Why do we define percentage by volume for an ideal gas in a closed container in spite of that we know the ideal gas will occupy the volume of the whole container?

I have seen many people who prove that % by volume is equal to the percentage by mol for an ideal gas (at constant temperature and pressure conditions). They follow the following procedure...

PV = nRT

V = nRT/P

(let RT/P = K as a constant as pressure and temperature are constants)


as volume is directly proportional to the number of moles and hence % by volume is equal to the percentage by mole...

But the doubt is here, when we write PV=nRT for a gas in a closed container then we write volume as the volume of the container then where do we define % volume for gas?


2 Answers 2


It is easy to remember that for ideal gases,

$$V_{\text{const }p} = \sum_{i=1}^{n}{V_i} = \sum_{i=1}^{n}{\frac{n_iRT}{p}}\tag{1}$$ $$p_{\text{const }V} = \sum_{i=1}^{n}{p_i} = \sum_{i=1}^{n}{\frac{n_iRT}{V}}\tag{2}$$

It can be also easily seen that

$$\frac{V_i}{n_i} = \left(\frac{RT}{p}\right)_\text{p = const}\tag{3}$$ $$\frac{p_i}{n_i} = \left(\frac{RT}{V}\right)_\text{V = const}\tag{4}$$

And the molar, pressure and volume ratios:

Constant pressure:

$$\frac{n_i}{n} = \frac{\frac{pV_i}{RT} }{ \frac{pV}{RT}} = \frac{V_i}{V}\tag{5}$$

Constant volume:

$$\frac{n_i}{n} =\frac{\frac{p_iV}{RT} }{ \frac{pV}{RT}}=\frac{p_i}{p}\tag{6}$$


Considering variables without subscripts represent total state functions (mixture), and variables with subscript $i$ represent the state functions of a single component $i$ in the mixture, at constant pressure and temperature for an ideal gas, according to Amagat's law, we have:



Dividing the first equation by the second:


Cancelling all equal terms:


Defining the volume fraction of $i$ and mole fraction of $i$, respectively:



It follows that both fractions are equal:


A second way of showing they are equal is using the relationship between both fractions:


Where $\bar{V}$ represents molar volume, $M$ molar mass, and $C$ molar concentration.

The molar volume of an ideal gas is equal to that of the mixture at constant temperature and pressure, so we get:


  • $\begingroup$ when you wrote PVi = nRT then why did you use total pressure instead of partial pressure i.e it should be PiVi = nRT and from that after dividing the two equations and applying dalton's law of partial pressure on gaseous mixture you will get Vi = V. $\endgroup$ Commented Nov 14, 2022 at 7:39
  • $\begingroup$ Amagat's law uses total $P$, while Dalton's law uses total $V$. $\endgroup$
    – Sam202
    Commented Nov 14, 2022 at 14:15

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