Meaning of the adiabatic constant

[Sunfire]

Main Question or Discussion Point

Hello

for adiabatic process,

pV^γ=const

What is the physical meaning of the constant at the Rhs of this equality?

Many thanks.

 

[pa5tabear]

Multiply the units of pressure and volume together and you get units of energy. This relationship is saying that there is no heat transfer into or out of the system. Therefore any change in pressure will be accounted for by an "equalizing" change in volume.

Note that there might still be work.

 

[Sunfire]

looking at the constant itself - it is pressure multiplied by volume^γ

Because all states of the system are equal to this constant, it has to have a meaning... if it is constant energy, the units won't match due to the gamma exponent

E.g. it is not energy, but something else. Any ideas?

 

[Sunfire]

gamma is related to the type of gas... (mono- diatomic etc) probably then the constant is

const = (pV) V^γ-1

e.g. energy multiplied by some volume fraction

probably energy density of sorts?

 

[Jasso]

The pV^γ=const describes the relation between the pressure and the volume of a gas while it undergoes an adiabatic process. The exponent γ is the ratio between the specific heats at constant pressure and volume, C_p / C_v. That means that the dimensions of the constant can be different for different systems.

In other words, if you put a system into a specific state with pressure P and volume V, then adiabatically change the pressure or volume, then the new pressure P` and volume V` will be related to the original by PV^γ = const = P`V`^γ.

 

[Studiot]

The other respondents are correct in that you should be focusing on the exponent, gamma, rather than the constant C.

It is an experimental fact that for many processes there is both change of pressure and of volume for the working substance (which does not have to be a gas)

Thus the laws relating to constant pressure or constant volume processes are not applicable.

If we experimentally plot a process on a PV diagram we find that very often it is of the form

PVⁿ = C

(Note here n is not gamma)

It is better to plot logP against logV this yields a straight line of slope -n

logP = log C - nlogV

Which is a straight line of slope -n and intercept logC on the logP axis.

So this is the meaning of the constant C - It is a quantity inherent in the system that is preserved in any such change.

This type of process is called the Polytropic Process.

You should look it up.

Please also note that

Putting n = 0 yields a constant pressure process (horizontal line on a PV diagram)

Putting n = ∞ yields a constant volume process (vertical line on a PV diagram)

It is usual for the change in pressure to be in the opposite direction from the change in volume, in which case n is positive and the slope is negative.
However it is possible to devise a system in which n is negative.

Gamma is a particular case of n for an adiabatic process. It can be shown that gamma is the ratio of the specific heats for an ideal gas.

γ = C_p/C_v

 

[Sunfire]

thank you, I appreciate the points above and they are valid.

Studiot,
"So this is the meaning of the constant C - It is a quantity inherent in the system that is preserved in any such change."

This is EXACTLY what I am interested in. What is this inherent quantity? Does it have a name? How can it be determined for a given system, etc. My feeling is that C is a fundamental quantity for a given system (or a gas) and understanding its meaning can be very useful in understanding all other states the system can be in.

 

[Studiot]

No I'm sorry it doesn't have a name or even a particular value.

It's just the antilog of the y intercept on a logp v logV diagram.

Since log1 = 0 the (antilog)value of C gives the base pressure for unit volume in the polytropic law.

Since both pressure and volume vary with the polytropic law you cannot assign anything further of significance to it as you could to the constant in the special case of Boyles law.

 

[boneh3ad]

Unless I am going crazy here, isn't the relationship in question actually for an isentropic process (adiabatic and reversible)?

 

[Studiot]

Unless I am going crazy here, isn't the relationship in question actually for an isentropic process (adiabatic and reversible)?

Is there something that need further explanation here? The adiabatic equation is just a special case of the more general polytropic process.

 

[Q_Goest]

Studiot pointed out that PVⁿ = constant is a polytropic process. It would help to point out that a polytropic process (almost?) always refers to a process that a gas undergoes. Wikipedia suggests that one could apply it to a liquid or a solid but I think that's very uncommon.

A polytropic process for a gas is one in which a gas is compressed or expanded, heated or cooled. RoyMech.co.uk has a decent article on it.

An adiabatic process is simply one that has no heat entering or leaving the gas, so the only energy going into or coming out of the gas is work. If there's no heat transfer into or out of the gas, and work is being done on or by the gas, it is also an isentropic process. For an adiabatic process, the polytropic exponent, n, becomes the ratio of specific heats for the gas. So the equation would become:
PV^k = constant

Note that k is the ratio of specific heats, C_p/C_v and also sometimes called "γ". This is a special case of the polytropic process since the exponent n, can be any value.

Some of the other values for n were pointed out by Studiot and can be found on the Wikipedia article. Another one that's of interest is when n = 1. In that case, the temperature of the process does not change. It is an isothermal process. So if the gas is being compressed or expanded and the gas remains at a constant temperature, heat has to be removed (during compression) or added (during expansion) to keep the temperature constant.

A polytropic process and the general equation that describes it (PVⁿ = C) is just a general equation that can be used to describe a wide variety of different processes depending on whether the gas is compressed or expanded, heated or cooled.