Some instruments differentiate individual quanta of electromagnetic radiation based on their energies. Assume such an instrument has been adjusted to detect quanta that have 3.75 × 10–16 J of energy. What is the wavelength of the detected radiation? Give your answer in nanometers.

To find the wavelength of the electromagnetic radiation given the energy of the quanta, we can use the relationship between energy and wavelength in the context of photons. The energy \( E \) of a photon is related to its wavelength \( \lambda \) by the equation:

\[
E = \frac{hc}{\lambda}
\]

where:
- \( E \) is the energy of the photon (in joules),
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \)),
- \( c \) is the speed of light in a vacuum (\( 3.00 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength (in meters).

Given that the energy \( E \) is \( 3.75 \times 10^{-16} \) J, we can rearrange the formula to solve for \( \lambda \):

\[
\lambda = \frac{hc}{E}
\]

Now, substitute the known values into the equation:

\[
\lambda = \frac{(6.626 \times 10^{-34} \, \text{J} \cdot \text{s})(3.00 \times 10^8 \, \text{m/s})}{3.75 \times 10^{-16} \, \text{J}}
\]

\[
\lambda = \frac{1.9878 \times 10^{-25} \, \text{J} \cdot \text{m}}{3.75 \times 10^{-16} \, \text{J}}
\]

\[
\lambda = 5.3013 \times 10^{-10} \, \text{m}
\]

To convert this wavelength to nanometers (1 nm = \( 10^{-9} \) m):

\[
\lambda = 5.3013 \times 10^{-10} \, \text{m} \times \frac{10^9 \, \text{nm}}{1 \, \text{m}}
\]

\[
\lambda = 530.13 \, \text{nm}
\]

**Therefore, the wavelength of the detected radiation is approximately 530.13 nanometers.**

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the frequency of electromagnetic radiation with a wavelength of 530.13 nm?
2. How does the energy of a photon change as its wavelength increases?
3. What part of the electromagnetic spectrum does a wavelength of 530.13 nm belong to?
4. How would the energy of a photon change if its wavelength was halved?
5. What is the energy of a photon with a wavelength of 700 nm?

**Tip:** When working with photon energy and wavelength, remember that as the wavelength decreases, the energy of the photon increases, and vice versa. This inverse relationship is crucial for understanding the behavior of electromagnetic radiation.

Learn how to calculate the wavelength of electromagnetic radiation given the energy of the photon (3.75 × 10–16 J). This problem involves applying the photon energy formula and understanding the relationship between energy and wavelength in quantum mechanics.

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