Update: this is a nice story, but somebody pointed out that the glass of the Sport Watch is of a different material. This is not taken into account in the calculation, which must be wrong.
Last week Apple revealed more details of the new Apple Watch. There has been an ongoing discussion about the materials used, especially regarding the gold model. Is it the new light weight gold that Apple patented? How much gold is used in the case? And what is the cost of the gold?
Apple has released information about the weight of the watches (without bands) (nicely summarized by Rob Griffiths). Using the available data a reasonable estimation of the density of the gold alloy can be made, making a comparison with traditional gold alloys possible.
With this Numbers spreadsheet you can play around with the numbers yourself.
Summary
The density of the rose gold alloy is calculated to be between 12.5 and 13.0 g/cc. This is lower than the typical density of rose gold alloy, 15.1 g/cc. The calculation assumes the volume of the metal parts of the different models to be the same. If the actual volume of the gold case is smaller, the density would be higher. To explain the difference between the calculated and typical density the volume has to be smaller by 15 %. It is unlikely that Apple wouldn’t take advantage of this reduction for the aluminium and steel watches as well. It is more likely that Apple reduced the volume by a smaller percentage, in combination with a less dense alloy, using the technique described in the Apple patent.
Gold alloys
An alloy is a mix of a metal with other metals/materials. Alloys are designed to have better properties than the constituent materials: stronger, more resistant (against rust), more beautiful, cheaper, lighter (less dense) etc. In this post we are interested in the last property: density.
Most gold we encounter is in an alloy because pure gold is rather soft. By adding silver and copper it is made stronger. The karat is used to indicate the purity of gold. 24 karat is 99.9 mass-% pure gold. 18 karat gold (used in the Apple Watch) contains between 75 and 79 m% gold (I’ll use 75 m% in this post). The remaining 25 m% is other material.
The density of alloys can be approximated using the densities and mass fractions of the original materials, as explained by Dr. Drang.
[A]ssume a conventional 18k gold alloy with 75% gold (19.3 g/cc), 15% silver (10.5 g/cc), and 10% copper (8.96 g/cc) by weight. The alloy will have a density of […] 15.6 g/cc
Different gold alloys have different densities. According to Wikipedia 18K rose gold (one of the two golds used by Apple) contains 75% gold, 22.25% copper and 2.75% silver. This has a density of 15.1 g/cc, 0.5 g/cc lower than the earlier example.
The patent application of Apple goes a step takes this to the extreme:
Now assume an 18k gold with 75 % gold and 25 % boron carbide by weight (that’s one of the ceramics mentioned in Apple’s patent). Boron carbide has a density of 2.52 g/cc, so a gold/boron carbide metal matrix composite would have a density of about […] 7.24 g/cc.
The idea is that the gold content (karat) is measured in mass-percentage, but that you have to fill a volume. Most of the volume is filled with boron carbide, which doesn’t weight very much. 25 m% boron carbide is an extreme example: 7.24 g/cc is less than stainless steel (8.0 g/cc) or copper (9.0 g/cc).
An interesting but inconclusive bit of information is the Jony Ive video from Apple. Ive says that “precise adjustments in the amount of silver, copper and palladium in the alloy result in very specific hues of yellow and rose gold”. Are these the only materials in the alloy, or are these the four materials responsible for the hues?
How do we know if Apple used a fancy alloy? Table 1 shows the densities for different gold alloys. The data for pink, rose and red gold are from Wikipedia. For the traditional gold, copper and maybe-something-else alloy the density is at least 15.0 g/cc (red gold). If the density of the gold alloy used by Apple is below that, they must have done something special.
Another thing we can learn from Table 1 is that to obtain rose gold a fairly high amount of copper is needed. The rose gold color is also very sensitive to the amount of copper, a 2 m% difference changes the color significantly. Therefore, it is not possible to add significant amounts of boron carbide, certainly not 25 m%. The lower part of the table shows possible alloys with 2 to 5 m% boron carbide. The choice for the silver and palladium content is arbitrary, I chose a low value because both silver and palladium are denser than copper.
The data in Table 1 gives us an idea of what we are looking for. If the density is around 15 g/cc it is unlikely that a fancy alloy was used. However, if a fancy alloy was used we shouldn’t expect to obtain a ridiculous density of 7.24 g/cc, but something in the range of 11-14 g/cc or so. In this post I will calculate the density and discuss how accurate the calculation is.
| Gold (m%) | Silver (m%) | Copper (m%) | Palladium (m%) | Boron carbide (m%) | Density (g/cc) | ||
|---|---|---|---|---|---|---|---|
| Pink gold | 75 | 5 | 20 | 0 | 0 | 15.16 | |
| Rose gold | 75 | 2.75 | 22.25 | 0 | 0 | 15.08 | |
| Red gold | 75 | 0 | 25 | 0 | 0 | 14.98 | |
| Possible alloys | 75 | 0.5 | 22 | 0.5 | 2 | 13.84 | |
| 75 | 0.5 | 21 | 0.5 | 3 | 13.32 | ||
| 75 | 0.5 | 20 | 0.5 | 4 | 12.83 | ||
| 75 | 0.5 | 19 | 0.5 | 5 | 12.38 | ||
| Density (g/cc) | 19.3 | 10.5 | 8.96 | 12.02 | 2.52 |
Table 1: Densities for different alloys. The data for pink, rose and red gold is from Wikipedia, the “possible alloys” are speculation.
