Calculating Frequency, Energy of Photons, and Electronic Transitions

Sample Exercise 1: Calculating Frequency from Wavelength

• Problem Statement: The yellow light emitted by a sodium vapor lamp has a wavelength of 589 nm (nanometers). We seek to calculate the frequency of this radiation.

  • Given Information:
  • Wavelength, λ = 589 nm
  • Required: Frequency, ν

• Analysis:

  • The relationship between wavelength (λ) and frequency (ν) is expressed by the equation:
    
    u = \frac{c}{oldsymbol{\lambda}}
  • Here, c is the speed of light, which is approximately 3.00 x 10^8 m/s (to three significant figures).

• Plan:

  • Rearrange the equation to solve for frequency, then substitute the known values of wavelength and speed of light.
  • Note that the units of length for wavelength will need conversion from nanometers to meters to cancel appropriately.

• Solution:

  • Convert the wavelength from nanometers to meters:
    $589 \text{ nm} = 589 \times 10^{-9} \text{ m}$
  • Now, substituting values into the frequency equation:
    $\nu = \frac{3.00 \times 10^8 \text{ m/s}}{589 \times 10^{-9} \text{ m}}$
  • Calculate Frequency:
  • The computation yields a frequency of approximately:
    $\nu \approx 5.08 \times 10^{14} \text{ Hz}$

• Check:

  • The resulting frequency is high, consistent with a short wavelength of light. The units are correct, as frequency is expressed in per second (Hz).

Sample Exercise 1: Practice Problems

• (a) A laser used in orthopedic spine surgery produces radiation with a wavelength of 2.10 μm (micrometers). Calculate the frequency of this radiation using the speed of light given to four significant figures.

• (b) An FM radio station broadcasts electromagnetic radiation at a frequency of 103.4 MHz (megahertz). Calculate the wavelength of this radio frequency radiation. Use the speed of light specified to four significant figures.

Sample Exercise 2: Energy of a Photon

• Problem Statement: Calculate the energy of one photon of yellow light with a wavelength of 589 nm.

• Analysis:

  • The task is to find the energy E of a photon based on its wavelength. We need to relate it to frequency using the earlier equation and then calculate the energy using Planck’s equation:
    $E = h <br /> u$
  • Here, h is Planck's constant, approximately 6.626 x 10^{-34} J imes s.

• Plan:

  • Convert the wavelength to frequency exactly as done in Sample Exercise 1 and calculate energy using:
    $E = h \nu$

• Solution:

  1. Frequency Calculation:
  • From the prior calculation, ν = 5.08 x 10^{14} Hz.
  1. Energy Calculation:
  • Substituting values:
    $E = (6.626 \times 10^{-34} J \times s)(5.08 \times 10^{14} Hz)$
  • This yields approximately:
    $E \approx 3.37 \times 10^{-19} J$

• Comment: If one photon supplies 3.37 x 10^{-19} J of radiant energy, then we can calculate the energy supplied by one mole of such photons using Avogadro's number, approximately 6.022 x 10^{23} mol^{-1}, resulting in:
  $E_{mole} = 3.37 \times 10^{-19} J \times 6.022 \times 10^{23} \text{ mol}^{-1}$

Sample Exercise 2: Practice Problems

• (a) A laser emits light with a frequency of 5.10 x 10^{14} Hz. What is the energy of one photon of this radiation?
• (b) If the laser emits a pulse containing 3.00 x 10^{10} photons, compute the total energy of that pulse.
• (c) When the laser emits 2.00 J of energy during a pulse, determine how many photons are emitted.

Sample Exercise 3: Electronic Transitions in the Hydrogen Atom

• Overview: Discuss the Bohr model for the hydrogen atom, where electrons occupy fixed orbits with defined radii.

• Key Concept: Transition of an electron from a higher energy level to a lower one results in photon emission.

  • Electrons transition from level n = 4 to n = 3, 2, or 1. The transition leading to the shortest wavelength photon must be determined.

• Questions:

  • (a) Which transition produces a photon with the shortest wavelength?
  • (b) What are the energy and wavelength of such a photon, including its placement in the electromagnetic spectrum?

