How much mass is converted into energy per day in Tarapur Nuclear Power Plant operated at 10$^6$ kW ?

Question

Accepted Answer

To determine how much mass is converted into energy per day in the Tarapur Nuclear Power Plant, we can use Einstein's mass-energy equivalence principle, expressed by the equation $E = mc^2$, where $E$ is energy, $m$ is mass, and $c$ is the speed of light in a vacuum ($c = 3 \times 10^8 \, \text{m/s}$).

1. **Calculate the energy produced per day:**

The power output of the plant is given as $10^6 \, \text{kW}$. To find the energy produced in one day, we convert this power into energy:

$$
   \text{Energy per day} = \text{Power} \times \text{Time} = 10^6 \, \text{kW} \times 24 \, \text{hours} \times 3600 \, \text{seconds/hour}
   $$

$$
   = 10^6 \times 24 \times 3600 \, \text{kJ}
   $$

$$
   = 8.64 \times 10^{10} \, \text{kJ}
   $$

Since $1 \, \text{kJ} = 10^3 \, \text{J}$, the energy in joules is:

$$
   = 8.64 \times 10^{13} \, \text{J}
   $$

2. **Calculate the mass converted into energy:**

Using $E = mc^2$, we solve for $m$:

$$
   m = \frac{E}{c^2} = \frac{8.64 \times 10^{13} \, \text{J}}{(3 \times 10^8 \, \text{m/s})^2}
   $$

$$
   = \frac{8.64 \times 10^{13}}{9 \times 10^{16}}
   $$

$$
   = 0.96 \, \text{g}
   $$

Thus, the mass converted into energy per day is 0.96 grams, which corresponds to Option 4.

**Explanation for why other options are incorrect:**

- **Option 1 (0.16 g):** This value is significantly lower than the calculated mass of 0.96 grams. It does not account for the full energy output of the plant as calculated using the mass-energy equivalence principle.

- **Option 2 (0.32 g):** This value is also lower than the calculated mass of 0.96 grams. It represents only a fraction of the energy conversion and does not match the energy output derived from the given power.

- **Option 3 (0.64 g):** While closer to the correct value, this option still underestimates the mass converted into energy. The precise calculation shows that 0.96 grams is the correct amount of mass converted.

In conclusion, Option 4 is the correct answer because it accurately reflects the mass converted into energy per day based on the given power output and the application of Einstein's mass-energy equivalence principle.