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What is $vdp$ work and when do I use it?

I've read this answer already and I understand that it is the work to add in more substance over a pressure difference.

When we do the derivation for $$PV^{gamma} = C$$, In on of the steps we write

$$ PV=nRT$$ and, then we write that in differential form

$$ dp V + P dv = nR dT$$

So what would this Vdp term suggest in this context for say an ideal gas trapped in a spherical balloon?

I know that $P dv$ Is the energy in expanding the boundary i.e enlarging the sphere and my guess is that $Vdp$ is the work needed to add in more air to the balloon

Qmechanic
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tryst with freedom
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1 Answers1

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$Vdp$ work is called flow work and applies to open systems, i.e., systems that permit mass to enter and exit the system. It’s the work required to move a volume of mass into and/or out of a system where a difference in pressure exists across the system boundary. It could apply when adding air to the balloon as you surmised. But it would not apply after the air is trapped in the balloon because then it becomes a closed system. Then only $pdV$ (boundary work) applies to the closed system.

That being said, $Vdp$ can have a different meaning than flow work in the case of a closed ideal gas system. Though not applicable to your balloon example (whose boundary is not rigid), it can refer to an isochoric (constant volume) heat addition or subtraction which results in an increase or decrease in the pressure and temperature of the gas.

The differential form of the ideal gas law applies regardless of the process, so it can be used for the derivation of any process involving an ideal gas. The potential physical significance of the $VdP$ term for a closed ideal gas system, where flow work and $PdV$ work is not involved, is for an isochoric process.

Hope this helps

Bob D
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  • So wouldn't that imply the Vdp term is 0 ? Then why does the derivation for $pv^{\gamma}$ work? – tryst with freedom Apr 20 '20 at 05:18
  • Sorry I missed you talking about adding more air to the balloon at the end of the post. That is $Vdp$ work because then it’s an open system. I will update my answer – Bob D Apr 20 '20 at 05:54
  • @DDD4C4U I have updated my answer after realizing my original answer didn't completely address your question – Bob D Apr 20 '20 at 08:50
  • How does it matter which step that we account that Q=0? isn't it a bit weird that the arguement would break if one changes the order of ideas? – tryst with freedom Apr 20 '20 at 08:54
  • @DDD4C4U The argument doesn't break. I didn't mean to imply that. The differential form of the ideal gas law applies regardless of the process. That form gives us a $VdP$ term. You asked for the physical significance of the term. I'm saying the term applies for an isochoric process, a constant volume heat addition/extraction. I will revise my answer to clarify. – Bob D Apr 20 '20 at 15:45
  • Why doesn't the VdP work show up in say an adiabatic expansion say?There is a change in pressure there too? – Schwarz Kugelblitz Apr 23 '20 at 21:33
  • @Schwarz Kugelblitz sorry just saw your comment. It doesn’t show up because $VdP$ work is called flow work and only applies to open systems, i.e., systems where work is involved moving mass into and out of the system. The OP’s question is about an adiabatic process for a closed system, i.e., no mass flow. – Bob D Apr 27 '20 at 20:50
  • @BobD isn't PV the flow work involved in an open system? – GRANZER Jul 11 '22 at 06:27
  • @GRANZER I don’t understand your question. PV is simply the product of pressure and volume – Bob D Jul 11 '22 at 08:59
  • Yes, and the product is the flow work done by an element entering a Open system/Control Volume. Ref:http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?topic=th&chap_sec=04.1&page=theory – GRANZER Jul 11 '22 at 09:02
  • @BobD My question is what is the difference b/w PV and Pdv if both are flow work? – GRANZER Jul 11 '22 at 09:03
  • @GRANZER $Pdv$ is not flow work. It is boundary work. It is the work done expanding or contracting the boundary of a closed system. Flow work is the work associated with moving fluid across the boundary of an open system where there is a pressure difference. – Bob D Jul 11 '22 at 14:04
  • @BobD I am sorry I meant Vdp. Difference b/w PV vs VdP as both are called flow work. – GRANZER Jul 11 '22 at 15:29