Additional Math Pages & Resources

Monday, December 19, 2011

Dilution and Density

In the most recent series of math blogs I have been talking about drinks - water, juices, tea, etc.

What are drinks?

Drinks are liquids that we consume to sustain or entertain ourselves. They are carefully-contrived blends of various substances (carbonation, fruit or vegetable juice, water, sugar, etc.) that are wet, drinkable and nutritious (or intoxicating), etc.

As I discussed earlier, whenever companies create blends or combinations, government gets interested. Blending may be used to disguise fraudulent or unhealthful products. Thus we regulate what vendors can claim or sell. We often need some elementary school math skills to understand these regulations and the products being controlled.

Today I would like to turn your attention to a different type of "blend". I invite you to compare cranberry juice and gold.

Pure cranberry juice is extremely tart, sour, bitter and acidic. It's nearly impossible to drink without dilution and sweetening. We need to mix the juice with sweeter fruit juices or add water and sugar. Most cranberry juice products end up being about 25-30% cranberry juice.

 Now let's make a big mental jump from red to gold.

Pure gold (24 carat) is very valuable, very heavy and very soft. It's nearly impossible to use without dilution with silver, copper, nickel, etc. We need to mix the gold with other stronger substances, which make gold much stronger, a bit cheaper, and change its color. Many gold products are 18k gold, which is a blend (alloy) of 75% gold and 25% other materials.

ALLOY: A metallic substance made by mixing and fusing two or more metals, or a metal and a nonmetal, to obtain desirable qualities such as hardness, lightness, and strength.

OK, with all that background established, this week a brand-new gold alloy has appeared. It's called MAGIC GOLD. It was developed by Hublot, a Swiss watch company. I have visited the company, and met with its dynamic president, Mr. J-C Biver.

To clarify the magic, I will use a trick question "Which is heavier, a pound of feathers or a pound of lead?"  Of course the answer is "A pound of one is the same weight as a pound of the other."

This reveals the MAGIC at the heart of MAGIC GOLD. It weighs less than any other alloy of gold, yet it is still 75% gold. Why? How?

Here is the pound of lead part of the question:

The Swiss Federal Customs Administration (FCA) manages any claims made about gold, and Hublot quotes them when saying:
  1. MAGIC GOLD contains 750 grams of pure gold per 1000 grams of alloy (proportions by mass), thus FCA gave us permission to use a hallmark of 750/1000 (18 carat)
  2. MAGIC GOLD is a composite (alloy) made of pure gold [density 19.32 g/cm3] and boron carbide ceramic [density 2.5 g/cm3]
  3. The density of this new material is found with this equation:
((% gold x mass of gold) + (% ceramic x mass of ceramic) = (mass of the alloy) 
((0.3 x 19.32) + (0.7 x 2.5)) = (5.796 + 1.75) = 7.546 g  so density = 7.546 g/cm3
Now here is the pound of feathers.
  1. MAGIC GOLD is by volume 30% pure gold and 70% boron carbide (proportions by volume)
  2. A watch case in 18k gold that weighs 100g contains 75g of pure gold (75%)
  3. The same watch case in MAGIC GOLD weighs 48.68g (about half of 100g) and contains 37.38g of pure gold (about half of 75g)
  4. So, for the same volume of material, the watch case contains half as much gold by mass
  5. Total weight of the case is also half, but the proportion of pure gold remains at 75%
Where is the MAGIC here? The MAGIC is that Hublot can make watch cases and cuff links, rings, etc. using half of the gold but adding lots of research, ceramics, temperature and pressure. The resulting substance is stronger, more scratch-resistant than the old stuff and it's still 18k gold.

Sweet, eh?  Just like cranberry juice punch!