Question

Place the following transitions of the hydrogen atom in order from longest to shortest wavelength of the photon emitted. Rank from longest to shortest wavelength. To rank items as equivalent, overlap them. n=7 to n=4 n=5 to n=3 n=4 to n=2 n=3 to n=2

Answer 1

Answer:

7 to 4     (higher  $\lambda$)

5 to 3

3 to 2

4 to 2   (lower  $\lambda$)

Explanation:

We can use the Rydberg formula which relates the wavelength of the photon emissions to the principle quantum numbers involved in the transition:

$\frac{1}{\lambda}=R((\frac{1}{n_1^2})-(\frac{1}{n_2^2}))$

with $n_1$ final n, and $n_2$ initial n

evaluating for each transition:

7 to 4 $\frac{1}{\lambda} = R((\frac{1}{4^2})-(\frac{1}{7^2}))= R(1/16-1/49)=R(0.042)$

5 to 3 $\frac{1}{\lambda}= R(1/9-1/25)=R(0.071)$

4 to 2 $\frac{1}{\lambda}= R(1/4-1/16)=R(0.1875)$

3 to 2 $\frac{1}{\lambda}= R(1/4-1/9)=R(0.139)$

Note that the above formula is written for $\frac{1}{\lambda}$, so lower $\frac{1}{\lambda}$ value obtained involves higher $\lambda$.

So we should order from lower to higher $\frac{1}{\lambda}$

7 to 4     (higher  $\lambda$)

5 to 3

3 to 2

4 to 2   (lower  $\lambda$)

Note: Take into account that longer wavelength involves lower energy ($E=\frac{hc}{\lambda}$).

Answer 2

The shortest wavelength corresponds to the transition involving the highest difference in energy.

According to the Bohr model of the atom, an electron can absorb energy and move from a lower to a higher energy level. The lower energy level is known as the ground state while the higher energy level is known as the excited state.

As the electron returns to ground state, energy is emitted in form of photons of light. Recall that the magnitude of energy absorbed by the electron as it transits between energy levels is equal to the difference in energy between the two energy levels. This amount of energy absorbed during during excitation is also the same amount of energy released as the electron returns to ground state.

This implies that the higher the energy difference between the ground state and the excited state, the shorter the wavelength of photon emitted. Recall that energy and wavelength of a photon are inversely proportional to each other. Based on this, the order of photons from longest to shortest is;

• n=3 to n=2
• n=4 to n=2
• n=5 to n=3
• n=7 to n=4

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