To solve the problem of determining the amount of fuel consumed by a nuclear reactor delivering a power of \(10^9\) W in one hour, we can follow these steps:
### Step 1: Understand the relationship between power, energy, and time.
The energy consumed by the reactor can be calculated using the formula:
\[
E = P \times t
\]
where:
- \(E\) is the energy consumed (in joules),
- \(P\) is the power (in watts),
- \(t\) is the time (in seconds).
### Step 2: Convert time from hours to seconds.
Since the power is given in watts (joules per second), we need to convert the time from hours to seconds.
1 hour = 3600 seconds.
### Step 3: Calculate the energy consumed in one hour.
Substituting the values into the energy formula:
\[
E = 10^9 \, \text{W} \times 3600 \, \text{s}
\]
Calculating this gives:
\[
E = 10^9 \times 3600 = 3.6 \times 10^{12} \, \text{J}
\]
### Step 4: Relate energy to mass using Einstein's equation.
According to Einstein's mass-energy equivalence principle, the energy can also be expressed in terms of mass:
\[
E = mc^2
\]
where:
- \(m\) is the mass (in kilograms),
- \(c\) is the speed of light (\(c \approx 3 \times 10^8 \, \text{m/s}\)).
### Step 5: Rearrange the equation to find mass.
We can rearrange the equation to find the mass:
\[
m = \frac{E}{c^2}
\]
### Step 6: Substitute the values into the mass equation.
Substituting the values we have:
\[
m = \frac{3.6 \times 10^{12} \, \text{J}}{(3 \times 10^8 \, \text{m/s})^2}
\]
Calculating \(c^2\):
\[
c^2 = (3 \times 10^8)^2 = 9 \times 10^{16} \, \text{m}^2/\text{s}^2
\]
Now substituting this back:
\[
m = \frac{3.6 \times 10^{12}}{9 \times 10^{16}} = 4 \times 10^{-5} \, \text{kg}
\]
### Step 7: Convert mass from kilograms to grams.
Since 1 kg = 1000 grams, we convert the mass:
\[
m = 4 \times 10^{-5} \, \text{kg} \times 1000 \, \text{g/kg} = 4 \times 10^{-2} \, \text{g}
\]
### Final Answer:
The amount of fuel consumed by the reactor in one hour is \(4 \times 10^{-2} \, \text{grams}\).
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