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Calculate the energy, in joules, required to excite a hydrogen atom by causing an electronic transition from the [tex]n=1[/tex] to the [tex]n=4[/tex] principal energy level.

A. [tex]2.19 \times 10^5 \, J[/tex]
B. [tex]2.04 \times 10^{-18} \, J[/tex]
C. [tex]3.27 \times 10^{-17} \, J[/tex]
D. [tex]2.07 \times 10^{-29} \, J[/tex]
E. [tex]2.25 \times 10^{-18} \, J[/tex]



Answer :

GinnyAnswer
To calculate the energy required to excite a hydrogen atom by causing an electronic transition from the [tex]\( n=1 \)[/tex] to the [tex]\( n=4 \)[/tex] principal energy level, we use the Rydberg formula for energy differences. The energy of a photon absorbed or emitted during a transition between levels [tex]\( n_1 \)[/tex] and [tex]\( n_4 \)[/tex] in a hydrogen atom is given by:

[tex]\[ E = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_4^2} \right) \][/tex]

where:
- [tex]\( R_H \)[/tex] is the Rydberg constant for hydrogen, with a value of [tex]\( 2.18 \times 10^{-18} \)[/tex] joules.
- [tex]\( n_1 \)[/tex] and [tex]\( n_4 \)[/tex] are the principal quantum numbers of the initial and final energy levels, respectively.

Substituting the given values:
[tex]\[ n_1 = 1 \][/tex]
[tex]\[ n_4 = 4 \][/tex]
[tex]\[ R_H = 2.18 \times 10^{-18} \text{ J} \][/tex]

The energy difference [tex]\( E \)[/tex] can be calculated as follows:
[tex]\[ E = 2.18 \times 10^{-18} \left( \frac{1}{1^2} - \frac{1}{4^2} \right) \][/tex]

We first calculate the individual fractions:
[tex]\[ \frac{1}{1^2} = 1 \][/tex]
[tex]\[ \frac{1}{4^2} = \frac{1}{16} = 0.0625 \][/tex]

Now, calculate the difference:
[tex]\[ 1 - 0.0625 = 0.9375 \][/tex]

Next, multiply this value by the Rydberg constant:
[tex]\[ E = 2.18 \times 10^{-18} \times 0.9375 = 2.04375 \times 10^{-18} \][/tex]

In scientific notation with appropriate significant figures, the energy required for the electronic transition is:
[tex]\[ 2.04 \times 10^{-18} \text{ J} \][/tex]

Therefore, the correct answer is:
[tex]\[ 2.04 \times 10^{-18} \text{ J} \][/tex]

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