Work Done During Adiabatic Expansion for IIT JEE

Definition

“An adiabatic system is a thermodynamic process in which heat does not enter or leave the system while expansion or compression of the fluid in the system.” An example of the adiabatic process is the vertical airflow in the atmosphere where the expansion of air and its cooling occurs. It then contracts and grows warmer as it decreases. Another such example of the adiabatic process is when the expansion and compression of interstellar gas takes place.

The changes in the adiabatic process are often accompanied by temperature changes. In most of the examples, truly adiabatic changes must take place in the intervals of short time, so that the heat content of the system remains same or else the system should be insulated as it is not practically possible. The production of the heat when a gas is compressed as such in the piston is approximately adiabatic.

The word ‘adiabatic efficiency’ is applicable to all the devices such as compressors, turbines, and nozzles. Let us take an example of adiabatic efficiency in steam turbine where it is found that it is the ratio of work done to the unit mass of steam and to the release of heat. It is theoretically capable of transforming into mechanical work via adiabatic expansion of a unit weight of steam.

Adiabatic Compression or Expansion

The question arises is that how does pressure will vary with respect to the changes in volume with respect to adiabatic expansion and compression in an ideal gas? The atmospheric pressure of air is nearly an ideal gas and the transmission of sound in air is very close to adiabatic for frequencies and transmission distances normally.

The first law of thermodynamics gives relation in the change of internal energy dU to the work done dW by the system and the added heat dQ into it.

dU = dQ – dW

According to the definition, in an adiabatic process, the heat is exchanged and we get zero. dQ = 0. After substituting the law and rearranging it, we get,

dQ = 0 = dU + dW

The second term in this is easy. The work done dW is when a system changes its volume V by dV is PdV.

The first term may be related to the specific heat which is defined as the added heat per unit temperature change as per mole of the substance. If we add heat at a constant volume, then the gas does not expand and so that it does not do the work. So, the heat added increases the internal energy U. Thus, it follows as per definition, that the specific heat at a constant volume is

Cv = dU/dT × 1/n

Where, n is the number of moles. And hence nCv = dU/dT × 1/n ---- (1)

Now the equation of the state of an ideal gas is,

nRT = PV ----- (2)

Where, R is the gas constant. Taking derivation gives us, 

nRdT = PdV + VdP -------(3)

We may now combine all equations (1) and (3) for eliminating T. (1) and (2) gives these expressions for nCvdT,

nCvdT = Cv/R(PdV + VdP) = n CvdT

Together with PdV and VdP terms, we get,

(1 + Cv /R)PdV + Cv/RVdP = 0

Thus, 

R + Cv /Cv. (dV/V) + dP/P

Now, if the gas is ideal, its internal energy is kinetic, thus the internal energy U depends only on T. From the equation of state of an ideal gas (2), we can evaluate the work dW done at constant pressure, it is PdV = nRdT. So, if we add heat to the ideal gas at a constant pressure, we need extra heat RdT for each mole of gas, which is beyond the heat that we have to add at a constant volume. Thus, the specific heat of an ideal gas at constant pressure is just Cv = Cv + R. the ration of specific heat is represented as a standard symbol: γ(dV/V) + (dP/P) ----(4)

This is the main results which we need to discuss the sound waves. We have just noticed that the change in pressure proportionally is γ times the change in volume proportionally or γ times the change in density proportionally.

Let us not stop here, because we are only a few steps away from an important result as we all know that d(ln x) = dx/x. Thus, we can write it as the following equation:

γ d(ln V) + d(ln P) = 0

And rearranging it gives out the following,

D(γ ln V + ln P) = d(lnPVγ )

Therefore, for an adiabatic process in an ideal gas,

PVγ = constant --------- (5)

On the reflection, you will agree that Cp>Cv: at constant volume, all the heat you gather goes in the increasing of the internal energy and thus raising the temperature. At constant pressure, you need to put not only the heat which raises the internal energy but also an amount of heat which is equal to the work done when the system undergoes the thermal expansion. For air, γ is around 1.4.

Obviously, we may now substitute into (5) from the equation of state (2) to determine P and T or T and V are correlated for an adiabatic process.

We mentioned before that ‘sound transmission in the air is very close to adiabatic for normal frequencies and transmission distances.’ But when does this fail? At the maximum pressure in a sound wave, the temperature is high. Thus, we expect that same heat will get diffused to the neighbouring minimum pressure. Of course, in a half period of a sound wave, there is not so much time for heat to get diffused to one half of wavelength. 

So, the time taken for heat to diffuse goes as the square of the distance. Thus, this process is very less negligible for higher frequencies, as the shorter half wavelength which more than compensates for the shorter half period. This diffusion takes the mechanical energy out of the wave especially at higher frequencies and the increase in efforts with the transmission distance. And for this reason or something others, distant sounds are not only less loud, but also brighter than close ones.

Applications of the Adiabatic Process in Real Life

An example of an adiabatic process is a working piston in a cylinder which is completely insulated. In the adiabatic process, energy is transferred through the work except the heat.

Let us take an example. When the air or other gases are compressed, the gas becomes hot. The opposite is also true that as it cools down, it quickly expands. Thus, the quick expansion or contraction of gas is not completely adiabatic. The change in temperature does not happen in an adiabatic process, but energy does not get transferred outside the system via temperature changing.

The turbines are also great examples of adiabatic systems which utilize heat as a source of producing work, so any lost heat to the surroundings will reduce their efficiency. As we say, the compression and expansion of air is due to the passage of sound waves is an adiabatic process. Newton popularly thought that it was an isothermal process. Later, scientists have corrected this hypothesis and evaluated the speed of sound which is based on the model of a sound wave as an adiabatic process. This speed of sound was closer to the experimental value rather than the one evaluated by Newton.

In aero engines, the tubes of services which carry lubrication oil are surrounded by high-temperature environment. An attempt to make it adiabatic is because the input of that to the tubes will change the property features of oils and interfere with its lubricating behaviour.

Also, in aero engines, the thermal process of stators occur in the compressors which are faster than the thermal process of the rotors because the rotors are bulkier. This poses the problem of clearance controlling that means the gap between the stator and rotor which keep on changing throughout the running engine.

We attack the problem directly from the stator side. We try it to make the inner side of the stator which receives heat as an adiabatic process as much as possible. We also put a shield of heat around that area. The shield of heat deducts the flow in that particular portion whereas cutting off the convection. It also cuts off radiation by reacting as a physical barrier. And obviously, the universe too is adiabatic.

Any process is considered as adiabatic if during the process, there is no exchange of heat between the system and its environment. Thus dQ = 0. During this the volume, temperature and pressure of an adiabatic process are changing. During an adiabatic compression, the temperature increases while during expansion, the temperature decreases.

Another real-time example of the adiabatic process is the pumping up the tyre of a bicycle with a manual pump. The compression of air is fast and can be recognized as adiabatic. The air in the pump becomes heated which can be felt by touching the bottom of the pump which is also an indication of escaping of heat due to the lack of insulation.