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You know this equation : $$ρ_{mix} =\frac{2 × ρ_1 × ρ_2} {ρ_1 + ρ_2} $$ But where is it derived from?

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    $\begingroup$ In itself the equation doesn't make sense for the general case of combining two substances that have different densities because the resulting density should depend on what the added proportions are and there is no mention of that in your equation. I'm guessing that your equation is for either (1) adding two equal masses of different densities or (2) adding two equal volumes of different densities. Do you know which it is? $\endgroup$
    – user93237
    Nov 13, 2015 at 19:46
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    $\begingroup$ @SamuelWeir: the title does say ...mixed liquids of the same mass, so it's clearly (1) ;) $\endgroup$
    – Kyle Kanos
    Nov 13, 2015 at 20:10
  • $\begingroup$ @Kyle- Oops. Didn't look at the title.... $\endgroup$
    – user93237
    Nov 13, 2015 at 20:11
  • $\begingroup$ @SamuelWeir: Happens to the best of us ;) $\endgroup$
    – Kyle Kanos
    Nov 13, 2015 at 20:12
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    $\begingroup$ Just a fair warning: the equation assumes $V_{mix} = V_1 + V_2$ which is only true for ideal solutions see e.g. this Chemistry.SE post for more info $\endgroup$
    – pentane
    Nov 13, 2015 at 20:32

2 Answers 2

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Suppose you have two masses M1(=M) and M2(=M) with volumes V1 and V2, respectively. Then the total density is the total mass divided by the total volume. So $\rho_{mix}$=2M/(V1+V2). V1=M/$\rho_1$ and V2=M/$\rho_2$ so $\rho_{mix}$=2M/(M/$\rho_1$+M/$\rho_2$), which after canceling the M's and simplifying the expression is equal to what you wrote above.

Additional Note

By the way, if instead of considering mixing two equal masses of different densities, you consider mixing two equal volumes of different densities then the resulting equation for the total density is much simpler. It's just $\rho_{mix}=\rho_1+\rho_2$.

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How do you derive the formula of density of mixed liquids with the same mass?

We know that: $\rho_1=m_1/V_1$, $\rho_2=m_2/V_2$

It is also known that:

$\rho_{mix}=(m_1+m_2) / (V_1+V_2)$

As $m_1=m_2$ it means that $\rho_1*V_1=\rho_2*V_2$ and $m_1+m_2 = 2 * \rho_1*V_1$

Finally you get:

$ρ_{mix} =\frac{2 * ρ_1 * ρ_2} {ρ_1 + ρ_2}$

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