The power obtained in a reactor using `U^(235)` disintergration is `1000kW`. The mass decay of `U^(235)` per hour is

Options

A. 1 microgram
B. 10 microgram
C. 20 microgram
D. 40 microgram

Answer

To solve the problem of finding the mass decay of \( U^{235} \) per hour given a power output of 1000 kW, we can follow these steps:

### Step 1: Understand the relationship between power, energy, and time
Power is defined as the rate of energy transfer. Therefore, the energy produced over a certain time can be calculated using the formula:
\[ E = P \times t \]
where \( E \) is energy, \( P \) is power, and \( t \) is time.

### Step 2: Convert the power from kilowatts to watts
Given that the power output is 1000 kW, we need to convert this to watts:
\[ P = 1000 \, \text{kW} = 1000 \times 10^3 \, \text{W} = 10^6 \, \text{W} \]

### Step 3: Calculate the time in seconds for one hour
Since we need to calculate the energy for one hour, we convert one hour into seconds:
\[ t = 1 \, \text{hour} = 60 \, \text{minutes} \times 60 \, \text{seconds/minute} = 3600 \, \text{seconds} \]

### Step 4: Calculate the total energy produced in one hour
Now we can calculate the total energy produced in one hour:
\[ E = P \times t = 10^6 \, \text{W} \times 3600 \, \text{s} = 3.6 \times 10^9 \, \text{J} \]

### Step 5: Use the mass-energy equivalence formula
According to Einstein's mass-energy equivalence principle, the energy can also be expressed in terms of mass decay using the formula:
\[ E = \Delta m \cdot c^2 \]
where \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)).

### Step 6: Rearrange the formula to find the mass decay
Rearranging the formula gives:
\[ \Delta m = \frac{E}{c^2} \]

### Step 7: Substitute the values into the equation
Substituting the values we have:
\[ \Delta m = \frac{3.6 \times 10^9 \, \text{J}}{(3 \times 10^8 \, \text{m/s})^2} \]
Calculating \( c^2 \):
\[ c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2 \]

Now substituting back:
\[ \Delta m = \frac{3.6 \times 10^9}{9 \times 10^{16}} = 4 \times 10^{-8} \, \text{kg} \]

### Step 8: Convert kilograms to micrograms
To convert kilograms to micrograms, we use the conversion factor \( 1 \, \text{kg} = 10^9 \, \mu\text{g} \):
\[ \Delta m = 4 \times 10^{-8} \, \text{kg} \times 10^9 \, \mu\text{g/kg} = 40 \, \mu\text{g} \]

### Final Answer
The mass decay of \( U^{235} \) per hour is:
\[ \Delta m = 40 \, \mu\text{g} \]