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Use Wien's Law to answer the following.

The brightest color emitted by a certain star has a wavelength of 240 nm. Given that a 6000 K star produces its peak radiation at a wavelength of 483 nm, what is the temperature of the star?

Formula: [tex]\lambda_1 \max \cdot T_1 = \lambda_2 \max \cdot T_2[/tex]

A. 12,900 K
B. 12,200 K
C. 12,150 K
D. 12,075 K


Sagot :

Certainly! Let's tackle this problem step by step using Wein's displacement law.

### Given Data:
1. Wavelength at maximum brightness for the certain star, [tex]\(\lambda_1 \max = 240\)[/tex] nm
2. Temperature of the reference star, [tex]\(T_1 = 6000\)[/tex] K
3. Wavelength at maximum brightness for the reference star, [tex]\(\lambda_2 \max = 483\)[/tex] nm

### Wein's Displacement Law:
The law states that [tex]\(\lambda \max * T = \text{constant}\)[/tex], or specifically for two different conditions of stars:
[tex]\[ \lambda_1 \max T_1 = \lambda_2 \max T_2 \][/tex]

### Goal:
Find the temperature [tex]\(T_2\)[/tex] of the certain star.

### Steps:
1. Start with the formula given by Wein's displacement law:
[tex]\[ \lambda_1 \max T_1 = \lambda_2 \max T_2 \][/tex]

2. Rearrange the equation to solve for [tex]\(T_2\)[/tex]:
[tex]\[ T_2 = \frac{\lambda_1 \max * T_1}{\lambda_2 \max} \][/tex]

3. Plug in the given values:
[tex]\[ T_2 = \frac{240 \text{ nm} * 6000 \text{ K}}{483 \text{ nm}} \][/tex]

4. Simplify the calculation:
[tex]\[ T_2 = \frac{1440000}{483} \][/tex]

5. The result of this division is approximately:
[tex]\[ T_2 \approx 2981.3664596273293 \text{ K} \][/tex]

Thus, given the calculations, the temperature of the star corresponding to the brightest wavelength of 240 nm is approximately [tex]\(2981.37\)[/tex] K. Unfortunately, this value does not match any of the options provided (12,900 K, 12,200 K, 12,150 K, 12,075 K). If there's a discrepancy here, you might want to double-check the problem statement or the given options, as they don't appear to align with the correct Webin's law calculation.