1 a) In the first century BC the Roman architect, Vitruvius, related a story of

Question

1

a)

In the first century BC the Roman architect, Vitruvius, related a story of how Archimedes uncovered a fraud in the manufacture of a golden crown commissioned by the king of Syracuse. The crown would have been in the form of a wreath. Suspecting that the goldsmith might have replaced some of the gold given to him by an equal weight of silver, the king asked Archimedes to determine whether the wreath was pure gold. And because the wreath was a holy object dedicated to the gods, he could not disturb the wreath in any way. How could he do this?

Archimedes is credited with the discovery of the concept of displacement. (The story being that he stepped into his bathtub and it overflowed, whereupon he jumped out of the tub naked and ran down the street naked, screaming “Eureka”.) So, here’s what he found: the mass of the crown is 1.00 kg; the design of the crown is such that a direct volume cannot be calculated; when placed in a cylinder containing 400.00 mL, the new volume reads 464.8 mL. If the density of gold is 19.3 g/mL and the density of silver is 10.5 g/mL, can you determine if the crown is gold or a mixture of gold & silver?
(source https://www.math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html)

b)

The density of a mixture of 2 materials can be found if the densities of the 2 materials are known. The calculation may seem easy – just average the 2 values, but it is not quite that straight forward. Density is mass/volume, so for any mixture the total mass of the components and the total volume must be found. Determining these numbers depends on what is known about the mixture. If the mixture was created by mixing a certain number of grams of the material, then the mass is immediately known. The volume can then either be measured or, if the densities are known, as a calculation from the mass and the density. If volumes were added together, then the volumes are assumed to be additive and the mass can be found from the density and volume.

This all works fine if masses and volumes are additive. It is not very often that you find that the masses are not additive, but there are times when volumes are not. For example, ethanol and water are so highly attracted to each other and so soluble, the final volume of a mixture is actually less than the 2 volumes combined.

Find the density of a mixture of 70 mL of water (density = 1.00 g/mL) and ethanol (.789 g/mL) if when added the final volume is reduced to 96.8 mL.

Explanation / Answer

Mass of the crown=1kg or 1000g

Volume of a material of irregular shape= volume of water displaced by it when it is completely immersed in water.

Therefore volume of the crown= v(displaced volume)=464.8mL-400mL

Density of gold(dg)= 19.3g/mL

Density of silver(ds)= 10.5g/mL

Density of crown=1000g/64.8mL=15.432g/mL

Since the density of the crown is less than the density of pure gold, it is a mixture of gold and silver

Hence the density of the mixture (Dmix) is given by the formula

Dmix=total mass/[mass a/density a +mass b+ density b]

Step 1 calculate total mass

Here let ‘a’ be water and ‘b’ be ethanol.

Total mass= mass of water+ mass of ethanol

Mass of water can be calculated from the volume taken and given density

Using the formula Density=mass/volume

We know that 1L of water= 1kg of water

Therefore 1000mL of water=1000g of water

Hence 70mL of water= 70g of water

Let us consider that 30mL of ethanol is added(total volume(70+30=100 which gets reduced to 96.8mL)

Density of ethanol= 0.789g/mL

1mL of ethanol= 0.789g

Therefore 30 mL of ethanol=0.789x30/1=0.0263g

Total mass of water + ethanol=70g+0.0263g = 70.0263g

Step 2 calculate mass a/density a

Mass of water/ density of water= 70g/1[g/mL]=70

Step 3 calculate mass b+ density

mass of ethanol/desity of ethanol= 0.0263g/0.0789g/mL=0.033

Step 4 calculate Dmix using the formula

Dmix=Total mass/70+0.033= 70.0263/70.033=0.9999g/mL

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