Equation of state
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. 
Re: Equation of state
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. 
Re: Equation of state
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. 
Re: Equation of state
For ideal gases, Pv=nRT and therefore PvnRT=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)=PvnRT. There is another function f* with f*(P,v,T)=1: f*(P,v,T)=PvnRT+1 As you can see, that function does not help  it just gives an additional constant. Nonideal gases have more complicated laws, but the idea is the same. Edit: Oh, forgot to add n. 
Re: Equation of state
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Re: Equation of state
Since you know PV= RT where R is the gas constant, with P, V and T variables, you know that
[tex]\frac{{PV}}{T} = R = {\rm{a}}\;{\rm{constant}}[/tex] We can rearrange this [tex]\left( {\frac{{PV}}{T}  R} \right) = {\rm{0}}[/tex] 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. 
Re: Equation of state
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R = universal gas constant = 8.3145 J/mol K http://hyperphysics.phyastr.gsu.edu...idegas.html#c1 http://hyperphysics.phyastr.gsu.edu/hbase/hframe.html You need to specify the number of moles. 
Re: Equation of state
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In which case you would use the equation PV = nRT. 
Re: Equation of state
Thank mfb ,I totally understand.
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Re: Equation of state
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What are you actually seeking to know? 
Re: Equation of state
Understand already . Thank you

Re: Equation of state
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The proper approach would be [itex]PV \propto T[/itex]. ##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. 
Re: Equation of state
##[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. 
Re: Equation of state
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Re: Equation of state
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. 
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