Adiabatic Process Derivation: Formula, Examples & Equation

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.

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.

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

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.