Method
The calculation is based on the assumption that the weight different between the different models (aluminium, steel, gold) is only due to the different density of the metal. The weight of everything that is not metal (internals like electronics etc) is the same for the models. The volume of the metal (the case, crown etc) is also the same for all models. Of course, the weight of the non-metal internals and the volume of the metal case are different for the two watch sizes.
Using the weight difference between the aluminium and steel models we can, using the density of the materials, calculate the weight of the internals and the volume of the case. Using the weight of the gold watches we can then calculate the density of the gold.
Material properties
To do the calculation we need to know the densities of the materials. The Apple website writes that 316L stainless steel is used for the steel watches, this has a density of 8.0 g/cc.
For the aluminium case it is more difficult. Greg Koenig writes how the Apple Watch is made. Regarding the aluminium alloy he writes:
‘With the Watch, Apple has upgraded from 6000 series alloy compositions (using magnesium and silicon) to a custom 7000 series alloy that relies on zinc. The closest commercial equivalents would be the 6061 aluminum alloy (the world’s most common manufacturing material) and 7075 aluminum – the comparison between the two tracks very precisely with Jony Ive’s language about having “custom designed a new alloy that it 60% stronger, but just as light.”’
The densities of these materials are 2.81 g/cc for 7075A and 2.7 g/cc for 6061 aluminium. I think 2.7 g/cc is a reasonable choice.
Calculation
To calculate the volume of the case, the weight m is divided by the density p:
V(case) = m(case) / p
We don’t know the weight of the case, but we know:
m(total) = m(case) + m(internals)
Apple has given m(total) and we assumed m(internals) was the same for cases of the same size. We get:
(m(total aluminium) – m(internals)) / p(aluminium) = (m(total steel) – m(internals)) / p(steel)
We can solve this for m(internals): 17.4 and 19.8 g for the 38 and 42 mm watches respectively. Presumably the larger watch has a larger battery.
This means the weight of the gold cases is 36.6 g and 47.2 g for the 38 and 42 mm sizes respectively. Using a gold price according Wolfram Alpha of 37.23 USD/g, this means the 18 karat gold cases cost about 1050 and 1350 USD.
The volume of the cases are 2.8 and 3.8 cc respectively. Given the weight of the gold cases and these volumes, the density of rose gold is between 12.5-13.0 g/cc, for yellow gold it is between 13.0-13.3 g/cc.
Discussion
In this section I will discuss how accurate the result is. Since I only have a reference density for rose gold, I will only use that in the discussion below, unless otherwise noted.
Different gold alloy densities for the two watch sizes
Because Apple rounded the watch weights to grams the gold alloy density differs slightly between the two watch sizes. We can get a feeling of the error by looking at the extremes: the cases where the weight difference between the aluminium and steel watch are the largest and where it is the lowest. This is shown in Table 2. A similar problem exists for the weight of the gold watches, although the effect of this is smaller, as shown in Table 3. A density between 12.5 and 13.0 g/cc seems realistic.
| Model | m(total aluminium) (g) | m(total steel) (g) | Gold density (g/cc) |
|---|---|---|---|
| 38 mm | 25 | 40 | 13.0 |
| 24.5 | 40.5 | 12.5 | |
| 25.5 | 39.5 | 13.5 | |
| 42 mm | 30 | 50 | 12.5 |
| 29.5 | 50.5 | 12.2 | |
| 30.5 | 49.5 | 12.9 |
Table 2: Variation in gold density because of rounding errors in watch weight.
| Model | m(total rose gold) (g) | Gold density (g/cc) |
|---|---|---|
| 38 mm | 54 | 13.0 |
| 53.5 | 12.8 | |
| 54.5 | 13.1 | |
| 42 mm | 67 | 12.5 |
| 66.5 | 12.4 | |
| 67.5 | 12.6 |
Table 3: Variation in gold density because of rounding errors in watch weight of the rose gold models.
