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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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Clemens Bartholdy
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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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