Estimate the temperature at which a body may appear blue or red. The values of `lambda_(mean)` for these are `5000` and `7500Å` respectively. [Given Wein's constant `b = 0.3 cmK]`

Explanation

To estimate the temperature at which a body may appear blue or red, we can use Wien's Displacement Law, which states that the wavelength at which the emission of a black body spectrum is maximized (λ_max) is inversely proportional to the temperature (T) of the body. The formula is given by:

\[ \lambda_{max} = \frac{b}{T} \]

where \( b \) is Wien's constant.

### Step-by-Step Solution:

1. **Identify Given Values:**
- For blue light: \( \lambda_{blue} = 5000 \, \text{Å} = 5000 \times 10^{-10} \, \text{m} = 5 \times 10^{-7} \, \text{m} \)
- For red light: \( \lambda_{red} = 7500 \, \text{Å} = 7500 \times 10^{-10} \, \text{m} = 7.5 \times 10^{-7} \, \text{m} \)
- Wien's constant \( b = 0.3 \, \text{cm} \cdot \text{K} = 0.3 \times 10^{-2} \, \text{m} \cdot \text{K} \)

2. **Calculate Temperature for Blue Light:**
- Using the formula, rearrange to find \( T \):
\[ T_{blue} = \frac{b}{\lambda_{blue}} \]
- Substitute the values:
\[ T_{blue} = \frac{0.3 \times 10^{-2}}{5 \times 10^{-7}} = \frac{0.3 \times 10^{-2}}{5 \times 10^{-7}} = 6000 \, \text{K} \]

3. **Calculate Temperature for Red Light:**
- Similarly, use the formula for red light:
\[ T_{red} = \frac{b}{\lambda_{red}} \]
- Substitute the values:
\[ T_{red} = \frac{0.3 \times 10^{-2}}{7.5 \times 10^{-7}} = \frac{0.3 \times 10^{-2}}{7.5 \times 10^{-7}} = 4000 \, \text{K} \]

4. **Final Results:**
- The temperature at which a body may appear blue is approximately **6000 K**.
- The temperature at which a body may appear red is approximately **4000 K**.

### Summary of Results:
- Temperature for blue light: **6000 K**
- Temperature for red light: **4000 K**