Calculate Rh Using The Combination Of Equations 1A

Calculate Rh Using The Combination Between The Equations 1 And 2 Based

Calculate Rh using the combination between the equations 1 and 2 based on 1/ λ = (Rh) (1/n_in² – 1/n_out²) and calculate the average of the values and the % error. Equation 1: E photon = |ΔE| = E_out – E_in = B(1/n_in² – 1/n_out²) Equation 2: λ = hc / E photon. Given: colour, wavelength obtained (nm), N(out), N(in). The data are: Red 644 nm, Turquoise 518 nm, Violet 438 nm, Faint Violet 385 nm. Determine the average Rydberg constant, m^-1. Show all steps.

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The objective of this analysis is to determine the Rydberg constant (Rh) by utilizing the provided spectral data and fundamental equations relating energy, wavelength, and quantum numbers. The process involves applying the Rydberg formula, combining it with energy calculations, and finally deriving an average Rh value along with the percentage error to assess accuracy.

First, let's recall the key equations. The Rydberg formula relates the wavelength (λ) of emitted light to the quantum numbers involved:

1/ λ = Rh * (1/n_in² – 1/n_out²)

where n_in is the initial energy level, and n_out the final energy level of the electron transition.

Additionally, the photon energy is given by the equation: E_photon = hc / λ, where h is Planck's constant (6.626 x 10^-34 Js), and c is the speed of light (3.00 x 10⁸ m/s). For calculations involving energy differences, the spectral wavelength must be converted into meters, noting that 1 nm = 1 x 10^-9 m.

The energy difference per transition, as per Equation 1, can be written as:

E_photon = B * (1/n_in² – 1/n_out²)

where B is a proportionality constant related to the Rydberg constant (Rh). The connection between the energy difference and the Rydberg constant involves the fundamental constants, thus enabling calculation of Rh from the measured wavelengths.

Given the data for the spectral lines, the spectral wavelengths and corresponding quantum numbers are considered. We typically assume the primary transitions involve n_in = 2 (for Balmer series in hydrogen), and n_out varies for different spectral lines, corresponding to the observed transitions. For simplicity, these quantum numbers are inferred based on known spectral series and the wavelength values:

• Red (644 nm): likely from n_in= 3 to n_out= 2
• Turquoise (518 nm): n_in= 4 to n_out= 2
• Violet (438 nm): n_in= 5 to n_out= 2
• Faint Violet (385 nm): n_in= 6 to n_out= 2

Using these, we compute the Rh for each transition as follows:

Step-by-step Calculations

1. Converting wavelengths to meters

λ in meters = wavelength in nm x 10^-9

• Red: 644 nm = 644 x 10^-9 m
• Turquoise: 518 nm = 518 x 10^-9 m
• Violet: 438 nm = 438 x 10^-9 m
• Faint Violet: 385 nm = 385 x 10^-9 m

2. Calculating Energy of Photons

Using E_photon = hc/λ, with h = 6.626 x 10^-34 J·s, c = 3.00 x 10⁸ m/s

• Red: E = (6.626 x 10^-34)(3.00 x 10⁸) / (644 x 10^-9) ≈ 3.09 x 10^-19 J
• Turquoise: ≈ 3.84 x 10^-19 J
• Violet: ≈ 4.55 x 10^-19 J
• Faint Violet: ≈ 5.17 x 10^-19 J

3. Calculating Rh Values

From the Rydberg formula:

Rh = 1 / (λ * (1/n_in² – 1/n_out²)))

Assuming n_in= 2 for all lines, and the destination levels as indicated, we substitute the known values.

Example Calculation for Red line (644 nm, n_out=2, n_in=3)

• Calculate (1/n_in² - 1/n_out²) = (1/3²) – (1/2²) = (1/9) – (1/4) = (4/36) – (9/36) = -5/36
• Note: Since the energy is emitted, the difference is positive, so take the absolute value: 5/36 ≈ 0.1389
• Calculate Rh: Rh = 1 / (λ * 0.1389) in SI units
• λ = 644 x 10^-9 m
• Rh ≈ 1 / (644 x 10^-9 * 0.1389) ≈ 1 / (8.956 x 10^-8) ≈ 1.116 x 10⁷ m^-1

Similarly, for the other lines

• Turquoise (518 nm, n_in=4, n_out=2):
• Calculate (1/4²) – (1/2²) = (1/16) – (1/4) = (1/16) – (4/16) = -3/16 ≈ 0.1875
• Rh ≈ 1 / (518 x 10^-9 * 0.1875) ≈ 1.02 x 10⁷ m^-1
• Violet (438 nm, n_in=5, n_out=2):
• (1/25) – (1/4) = 0.04 – 0.25 = -0.21, absolute value 0.21
• Rh ≈ 1 / (438 x 10^-9 * 0.21) ≈ 1.09 x 10⁷ m^-1
• Faint Violet (385 nm, n_in=6, n_out=2):
• (1/36) – (1/4)= 0.0278 – 0.25 = -0.2222, absolute value 0.2222
• Rh ≈ 1 / (385 x 10^-9 * 0.2222) ≈ 1.17 x 10⁷ m^-1

4. Calculating the Average Rh and Percentage Error

The calculated Rh values are approximately:

• Red: 1.116 x 10⁷ m^-1
• Turquoise: 1.02 x 10⁷ m^-1
• Violet: 1.09 x 10⁷ m^-1
• Faint Violet: 1.17 x 10⁷ m^-1

The average value of Rh is then:

Average Rh = (1.116 + 1.02 + 1.09 + 1.17) x 10⁷ / 4 ≈ (4.396 x 10⁷) /4 ≈ 1.099 x 10⁷ m^-1

For theoretical purposes, the accepted value of the Rydberg constant for hydrogen is approximately 1.097 x 10⁷ m^-1. The percentage error is calculated as:

% Error = |(Experimental Rh – Accepted Rh) / Accepted Rh| * 100

% Error ≈ |(1.099 x 10⁷ – 1.097 x 10⁷) / 1.097 x 10⁷| * 100 ≈ 0.182%

This small percentage difference indicates good experimental accuracy in the calculation of the Rydberg constant from the spectral data.

Conclusion

By applying the Rydberg formula and calculating each spectral transition's corresponding Rh, the average Rydberg constant is approximately 1.099 x 10⁷ m^-1. The minor deviation from the accepted value (~1.097 x 10⁷ m^-1) underscores the reliability of the spectroscopic measurements and the method employed. Such calculations are fundamental in atomic physics, affirming the quantum theoretical models of atomic structure and spectral emissions.

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