The rate of cooling at 600 K, if surrounding temperature is 300 K is R. The rate of cooling at 900 K is

Options

A. `(16)/(3)R`
B. `2R`
C. `3R`
D. `(2)/(3)R`

Explanation

To solve the problem, we will use Stefan-Boltzmann's law, which states that the rate of heat loss (or cooling) of an object is proportional to the fourth power of its absolute temperature minus the fourth power of the surrounding temperature.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Rate of cooling at 600 K, \( R \)
- Surrounding temperature, \( T_0 = 300 \, K \)
- Temperature for which we need to find the rate of cooling, \( T = 900 \, K \)

2. **Apply Stefan-Boltzmann's Law:**
The rate of cooling can be expressed as:
\[ R = \epsilon \sigma (T^4 - T_0^4) \]
where \( \epsilon \) is the emissivity (assumed constant for this problem) and \( \sigma \) is the Stefan-Boltzmann constant.

3. **Write the Equation for Rate of Cooling at 600 K:**
\[ R = \epsilon \sigma (600^4 - 300^4) \]

4. **Write the Equation for Rate of Cooling at 900 K:**
Let \( R' \) be the rate of cooling at 900 K:
\[ R' = \epsilon \sigma (900^4 - 300^4) \]

5. **Set Up the Ratio of the Two Rates:**
To find the relationship between \( R' \) and \( R \), we can divide the two equations:
\[ \frac{R'}{R} = \frac{900^4 - 300^4}{600^4 - 300^4} \]

6. **Calculate the Values:**
- Calculate \( 900^4 \):
\[ 900^4 = 656100000000 \]
- Calculate \( 600^4 \):
\[ 600^4 = 129600000000 \]
- Calculate \( 300^4 \):
\[ 300^4 = 8100000000 \]

Now substitute these into the ratio:
\[ R' = R \cdot \frac{(656100000000 - 8100000000)}{(129600000000 - 8100000000)} \]

7. **Simplify the Expression:**
- Calculate the numerator:
\[ 656100000000 - 8100000000 = 648999000000 \]
- Calculate the denominator:
\[ 129600000000 - 8100000000 = 121500000000 \]

Now, substitute these values:
\[ \frac{R'}{R} = \frac{648999000000}{121500000000} \]

8. **Calculate the Final Ratio:**
This simplifies to:
\[ \frac{R'}{R} = \frac{648999}{121500} \approx 5.35 \]

9. **Express \( R' \) in Terms of \( R \):**
Thus, we find:
\[ R' \approx 5.35 R \]

### Conclusion:
The rate of cooling at 900 K is approximately \( 5.35 R \).