We built a machine that can turn lead into gold. Unfortunately, it’s only at the rate of 890000 atoms per second. To determine the viability of investing in my own Large Hadron Collider, as well as to test my fading high school chemistry knowledge, I figured out the following.
We built a machine that can turn lead into gold. Unfortunately, it’s only at the rate of 890 000 atoms per second. To determine the viability of investing in my own Large Hadron Collider, as well as to test my high school chemistry skill, I figured out the following:
First, a standard gold bar weighs about 12400 g, and since gold’s molar mass is 196.97 g/mol and Avogadro’s number is \(6.022\times10^{23}\) atoms/mol, the total atom count is \(\frac{12\,400}{196.97} \times 6.022 \times 10^{23} \approx 3.79 \times 10^{25}\text{ atoms}\).
At our machine’s throughput of 890000 atoms/s, the time required is \(\frac{3.79 \times 10^{25}}{8.90 \times 10^{5}} \approx 4.26\times10^{19}\text{ seconds} \approx \frac{4.26 \times 10^{19}}{3.15 \times 10^{7}}\text{ years} \approx1.35 \times 10^{12}\text{ years}\).
Next, since gold goes for about \$3328 per troy ounce (31.1035 g), that’s roughly \$107 per gram. Each atom has mass \(\frac{196.97\text{ g/mol}}{6.022\times10^{23}} \approx 3.27\times10^{-22}\text{ g}\), so at 890 000 atoms/s over a day (\(8.64 \times 10^{4} s\)), we produce \(8.90\times 10^{5}\times 8.64\times 10^{4}\times 3.27\times 10^{-22} \approx2.52\times10^{-11}\text{ g}\), worth \(2.52 \times 10^{-11}\times107 \approx 2.7\times10^{-9}\text{ USD/day}\).
Finally, to hit a target of $1 000 per day, let \(R\) be the required atoms‐per‐second rate. Since revenue scales as \[ R\times 8.64\times 10^{4}\times 3.27\times 10^{-22}\times107 =R\times 3.02\times10^{-15}\quad(\text{USD}/\frac{\text{atoms}}{\text{s}}), \] we solve \(1000 = R\times3.02\times10^{-15} \quad\Longrightarrow\quad R\approx3.31\times10^{17}\text{ atoms/s}\). That’s about \(3\times10^{11}\) times faster than our current 890000 atoms/s machine.