Question

A liquid mixture of volume v has two liquids as ingredients with densities $\alpha ,\beta $. If the density of the mixture is $\sigma $, then the mass of the first liquid in the mixture is:
$\begin{align}
  & a)\dfrac{\alpha V(\sigma \beta +1)}{\beta (\alpha +\sigma )} \\
 & b)\dfrac{\alpha V(\sigma -\beta )}{(\sigma +\beta )} \\
 & c)\dfrac{\alpha V(\beta -\sigma )}{\beta -\alpha } \\
 & d)none \\
\end{align}$

Answer 1

Hint: Assume the mass of the two liquids. Then we can write the mass in terms of the density of the liquid mixture. Next, calculate the density of the mixture as the total mass divided by the total volume. Here, mass is the same as the volume should be converted in terms of mass and density.
Formulas used:
$d=\dfrac{m}{v}$

Complete answer:
Let us assume the masses of the two liquids as ${{m}_{1}},{{m}_{2}}$ respectively. The total volume and net density if the mixture is given as $V,\sigma $ respectively.
Now let’s calculate the total mass,
\[\begin{align}
  & {{M}_{1}}+{{M}_{2}} \\
 & V\sigma ={{M}_{1}}+{{M}_{2}} \\
 & {{M}_{2}}=V\sigma -{{M}_{1}} \\
\end{align}\]
Now, the total density is calculated as,
$\begin{align}
  & T=\dfrac{{{M}_{1}}+{{M}_{2}}}{\dfrac{{{M}_{1}}}{\alpha }+\dfrac{{{M}_{2}}}{\beta }} \\
 & \\
\end{align}$
Substitute the value of second liquids mass in the above equation, we get,
$\begin{align}
  & \sigma =\dfrac{{{M}_{1}}+(V\sigma -{{M}_{1}})}{\dfrac{{{M}_{1}}}{\alpha }+\dfrac{V\sigma -{{M}_{1}}}{\beta }} \\
 & {{M}_{1}}=\dfrac{\alpha V(\beta -\sigma )}{\beta -\alpha } \\
\end{align}$

Therefore, the correct option is option c.

Additional information:
When two liquids of two different densities are mixed, they separate when we stop mixing them. The heavier liquid goes to the bottom as the density is high and the lighter liquid will deposit at the top layer as its density is less. The densest liquid is molasses and the least dense is the alcohol. The density of the final mixture after mixing different liquids with different densities, will be the ration between the total mass of the liquids and the total volume occupied by the liquids. After settling down, we will observe that the least dense liquid will stay on the top and the densest one at the bottom.

Note:
If two liquids have the same volume, it doesn’t mean they have the same density. The density is equal to the mass by the volume. So, if the masses of the liquids are also the same, then we can say the densities of the liquids are the same. In the above question, we found the mass of the liquid in known terms only.