Estimate the Doppler width of an emission line of wavelength 500A emitted by organ M=40 and the temperature is 300 Kelvin.

To estimate the Doppler width of an emission line, we can use the formula that relates the width of the line to the temperature of the emitting gas and the mass of the particles involved. The Doppler width is influenced by the thermal motion of the particles, which is determined by their temperature and mass. Let's break this down step by step.

Understanding the Doppler Width

The Doppler width (Δλ) of an emission line can be calculated using the following formula:

Δλ = λ * √(2kT/mc²)

Where:

• Δλ = Doppler width
• λ = wavelength of the emission line (in meters)
• k = Boltzmann's constant (1.38 x 10⁻²³ J/K)
• T = temperature (in Kelvin)
• m = mass of the emitting particle (in kilograms)
• c = speed of light (approximately 3 x 10⁸ m/s)

Given Values

In this case, we have:

• Wavelength (λ) = 500 Å = 500 x 10⁻¹⁰ m
• Mass (M) = 40 (assuming this is in atomic mass units, amu)
• Temperature (T) = 300 K

Converting Mass to Kilograms

First, we need to convert the mass from atomic mass units to kilograms. The conversion factor is:

1 amu = 1.66 x 10⁻²⁷ kg

Thus, for M = 40 amu:

m = 40 x 1.66 x 10⁻²⁷ kg = 6.64 x 10⁻²⁶ kg

Calculating the Doppler Width

Now we can substitute the values into the Doppler width formula:

Δλ = (500 x 10⁻¹⁰ m) * √(2 * (1.38 x 10⁻²³ J/K) * (300 K) / ((6.64 x 10⁻²⁶ kg) * (3 x 10⁸ m/s)²))

First, calculate the term inside the square root:

2 * (1.38 x 10⁻²³) * (300) = 8.28 x 10⁻²¹ J

Now, calculate the denominator:

(6.64 x 10⁻²⁶ kg) * (9 x 10¹⁶ m²/s²) = 5.976 x 10⁻⁹ kg·m²/s²

Now, we can compute the entire expression:

√(8.28 x 10⁻²¹ / 5.976 x 10⁻⁹) ≈ √(1.384 x 10⁻¹²) ≈ 1.176 x 10⁻⁶

Finally, substituting back into the Doppler width formula:

Δλ = (500 x 10⁻¹⁰) * (1.176 x 10⁻⁶) ≈ 5.88 x 10⁻¹⁴ m

Final Result

To express this in more familiar units, we can convert it back to angstroms:

Δλ ≈ 5.88 x 10⁻¹⁴ m = 5.88 x 10⁴ Å

Thus, the estimated Doppler width of the emission line at a wavelength of 500 Å, emitted by particles with a mass of 40 amu at a temperature of 300 K, is approximately 5.88 x 10⁴ Å. This width gives us an idea of how broad the emission line will be due to the thermal motion of the particles involved.