RTG

Estimate the heat and electrical energy output of a radioisotope thermoelectric generator (RTG) using 5 kg of Pu-238 over 50 years at 5% efficiency:

🔬 Step 1: Key Properties of Pu-238

Half-life: 87.7 years
Specific power: ~0.57 watts/gram (thermal)
Decay mode: Alpha decay (no significant gamma or neutron radiation)
Energy per decay: ~5.593 MeV

⚙️ Step 2: Initial Thermal Power

Pu-238 has a specific thermal power of:

$0.57 \, \text{W/g} \times 5000 \, \text{g} = 2850 \, \text{W}$

This is the initial thermal power at time $t = 0$ .

📉 Step 3: Decay Over Time

The power decreases exponentially due to radioactive decay:

$P(t) = P_0 \cdot e^{-\lambda t}$

Where:

$\lambda = \frac{\ln(2)}{T_{1/2}} = \frac{\ln(2)}{87.7} \approx 0.0079 \, \text{yr}^{-1}$
$t = 50$ years

So:

$P(50) = 2850 \cdot e^{-0.0079 \cdot 50} \approx 2850 \cdot 0.673 \approx 1917 \, \text{W}$

🔥 Step 4: Total Thermal Energy Over 50 Years

Integrate the power over time:

$E_{\text{thermal}} = \int_0^{50} P_0 \cdot e^{-\lambda t} \, dt = \frac{P_0}{\lambda} \left(1 - e^{-\lambda \cdot 50} \right)$

Plug in values:

$E_{\text{thermal}} = \frac{2850}{0.0079} \cdot (1 - 0.673) \approx 360,759 \cdot 0.327 \approx 118,000 \, \text{kWh}$

⚡ Step 5: Electrical Energy at 5% Efficiency

$E_{\text{electric}} = 0.05 \cdot E_{\text{thermal}} \approx 0.05 \cdot 118,000 \approx 5900 \, \text{kWh}$

✅ Summary

Quantity	Value
Initial thermal power	2850 W
Thermal power after 50 yrs	~1917 W
Total thermal energy	~118,000 kWh
Total electrical energy	~5900 kWh