How to define percentage by volume for ideal gas and Why percentage by volume is equal to percentage by mole? [closed]








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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)


V=Kn


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?




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:


$$\frac{n\_i}{n} =\frac{\frac{p\_iV}{RT} }{ \frac{pV}{RT}}=\frac{p\_i}{p}\tag{6}$$


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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:


$$PV\_i=n\_iRT$$


$$PV=nRT$$


Dividing the first equation by the second:


$$\frac{PV\_i}{PV}=\frac{n\_iRT}{nRT}$$


Cancelling all equal terms:


$$\frac{V\_i}{V}=\frac{n\_i}{n}$$


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


$$\upsilon\_i=\frac{V\_i}{V}$$


$$x\_i=\frac{n\_i}{n}$$


It follows that both fractions are equal:


$$\upsilon\_i=x\_i$$


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


$$\upsilon\_i=x\_i\frac{M\_i}{M}\frac{\rho}{\rho\_i}=x\_i\frac{C}{C\_i}=x\_i\frac{\bar{V\_i}}{\bar{V}}$$


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:


$$\frac{V}{n}=\frac{V\_i}{n\_i}\implies\bar{V}=\bar{V\_i}\implies\upsilon\_i=x\_i$$


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