Tag: Zero-practical-application

A solid $7\text{ kg}$ sample of Astatine-210 (the most stable isotope) would generate a thermal output of approximately $233\text{ MW}$ $(233,000,000\text{ watts})$ at the moment of its creation. If the sample consisted of Astatine-211—the isotope commonly created in particle accelerators—the thermal output would climb even higher, reaching roughly $495\text{ MW}$ $^{[1, 2]}$.

To put this scale into perspective, a 1-litre block of astatine would produce as much heat as a small commercial nuclear reactor, entirely concentrated within a handful of matter.

But don’t take my word for it; LET’S USE FIRST PRINCIPLES!!!.

1. Count the Atoms

To find out how much heat is released, we first determine how many atoms are present in a $7\text{ kg}$ ($7,000\text{ g}$) sample using Avogadro’s number:

$$N = \frac{\text{Mass}}{\text{Molar Mass}} \times N_A$$

Using Astatine-210 ($210\text{ g/mol}$) $^{[2]}$:

$$N = \frac{7,000\text{ g}}{210\text{ g/mol}} \times 6.022 \times 10^{23}\text{ atoms/mol}$$

$$\approx 2.0073 \times 10^{25}\text{ atoms}$$

2. Determine the Decay Constant

Next, we calculate the decay constant ($\lambda$) using the half-life of Astatine-210, which stands at $8.1\text{ hours}$ ($29,160\text{ seconds}$) $^{[3]}$:

$$\lambda = \frac{\ln(2)}{t_{1/2}} = \frac{0.69315}{29,160\text{ s}}$$

$$\approx 2.377 \times 10^{-5}\text{ s}^{-1}$$

3. Calculate Total Radioactivity

The total radioactivity, or activity ($A$), measured in Becquerels ($\text{Bq}$, decays per second) is the product of the number of atoms and the decay constant:

$$A = \lambda \times N$$

$$A = (2.377 \times 10^{-5}\text{ s}^{-1}) \times (2.0073 \times 10^{25})$$

$$\approx 4.771 \times 10^{20}\text{ Bq}$$

4. Convert Energy to Watts

Every individual decay of Astatine-210 releases an average combined radiation energy—comprising alphas, electrons, and photons—of approximately $3.0515\text{ MeV}$. We convert this particle energy into Joules ($1\text{ MeV} = 1.6022 \times 10^{-13}\text{ J}$) $^{[3, 4]}$:

$$\text{Energy per decay} = 3.0515\text{ MeV} \times 1.6022 \times 10^{-13}\text{ J/MeV}$$

$$\approx 4.889 \times 10^{-13}\text{ J}$$

Multiplying the total decays per second by the energy per decay yields the absolute power output:

$$P = A \times \text{Energy per decay}$$

$$P = (4.771 \times 10^{20}\text{ Bq}) \times (4.889 \times 10^{-13}\text{ J})$$

$$\approx 2.333 \times 10^8\text{ W}$$

5. Final Thermal Output Result

The hypothetical initial thermal output of a solid $7\text{ kg}$ sample of Astatine-210 sits at $2.33 \times 10^8\text{ watts}$ ($233\text{ MW}$).

For those who prefer programmatic verification, here is the Python script used to model these calculations based on specific activity and mean energy.

# Calculate the thermal power of Astatine-210 based on # specific activity and mean energy
#
# Specific activity # = 6.817e16 Bq/g
# Mass = 7000 g
# Mean energy per decay = 0.0097 + 0.07962 + 2.96215 = # 3.05147 MeV

spec_activity = 6.817e16  # Bq/g
mass = 7000  # g
total_activity = spec_activity * mass  # Bq

mean_energy_mev = 3.05147
energy_j = mean_energy_mev * 1.60218e-13

power_w = total_activity * energy_j
print(f"At-210 Power: {power_w:.4e} Watts")