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The ideal gas equation is $$pV=nRT$$

Show that the pressure exerted by a fixed mass of gas is given by the equation $$p=\frac{\rho RT}{M}$$

Where $\rho$ is the density of the gas and $M$ is the mass of one mole of gas.

Proof: $\rho = \frac{m}{V}$ where $m,V$ is mass and volume of the gas respectively.

$$p=\frac{nRT}{V}=\frac{n\rho RT}{m}$$

But we know that, $m=nM \implies \frac{n}{m}=\frac{1}{M}$

$$\therefore p=\frac{\rho RT}{M}$$

(b) The Earth’s atmosphere may be treated as an ideal gas whose density, pressure and temperature all decrease with height.

In 1924, Howard Somervell and Edward Norton set a new altitude record when attempting to climb Mount Everest. They managed to climb to a vertical height of $8570 m$ above sea level by breathing in natural air. At this height, the air pressure was $0.35$ times the pressure at sea level and the temperature was $−33 °C$. At sea level, air has a temperature $20 °C$ and density $1.3~kg~m^{-3}$

Calculate the density of the air at a height of $8570 m$ at the time the record was set.

As I was just asked to prove the above equation, I am going to assume that it will be useful for this part of the problem.

So, we have (at sea level) $$p=\frac{1.3 \times R \times 293}{M_1}$$

and at our height of $8570m$, $$p=\frac{\rho \times R \times 240}{M_2}$$

Now onto my question. Would $M's$ in my two equations be equal? I.e. would the mass of one mole of air at sea level and at our height of $8750m$ be equal? If so, could I have an explanation why please?

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1 Answer 1

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Would the mass of ten apples change if you moved the apples to another location? The principle is essentially the same here - a mole of a gas is just a particular number of gas atoms.

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  • $\begingroup$ It is plausible that the composition of the atmosphere changes with altitude... $\endgroup$
    – Tofi
    May 21, 2022 at 5:56
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    $\begingroup$ @Tofi That's true. However, given the context and the apparent level of the question, I didn't think that was a possibility the questioner intended the students to explore. More importantly though, while what you say is plausible, it turns out not to be true; mixing in the lower atmosphere keeps the atmospheric composition essentially uniform up to an altitude around 100 km (see my answer here), far above the peak of Everest. $\endgroup$
    – J. Murray
    May 21, 2022 at 6:03

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