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QUESTION

# Consider butter (density=0.860g/ml) and sand (density=2.28g/ml). If 1.00ml of butter were mixed with 1.00ml of sand and mixed as thoroughly as possible, what would be the density of the resulting mixture?

"1.57 g/mL"

The idea here is that the of the mixture will reflect the contribution the volume of each substance has in the total volume of the mixture.

In your case, you have equal volumes of butter and of sand, which means that the density of the mixture will be split half way between the two densities of the components of the mixture.

Mathemcatically, you can prove this using

rho_1 = m_1/V -> the density of butter

and

rho_2 = m_2/V -> the density of sand

The volume V is the same for both substances, since you're mixing "1.00 mL" of each.

Now, the density of the mixture will be

rho_"mixture" = m_"total"/V_"total" " ", where

m_"total" - the total mass of the mixture, equal to m_1 + m_2; V_"total" - the total volume of the mixture, equal to V + V

This means that you have

rho_"mixture" = (m_1 + m_2)/(V + V) = (m_1 + m_2)/(2V)

rho_"mixture" = 1/2 * (m_1 + m_2)/V = 1/2 * (m_1/V + m_2/V)

But m_1/V = rho_1 and m_2/V = rho_2, so you have

rho_"mixture" = 1/2 * (rho_1 + rho_2)

Therefore,

rho_"mixture" = 1/2 * (0.860 + 2.28)"g/mL" = color(green)("1.57 g/mL")