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So in a volume of Resnick halladay's book adapted by Amit Gupta. There was a following derivation of variation of atmospheric pressure with height.

here it is assumed temperature doesn't not vary with altitude

In the page 849, it was written from the ideal gas law $pV=RT $. Why not $nRT$? Wouldn't that be a more general case instead of using $n=$1 mole? Or does that mean anything about the slice of air being infinitesimally thin? I can't follow up

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    $\begingroup$ Either $V$ must be defined as the volume per mole or (more likely) this is an error in the book. $\endgroup$ Commented Jul 31 at 15:39
  • $\begingroup$ There was nothing specified about it . So I guess it was an error. $\endgroup$ Commented Jul 31 at 16:19
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    $\begingroup$ In that equation, V is supposed to be the volume per mole. This is the inverse of the molar density. The mass density $\rho$ is M/V. $\endgroup$ Commented Aug 1 at 10:39

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Be careful how each variable is defined. Ideal gas law is notorious for having a huge variety of forms.

It appears to mean

$$P \tilde V = R_u T$$

Where $\tilde V$ is molar volume (m$^3$/mol) or $V/n$ in more usual usage. Which leads to the next line

$$P \frac {\tilde M}{\rho} = R_u T$$

With molecular weight $\tilde M$ in kg/mol and density $\rho$ in kg/m$^3$.

As you said it's equivalent to setting $n=1$ mol, but that is still a general formula as long as you recall the appropriate mass or volume term is "per mole."

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