Question

What must be the temperature of an ideal blackbody so that photons of its radiated light having the peak-intensity wavelength can excite the electron in the Bohr-model hydrogen atom from the ground level to the $n$ = 4 energy level?

Short answer

The temperature must be approximately 29,810 K.

Step-by-step solution

Find the Energy Required for Transition

Using the Bohr model, the energy required to excite an electron from the ground state (n=1) to the fourth energy level (n=4) is given by the formula: $E=-13.6(\frac{1}{n_1^2}-\frac{1}{n_2^2})$ eV. Here, $n_1=1$ and $n_2=4$. Substitute these values into the formula to find the energy. We get: $E=-13.6(\frac{1}{1^2}-\frac{1}{4^2})=12.75$ eV.

Convert Energy to Wavelength

Using the photon energy equation $E=\frac{hc}{\lambda }$, where $h$ is Planck's constant $6.626\times {10}^{-34}Js$ and $c$ is the speed of light $3\times {10}^{8}m/s$, convert the energy from eV to joules $(1eV=1.602\times {10}^{-19}J)$. Calculate the wavelength. $12.75eV$ becomes $2.043\times {10}^{-18}J$ and $\lambda =\frac{6.626\times {10}^{-34}\times 3\times {10}^8}{2.043\times {10}^{-18}}=9.72\times {10}^{-8}m$.

Apply Wien's Displacement Law

Wien's Displacement Law relates the peak wavelength ${\lambda }_{max}$ of radiation from a blackbody to its temperature $T$ as ${\lambda }_{max}T=2.898\times {10}^{-3}m\cdot K$. Solve for temperature $T$ using the wavelength from Step 2. Substitute ${\lambda }_{max}=9.72\times {10}^{-8}m$, $T=\frac{2.898\times {10}^{-3}}{9.72\times {10}^{-8}}K=29810K$.

Key concepts

Bohr model and Energy Levels

The Bohr model of the atom is a revolutionary concept that helps explain how electrons are arranged around an atomic nucleus. Imagine it like a mini solar system where electrons orbit the nucleus in distinct levels or shells. These shells are named by numbers such as n=1, n=2, and so on. Each of these levels has a specific energy.

• The ground state is the lowest energy level, which is n=1.
• To move an electron from a lower to a higher energy level, energy must be added; this is called excitation.

In this exercise, the electron moves from n=1 to n=4. The energy needed for this transition is calculated by considering the differences in energy for each level. The formula used is:$$E=-13.6(\frac{1}{n_1^2}-\frac{1}{n_2^2})\text{ eV}$$ Substituting n=1 and n=4 into this equation gives us the energy required to complete the excitation.

Photon Energy and Wavelength

Photon energy plays a crucial role in understanding how light interacts with matter. A photon is essentially a packet of light energy, and its energy is inversely proportional to its wavelength. This is expressed by the equation:$$E=\frac{hc}{\lambda }$$ where $E$ is the energy of the photon, $h$ is Planck's constant, and $c$ is the speed of light.

• Planck's constant $h=6.626\times {10}^{-34}\text{ Js}$.
• The speed of light $c=3\times {10}^8\text{ m/s}$.

When an electron is excited, it often absorbs or emits a photon corresponding to these energies. In Step 2 of the solution, we convert the energy from electron volts to joules and use this relationship to find the wavelength of light that has this energy.

Understanding Wien's Displacement Law

Wien's Displacement Law gives us a wonderful way to connect temperature and the wavelength of peak emission in blackbody radiation. Blackbodies are ideal emitters of thermal radiation. The law is articulated through the formula:$${\lambda }_{max}T=2.898\times {10}^{-3}\text{ m}\cdot \text{K}$$ Here, ${\lambda }_{max}$ is the wavelength at which the emission is strongest, and $T$ is the temperature.

• According to Wien's Law, as temperature increases, the peak wavelength decreases, meaning hotter blackbodies emit light at shorter wavelengths.
• This principle helps astrophysicists determine the temperature of distant stars by observing their light.

Thus, by using the wavelength derived from photon energy, this law lets us calculate the temperature needed for an ideal blackbody.