• Solution Steps:

  1. Determine which transition corresponds to the highest energy, thus the shortest wavelength:
  • Transition from n = 4 to n = 1 has the highest energy.
  1. Calculate energy for this photon using:
     $E = -13.6 \times \frac{1}{n<em>{final}^2} + 13.6 \times \frac{1}{n</em>{initial}^2}$
  2. Corresponding wavelength can be derived using
     $\lambda = \frac{hc}{E}$

• Results:

  • The emitted light is in the ultraviolet range due to the high energy transition.

• Practice Problem: Determine for each given transition (n = 3 to n = 1 and n = 2 to n = 4) whether a photon is emitted or absorbed, and provide the sign of the change in energy.

Sample Exercise 5: Subshells of the Hydrogen Atom

• Overview: Understanding subshell structures within the fourth shell (n = 4).

• Questions:

  • (a) Predict the number of subshells for n = 4.
  • (b) Identify each subshell's designation.
  • (c) How many orbitals exist within each subshell?

• Solution:

  • The possible values for l are 0, 1, 2, 3, leading to subshells 4s, 4p, 4d, and 4f. The orbitals are as follows:
  • 4s: 1 orbital
  • 4p: 3 orbitals
  • 4d: 5 orbitals
  • 4f: 7 orbitals

• Total Count of Orbitals in Shell:

  • $Total = 1 + 3 + 5 + 7 = 16 orbitals$

• Practice Problem: Identify the designation for the subshell with n = 5 and l = 1, determine how many orbitals are present, and list the values for m_l for each orbital.

To input the calculations from the notes into a scientific calculator, you'll primarily use the EXP or EE key for scientific notation and parentheses () for order of operations.

Here’s a general guide for the types of calculations shown:

General Calculator Tips:

• Scientific Notation (e.g., $3.00 \times 10^8$): Use the EXP or EE button (often located near the log or x^y keys). You would enter 3.00 EXP 8 or 3.00 EE 8. For negative exponents, you'd then a negative sign, e.g., 5.89 EXP -7.
• Parentheses (): Crucial for ensuring the correct order of operations, especially in division or complex expressions. Always enclose the numerator and denominator in separate parentheses if they contain multiple terms or operations.

Specific Examples from the Note:

1. Frequency Calculation (e.g., $\nu = \frac{c}{\lambda}$):

   • For $\nu = \frac{3.00 \times 10^8 \text{ m/s}}{589 \times 10^{-9} \text{ m}}$
   • Input: (3.00 EE 8) / (589 EE -9)

2. Energy Calculation (e.g., $E = h \nu$):

   • For $E = (6.626 \times 10^{-34} J \times s)(5.08 \times 10^{14} Hz)$
   • Input: (6.626 EE -34) * (5.08 EE 14)

3. Energy per Mole Calculation (e.g., $E_{mole} = E \times N_A$):

   • For $E_{mole} = 3.37 \times 10^{-19} J \times 6.022 \times 10^{23} \text{ mol}^{-1}$
   • Input: (3.37 EE -19) * (6.022 EE 23)

4. Electronic Transitions Energy Calculation (e.g., $E = -13.6 \times \frac{1}{n_{final}^2} + 13.6 \times \frac{1}{n_{initial}^2}$):

   • For a transition from $n=4$ to $n=1$:
   • Input: (-13.6 * (1 / (1^2))) + (13.6 * (1 / (4^2)))
   • Note: Many calculators allow you to directly input 1^2 or 4^2 using an x^2 button or x^y button. Remember $1^2=1$ and $4^2=16$. So, you could simplify to (-13.6) + (13.6 / 16) if you prerun the squares.

5. Wavelength from Energy (e.g., $\lambda = \frac{hc}{E}$):

   • If $h = 6.626 \times 10^{-34}$, $c = 3.00 \times 10^8$, and E is a calculated value (e.g., $3.37 \times 10^{-19} J$ as in Sample Exercise 2):
   • Input: (6.626 EE -34 * 3.00 EE 8) / (3.37 EE -19)

Always double-check your input and use the correct decimal points and signs.