Equation of State: Understanding f(P,v,T)=0

[Outrageous]

I know Pv=RT, but I don't understand why my book said f(P,v,T)=0 mean?
Why the function of P,v,T equals to zero? Why not equal to a constant?
Thank you.

 

[mfb]

You could use any constant (with a modified function), it would not change anything. f(P,v,T)=0 implies f*(P,v,T)=c where f*=f+c.
=0 is chosen as 0 is a convenient value.

 

[Outrageous]

I am sorry.
I still don't understand why f(P,v,T)=c=0 ?
Can you give simple example which did not involve f()... or normal equation that show c=0?
Thank you.

 

[mfb]

For ideal gases, Pv=nRT and therefore Pv-nRT=0
That is a simple equation, and it shows that there is a function f with f(P,v,T)=0, namely f(P,v,T)=Pv-nRT.

There is another function f* with f*(P,v,T)=1:
f*(P,v,T)=Pv-nRT+1
As you can see, that function does not help - it just gives an additional constant.

Non-ideal gases have more complicated laws, but the idea is the same.

Edit: Oh, forgot to add n.

 

[Hetware]

I know Pv=RT, but I don't understand why my book said f(P,v,T)=0 mean?
Why the function of P,v,T equals to zero? Why not equal to a constant?
Thank you.

Please define you variables. From the definitions I am familiar with, you expression is wrong, unless you have made an unstated assumption.

 

[Studiot]

Since you know PV= RT where R is the gas constant, with P, V and T variables, you know that


$$\frac{{PV}}{T} = R = {\rm{a}}\;{\rm{constant}}$$


We can rearrange this


$$\left( {\frac{{PV}}{T} - R} \right) = {\rm{0}}$$

Isn't this now in the format you seek?

However noting your other threads about Van der Waal's equation I wonder if your book was leading up to some more complicated function of P, V and T such as VDW.

Incidentally the answer to your question about P, is that P is the real pressure exerted by the gas, not some equivalent pressure of an ideal or other gas.

 

[Hetware]

Since you know PV= RT where R is the gas constant, with P, V and T variables, you know that$$\frac{{PV}}{T} = R = {\rm{a}}\;{\rm{constant}}$$We can rearrange this$$\left( {\frac{{PV}}{T} - R} \right) = {\rm{0}}$$

Isn't this now in the format you seek?

However noting your other threads about Van der Waal's equation I wonder if your book was leading up to some more complicated function of P, V and T such as VDW.

Are you talking about the universal gas constant R? If so, you units are off.

R = universal gas constant = 8.3145 J/mol K

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.htmlYou need to specify the number of moles.

 

[Studiot]

You need to specify the number of moles.

Only if you are talking about a number of moles.

In which case you would use the equation PV = nRT.

 

[Outrageous]

Thank mfb ,I totally understand.

Please define you variables. From the definitions I am familiar with, you expression is wrong, unless you have made an unstated assumption.

Thank you Hetware, I wonder what is the definition.

However noting your other threads about Van der Waal's equation I wonder if your book was leading up to some more complicated function of P, V and T such as VDW..

Thanks. What do you mean by more complicated ? VDW=van der Waals?

 

[Studiot]

VDW=van der Waals?

Of course

What do you mean by more complicated ? van der Waals?

Isn't van der Waals complicated enough for you?

What are you actually seeking to know?

 

[Outrageous]

Understand already . Thank you

 

[Hetware]

Only if you are talking about a number of moles.

In which case you would use the equation PV = nRT.

Your units don't come out right if you use the standard definitions. Feynman makes that mistake in Vol 1, eq. 54.13 where it is of little consequence, but he uses the same flawed understanding in the discussion preceding Vol 1, eq. 47.24 where his statements are simply wrong. It's sloppy to omit variables without justifying doing so.

The proper approach would be $PV \propto T$. $PV=RT$ where $R$ is the universal gas constant is simply an incorrect proposition. One could also write $PV=CT$ declaring $C$ to be a constant.

Yes, these things do matter. I recall reading that chapter in Feynman back when I was first learning about thermodynamics and sound propagation. I was stumped by $PV=RT$. I didn't understand the justification for omitting the number of moles. I now realize that is because he never gave a justification for it.

 

[mfb]

$[R]=\frac{J}{mol K}$
$[PV]=\frac{J}{m^3}m^3=J$
The units match, if n is given as "x mol".

You can use the more fundamental Boltzmann constant, of course:
$k_b=\frac{R}{N_A}$ with the Avogadro number N_A.

$PV=Nk_bT$ where N is the number of molecules.

All equations are for ideal gases.

 

[Hetware]

$[R]=\frac{J}{mol K}$
$[PV]=\frac{J}{m^3}m^3=J$
The units match, if n is given as "x mol".

You can use the more fundamental Boltzmann constant, of course:
$k_b=\frac{R}{N_A}$ with the Avogadro number N_A.

$PV=Nk_bT$ where N is the number of molecules.

All equations are for ideal gases.

Thank you for supporting my position.

 

[Studiot]

Perhaps I should have specified R more thoroughly, but my excuse is that I was taught, nearly fifty years ago, that the gas constant, R is

Strictly R is the molar gas constant

(Formula edited to add missing index.)

R = 8.31 Joules °K^-1 moles^-1

and have been using it successfully ever since.

I see no inconsistency, perhaps you are referring to a different constant?

The dimensions of R have no bearing upon the answer to the original question which is why did the book state that a function of pressure, volume and temperature is zero rather than a constant.

I have shown a very simple mathematical manipulation of the ideal gas law to achieve this.

A physics motivation for this would be that R is the intercept on the PV/T axis of the PV/T v T graph ie the value of the constant at absolute zero. We cannot, of course, measure at this value but have to infer it from measurements at other temperatures an extrapolate back to the axis.