The volume for the case is the same for all models
The central assumption in my calculation is that the volume of the cases is exactly the same. This is however incorrect for the gold case. At the end of Greg Koenigs article you can clearly see that the inside of the aluminium and gold cases are not the same. The video for the steel watch shows that the case is very similar to the aluminium one. Table 4 shows that differences in case volume have a significant effect on the gold alloy density. This makes the question whether the volume of the gold is larger or smaller relevant.
| Gold density for case volumes | ||||||
|---|---|---|---|---|---|---|
| Model | 85% | 90% | 95% | 100% | 105% | 110% |
| 38 mm | 15.2 | 14.4 | 13.6 | 13.0 | 12.3 | 11.8 |
| 42 mm | 14.7 | 13.9 | 13.2 | 12.5 | 11.9 | 11.4 |
Table 4: Variation in gold density because of differences in case volume. For example: if the case volume is reduced to 85 % of the aluminium/steel case, the rose gold density would be 15.2 g/cc instead of 13.0 g/cc for the 38 mm watch.
Apple may have added more material to make the watch stiffer. This would increase the volume of the case and decrease the calculated density of the gold alloy. Most of the material seems to be in the sides of the case, not the underside. This would mean that the volume increase is relatively small.
On the other hand, it makes sense to reduce the volume of the gold case. Gold is a dense (and expensive) material and removing some of it makes the watch lighter (and cheaper). The question is why Apple wouldn’t try to achieve the same volume reduction for the other watches. It may have to do with the complexity/cost of the production, but it is also less relevant for less dense materials. Table 5 shows how a change in case volume would affect the weight of the total watch. For the 38 mm aluminium watch a 15 % reduction in case volume would result in a 1.1 g lighter watch (4.4 %), the 42 mm steel watch would be 4.5 g lighter (9.0 %).
| Watch weight for case volumes | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| M(internals) (g) | M(case) (g) | 85% | 90% | 95% | 100% | 105% | 110% | ||
| Aluminium | 38 mm | 17.4 | 7.6 | 23.9 | 24.2 | 24.6 | 25.0 | 25.4 | 25.8 |
| 42 mm | 19.8 | 10.2 | 28.5 | 29.0 | 29.5 | 30.0 | 30.5 | 31.0 | |
| Steel | 38 mm | 17.4 | 22.6 | 36.6 | 37.7 | 38.9 | 40.0 | 41.1 | 42.3 |
| 42 mm | 19.8 | 30.2 | 45.5 | 47.0 | 48.5 | 50.0 | 51.5 | 53.0 | |
Table 5: The weight of the aluminium and steel watches for a variation of case volumes. For example: if the case can be reduced by 15 %, the total weight of the watch (m(internals) + 0.85 x m(case)) would be 23.9 g instead of 25 grams.
One unknown is the crown. The aluminium crown seems to be from solid aluminium. The crowns of the steel and gold watches have a colored… thingy on the side. It is unclear what is on the inside. I don’t know how this affects the weight.
Density of aluminium
A third unknown is the actual density of the aluminium. I used 2.7 g/cc. Table 6 shows how the gold alloy density varies aluminium density. Given the other errors I think it is negligible.
| Model | Aluminium density (g/cc) | Gold density (g/cc) |
|---|---|---|
| 38 mm | 2.7 | 12.95 |
| 2.6 | 13.04 | |
| 2.8 | 12.85 | |
| 42 mm | 2.7 | 12.51 |
| 2.6 | 12.59 | |
| 2.8 | 12.42 |
Table 6: Variation in gold density because of differences in aluminium density.
Conclusions
The calculations show that there is a difference between the weight of the internals of the 38 and 42 mm watches: 17.4 and 19.8 g respectively. The gold cases weigh 36.6 and 47.2 g, at the current gold price this means about 1050 and 1350 USD of pure gold is used. The density of the rose gold alloy is calculated to be between 12.5 and 13.0 g/cc. The calculated density is significantly lower than the 15.1 g/cc of typical rose gold.
The main uncertainty in the density calculation is the volume of the gold case. You can equally argue that Apple made the volume smaller (to reduce the weight) or larger (for extra stiffening). To explain the obtained values using the volume alone, Apple would have to make the gold case volume 15 % smaller than for the steel and aluminium models. Because aluminium and steel are less dense than gold a small decrease in volume is not as important, but if they could have shaved of 15 % of the case of the 38 mm aluminium model, it would be 1.1 g lighter (4.4 %). The 42 mm steel case would be 4.5 g lighter (9.0 %). It is unlikely that Apple would not take advantage of such a possibility to reduce the weight.
Alternatively, Apple could have used something like boron carbide in the alloy. To reach a density between 12.5 and 13.0 g/cc about 4 m% boron carbide is needed (Table 1). This would mean the copper content has to be low, making the gold more pink than rose.
The most likely explanation is that Apple used a combination of volume reduction for the gold case and a fancy gold alloy.