The Doppler broadening of a spectral line increases with the rms speed of the atoms in the source of light. What should give narrower spectral lines: a mercury-198 lamp at \(300 \mathrm{~K}\) or a krypton-86 lamp at \(77 \mathrm{~K}\) ?

The root mean square (rms) speed of gas molecules is a way of expressing the average speed of particles in a gas. It considers the speeds of all the particles and provides a single value representing their kinetic energy. The formula for the rms speed is: \[ v_{rms} = \sqrt{\frac{3k_{B}T}{m}} \] where:

• \( v_{rms} \) is the rms speed,
• \( k_B \) is the Boltzmann constant (approximately \( 1.38 \times 10^{-23} \mathrm{J/K} \),
• \( T \) is the absolute temperature in Kelvin,
• \( m \) is the mass of a single atom of the gas.

Higher temperatures or lighter atoms result in higher rms speeds. In the context of Doppler broadening, gases with higher rms speeds lead to broader spectral lines.To better understand:

• Mercury-198 at 300 K has an rms speed of approximately 157 m/s.
• Krypton-86 at 77 K has an rms speed of approximately 139 m/s.

When we observe the light from a source, we often see it as a spectrum, consisting of various lines at different wavelengths. These lines are called spectral lines, and they correspond to specific energies that electrons in atoms can have. Spectral lines are unique to each element, which allows us to identify materials by their spectra. However, various effects can broaden these lines, making them wider and less sharp.One significant factor that causes broadening is the Doppler effect. When atoms move towards or away from an observer, the wavelength of the light they emit appears shorter or longer, respectively. This effect is more pronounced at higher temperatures because the atoms move faster (higher rms speed). Consequently, gases at higher temperatures, like mercury-198 at 300 K, exhibit broader spectral lines compared to gases at lower temperatures, like krypton-86 at 77 K.

The Boltzmann constant \( k_B \) plays a crucial role in understanding the connection between temperature and energy in physical systems. It is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. Mathematically, it is defined as: \[ k_B = 1.38 \times 10^{-23} \mathrm{J/K} \] In the rms speed formula, \( k_B \) ensures that we correctly account for the thermal energy of the gas particles. A higher value of \( k_B \) would mean greater energy at a given temperature. To apply this in the context of our example:

• For mercury-198 at 300 K and a mass of approximately \( 3.3019 \times 10^{-25} \mathrm{kg} \), the equation results in an rms speed of approximately 157 m/s.
• Similarly, for krypton-86 at 77 K and a mass of approximately \( 1.430 \times 10^{-25} \mathrm{kg} \), the rms speed comes out to be around 139 m/s.

By using the Boltzmann constant in these calculations, we can precisely determine how much energy the gas particles have based on their temperature, which in turn influences the broadening of spectral lines.