Hydrogen in interstellar space cannot be excited electronically by starlight. However, the cosmic background radiation, equivalent to `2.7 K` can cause the hyperfine transition. Calculate the temperature of a blackbody whose peak intensity corresponds to the `1420 MHz` transition.

Answer

To solve the problem of calculating the temperature of a blackbody whose peak intensity corresponds to the `1420 MHz` transition, we will use Wien's Displacement Law. Here’s a step-by-step solution:

### Step 1: Understand Wien's Displacement Law
Wien's Displacement Law states that the wavelength at which the intensity of radiation from a blackbody is maximized (λ_peak) is inversely proportional to the temperature (T) of the blackbody. The law is given by the formula:
\[ \lambda_{\text{peak}} \cdot T = b \]
where \( b \) is a constant approximately equal to \( 2.9 \times 10^{-3} \) m·K.

### Step 2: Convert Frequency to Wavelength
We need to convert the frequency of the transition (1420 MHz) to wavelength. The relationship between wavelength (λ), frequency (ν), and the speed of light (c) is given by:
\[ \lambda = \frac{c}{\nu} \]
where:
- \( c \) is the speed of light (\( 3 \times 10^8 \) m/s),
- \( \nu \) is the frequency in Hz.

First, convert 1420 MHz to Hz:
\[ 1420 \text{ MHz} = 1420 \times 10^6 \text{ Hz} \]

Now, calculate the wavelength:
\[ \lambda = \frac{3 \times 10^8 \text{ m/s}}{1420 \times 10^6 \text{ Hz}} \]

### Step 3: Calculate the Wavelength
Calculating the above expression:
\[ \lambda = \frac{3 \times 10^8}{1420 \times 10^6} \approx 0.211 \text{ m} \]

### Step 4: Apply Wien's Displacement Law
Now that we have the wavelength, we can use Wien's Displacement Law to find the temperature:
\[ T = \frac{b}{\lambda} \]
Substituting the values:
\[ T = \frac{2.9 \times 10^{-3} \text{ m·K}}{0.211 \text{ m}} \]

### Step 5: Calculate the Temperature
Now we perform the calculation:
\[ T \approx \frac{2.9 \times 10^{-3}}{0.211} \approx 0.0137 \text{ K}^{-1} \]
To find T:
\[ T \approx 137.0 \text{ K} \]

### Final Answer
The temperature of the blackbody whose peak intensity corresponds to the `1420 MHz` transition is approximately **137.0 K**.

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