I got stuck on this problem and I cannot figure out what to do or explain this problem. Please help me with this problem.
The emission lines of one-electron atoms and ions can all be fit to the equation describing the spectrum of the hydrogen atom:
E= -(2.18X10^-18J)Z^2(1/n^2final-1/n^2initial) where Z is the atomic number.
a) calculate the energy (in J) of one photon associated with the transition of the electron in He^+2 from n=2 to n=1.
b) As the value of Z increases, does the wavelength of the photon associated with the transition from n=2 to n=1 increase or decrease? Explain.
a) you need to plug the values into the equation to find E, where Z is the atomic number of helium.
b) if Z increases what happens to E? A larger value for Z will give a larger value for E. What is the relationship between E and wavelength? You also need this for your other posting.
To solve this problem, we need to plug in the given information into the equation and perform the necessary calculations. Let's start with part A:
a) To calculate the energy (E) of one photon associated with the transition of the electron in He^+2 from n=2 to n=1, we need to use the equation given:
E = -(2.18 × 10^-18 J) × Z^2 × (1/n^2final - 1/n^2initial)
For He^+2, Z (atomic number) is equal to 2.
Plugging in the values:
E = -(2.18 × 10^-18 J) × (2^2) × (1/1^2 - 1/2^2)
Simplifying the equation:
E = -(2.18 × 10^-18 J) × 4 × (1/1 - 1/4)
E = -(2.18 × 10^-18 J) × 4 × (1 - 1/4)
Calculating further:
E = -(2.18 × 10^-18 J) × 4 × (4/4 - 1/4)
E = -(2.18 × 10^-18 J) × 4 × (3/4)
E = -(2.18 × 10^-18 J) × 3
Evaluating the expression:
E ≈ -6.54 × 10^-18 J
Therefore, the energy of one photon associated with the transition of the electron in He^+2 from n=2 to n=1 is approximately -6.54 × 10^-18 J.
Now let's move on to part B:
b) As the value of Z increases, the wavelength of the photon associated with the transition from n=2 to n=1 decreases. This can be explained using the equation:
E = -(2.18 × 10^-18 J) × Z^2 × (1/n^2final - 1/n^2initial)
From the equation, we can see that as Z increases, the term Z^2 increases. Since Z is squared, any increase in atomic number Z is magnified by the square function.
The effect of Z^2 becoming larger is that the overall energy E becomes larger (the value inside the parentheses becomes more negative). The energy of a photon is directly proportional to its frequency, which means that as the energy increases, the frequency of the photon increases.
Wavelength (λ) and frequency (ν) are inversely related, meaning that as frequency increases, wavelength decreases. Therefore, as Z increases, the wavelength of the photon associated with the transition from n=2 to n=1 decreases.
In conclusion, as the atomic number Z increases, the wavelength of the photon associated with the transition from n=2 to n=1 decreases.