How much energy does an X-ray with an [tex]$8 nm\left(8 \times 10^{-9} m\right)$[/tex] wavelength have?
A. [tex]$2.48 \times 10^{-17} J$[/tex]
B. [tex]$8.28 \times 10^{-26} J$[/tex]
C. [tex]$3.33 \times 10^{16} J$[/tex]
D. [tex]$1.99 \times 10^{-25} J$[/tex]
Answer (2)
The energy of the X-ray with a given wavelength of 8 nm is calculated to be approximately 2.48 × 1 0 − 17 J . Therefore, the correct answer is option A. This is derived using the formulas for frequency and energy of a photon.
;
Calculate the frequency f using the formula f = λ c , where c = 3.0 × 1 0 8 m / s and λ = 8 × 1 0 − 9 m .
Calculate the energy E using the formula E = h × f , where h = 6.626 × 1 0 − 34 J s .
Substitute the values to find E = ( 6.626 × 1 0 − 34 ) × 8 × 1 0 − 9 3.0 × 1 0 8 = 2.48475 × 1 0 − 17 J .
The energy of the X-ray is approximately 2.48 × 1 0 − 17 J , so the final answer is 2.48 × 1 0 − 17 J .
Explanation
Problem Analysis We are asked to calculate the energy of an X-ray given its wavelength. We will use the formula relating energy, Planck's constant, the speed of light, and wavelength to find the answer.
Energy Formula The formula to calculate the energy (E) of a photon (like an X-ray) is: E = h × f where:
E is the energy of the photon in Joules (J),
h is Planck's constant, which is approximately 6.626 × 1 0 − 34 Joule-seconds (Js),
f is the frequency of the photon in Hertz (Hz).
Frequency Calculation We are given the wavelength ( λ ) of the X-ray, not the frequency ( f ). However, we know that the speed of light ( c ) is related to the wavelength and frequency by the formula: c = λ × f where:
c is the speed of light, which is approximately 3.0 × 1 0 8 meters/second (m/s),
λ is the wavelength in meters (m),
f is the frequency in Hertz (Hz). We can rearrange this formula to solve for the frequency: f = λ c
Energy Calculation Now, substitute the expression for f into the energy equation: E = h × λ c We are given:
λ = 8 nm = 8 × 1 0 − 9 m
h = 6.626 × 1 0 − 34 J s
c = 3.0 × 1 0 8 m / s Plug in these values to calculate the energy: E = ( 6.626 × 1 0 − 34 J s ) × 8 × 1 0 − 9 m 3.0 × 1 0 8 m / s E = 8 × 1 0 − 9 ( 6.626 × 1 0 − 34 ) × ( 3.0 × 1 0 8 ) J E = 8 × 1 0 − 9 1.9878 × 1 0 − 25 J E = 2.48475 × 1 0 − 17 J
Final Answer The energy of the X-ray is approximately 2.48 × 1 0 − 17 J . Comparing this to the given options, we see that option A matches our calculated value.
Examples
X-rays are used in medical imaging to visualize bones and internal organs. The energy of the X-rays determines their ability to penetrate different tissues. Higher energy X-rays can penetrate denser materials, allowing doctors to see structures that would otherwise be hidden. Understanding the relationship between wavelength and energy is crucial for controlling the dosage and image quality in medical X-ray procedures. Also, X-rays are used in airport security to scan luggage for prohibited items. The energy of the X-rays must be carefully controlled to ensure effective screening without damaging the contents of the luggage.
