layer can trap cooler air beneath it, creating a temperature inversion. Understanding this process is crucial for weather forecasting and air quality management.

47. What is the adiabatic approximation in quantum mechanics?

The adiabatic approximation in quantum mechanics assumes that a system adjusts its configuration instantly in response to a slow change in some parameter. This is analogous to classical adiabatic processes and is used in various areas of quantum physics, including molecular dynamics.

48. How does the adiabatic process relate to the concept of entropy in thermodynamics?

In a reversible adiabatic process, the entropy of the system remains constant because no heat is exchanged with the surroundings. However, in irreversible adiabatic processes, the entropy of the system increases due to internal irreversibilities, in accordance with the second law of thermodynamics.

49. What is the significance of the adiabatic demagnetization process in low-temperature physics?

Adiabatic demagnetization is a technique used to achieve very low temperatures. It involves aligning magnetic moments in a material using a strong magnetic field, then removing the field adiabatically. This causes the material to cool as energy is transferred from thermal motion to magnetic disorder.

50. How do adiabatic processes affect the behavior of gases in the upper atmosphere?

In the upper atmosphere, gases can undergo rapid expansion or compression due to changes in pressure or solar radiation. These processes are often adiabatic due to the low density of the atmosphere, affecting temperature profiles, atmospheric chemistry, and the behavior of spacecraft re-entering the atmosphere.

51. What is the role of adiabatic processes in the formation of hail?

Hail forms in strong updrafts within thunderstorms. As water droplets are carried upwards, they cool adiabatically. If lifted high enough, they freeze. The hailstones then grow as they move up and down in the updraft, experiencing alternating adiabatic cooling and heating cycles.

52. How does the adiabatic process contribute to the efficiency of turbochargers in engines?

Turbochargers compress intake air to increase engine power. The compression process is approximately adiabatic due to its rapid nature. Understanding this adiabatic compression is crucial for designing efficient turbochargers and predicting the temperature increase of the compressed air.

53. What is the significance of the adiabatic theorem in quantum mechanics?

The adiabatic theorem in quantum mechanics states that a system will remain in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum. This has applications in quantum computing and quantum adiabatic optimization.

54. How do adiabatic processes affect the behavior of plasmas in fusion reactors?

In fusion reactors, plasma confinement often involves rapid changes in magnetic fields. These changes can be approximated as adiabatic processes, affecting the plasma's temperature, pressure, and confinement properties. Understanding these adiabatic effects is crucial for designing and operating fusion devices.

55. What is the relationship between adiabatic processes and the speed of sound in a medium?

The speed of sound in a medium is related to how pressure changes propagate through it