Is this relation in case of a mixture of ideal gases valid?

Question

I came across the following relation in a book (written as a fact, with no proof), for a mixture of ideal gases both at the same pressure and temperature. The final volume of the mixture of gases after mixing is: $$ V_\text{mix} = \frac {M_1V_1 + M_2V_2}{M_1+M_2} $$

where $M_1$, $M_2$ are the molar masses of the individual gases and $V_1$, $V_2$ are the respective volumes. I could not figure out its proof. Is this relation valid at all? Please do not confuse with a similar looking relation,with $ M $ corresponding to molarity, in case of a solution mixture.

Answer 1

The relation appears to be incorrect.

For ideal gas law states:

$$ pV = nRT = \frac{m}{M} RT $$ This can be expressed as: $$ VM = m \left(\frac{RT}{p}\right) $$

Now for mixture of two ideal gases at same pressure $(p)$ and temperature $(T)$: $$ m_\text{mix} = m_1 + m_2 \tag{mass conservation}$$ $$ m_\text{mix} \left(\frac{RT}{p}\right) = m_1 \left(\frac{RT}{p}\right) + m_2 \left(\frac{RT}{p}\right) $$ $$ V_\text{mix} M_\text{mix} = V_1 M_1 + V_2 M_2 $$ $$ V_\text{mix} = \frac{V_1 M_1 + V_2 M_2}{M_\text{mix}} \tag{1}$$

This is not same as the mentioned relation, $$ V_\text{mix} = \frac {M_1V_1 + M_2V_2}{M_1+M_2} \tag{2}$$

because this require $ M_\text{mix} = M_1 + M_2 $. Which is not correct!

To understand why, imagine mixting two gases such that $ M_1 = M_2 $ and $ V_1 = V_2 $, then equation (2) gives $ V_\text{mix} = V_1 = V_2 $ instead of the correct final volume of $ V_\text{mix} = 2 V_1 = 2 V_2 $

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I think the relation should have read:

$$ M_\text{mix} = \frac { M_1 V_1 + M_2 V_2 }{ V_1 + V_2 } $$ which follows from equation (1) and Amagat's law which states $V_\text{mix} = V_1 + V_2$.

Answer 2

This relation is not valid if the gases are assumed to be ideal. see Amagat's law of partial volumes Moreover, if the gases are non ideal, the relations between the volumes of the combining gases and the final resultant volume is very complex and the given relation is too simple and includes too less variables to even begin describing the actual non-ideal gas mixing.
It, however unlikely, might represent a certain special case that can only be judged from the context wherein the aforementioned equation was taken from.