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

Question

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?

Answer 1

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:

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

Answer 2

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:

$$\frac{V}{n}=\frac{V_i}{n_i}\implies\bar{V}=\bar{V_i}\implies\upsilon_i=x_i$$