Wavelength Practice Problems Worksheet: Complete with ease ... - Free Printable

Let's solve each of these practice problems step by step using the fundamental relationships between wavelength, frequency, and energy in wave-particle physics.

We will use the following key equations:

1. Wave equation:
$$
c = \lambda \nu \quad \text{(speed of light = wavelength × frequency)}
$$

2. Energy of a photon:
$$
E = h\nu = \frac{hc}{\lambda}
$$
where:
- $ c = 3.00 \times 10^8 \, \text{m/s} $
- $ h = 6.626 \times 10^{-34} \, \text{J·s} $

---

Problem 1:

*Ultraviolet radiation has a frequency of $ 3.00 \times 10^{15} \, \text{Hz} $. Calculate the energy, in joules, of the photon.*

Given:
- $ \nu = 3.00 \times 10^{15} \, \text{Hz} $
- $ h = 6.626 \times 10^{-34} \, \text{J·s} $

Use:
$$
E = h\nu
$$

$$
E = (6.626 \times 10^{-34}) \times (3.00 \times 10^{15})
$$

$$
E = 1.9878 \times 10^{-18} \, \text{J}
$$

✔ Answer: $ \boxed{1.99 \times 10^{-18}} \, \text{J} $ (rounded to 3 significant figures)

---

Problem 2:

*Calculate the energy of microwave radiation with a frequency of $ 3.0 \times 10^9 \, \text{Hz} $.*

Given:
- $ \nu = 3.0 \times 10^9 \, \text{Hz} $

$$
E = h\nu = (6.626 \times 10^{-34}) \times (3.0 \times 10^9)
$$

$$
E = 1.9878 \times 10^{-24} \, \text{J}
$$

✔ Answer: $ \boxed{1.99 \times 10^{-24}} \, \text{J} $

---

Problem 3:

*A sodium vapor lamp emits light photons with a wavelength of $ 5.89 \times 10^{-7} \, \text{m} $. Calculate the energy of these photons.*

Given:
- $ \lambda = 5.89 \times 10^{-7} \, \text{m} $
- $ c = 3.00 \times 10^8 \, \text{m/s} $
- $ h = 6.626 \times 10^{-34} \, \text{J·s} $

Use:
$$
E = \frac{hc}{\lambda}
$$

First calculate $ hc $:
$$
hc = (6.626 \times 10^{-34}) \times (3.00 \times 10^8) = 1.9878 \times 10^{-25} \, \text{J·m}
$$

Now:
$$
E = \frac{1.9878 \times 10^{-25}}{5.89 \times 10^{-7}} = 3.375 \times 10^{-19} \, \text{J}
$$

✔ Answer: $ \boxed{3.38 \times 10^{-19}} \, \text{J} $ (rounded to 3 sig figs)

---

Problem 4:

*One of the electron transitions in a hydrogen atom produces infrared light with a wavelength of $ 7.404 \times 10^{-7} \, \text{m} $. Calculate the amount of energy lost during this transition.*

This is the same as calculating the energy of the emitted photon.

$$
E = \frac{hc}{\lambda}
$$

$$
E = \frac{1.9878 \times 10^{-25}}{7.404 \times 10^{-7}} = 2.684 \times 10^{-19} \, \text{J}
$$

✔ Answer: $ \boxed{2.68 \times 10^{-19}} \, \text{J} $

---

Problem 5:

*Calculate the energy, in joules, for an X-ray photon with a frequency of $ 4.4 \times 10^{18} \, \text{Hz} $.*

$$
E = h\nu = (6.626 \times 10^{-34}) \times (4.4 \times 10^{18})
$$

$$
E = 2.915 \times 10^{-15} \, \text{J}
$$

✔ Answer: $ \boxed{2.92 \times 10^{-15}} \, \text{J} $

---

Problem 6:

*A ruby laser produces red light having a wavelength of $ 6.93 \times 10^{-7} \, \text{m} $. Calculate the energy, in joules.*

$$
E = \frac{hc}{\lambda} = \frac{1.9878 \times 10^{-25}}{6.93 \times 10^{-7}}
$$

$$
E = 2.868 \times 10^{-19} \, \text{J}
$$

✔ Answer: $ \boxed{2.87 \times 10^{-19}} \, \text{J} $

---

Problem 7:

*Calculate the frequency of UV light that has an energy of $ 2.30 \times 10^{-19} \, \text{J} $.*

Use:
$$
E = h\nu \Rightarrow \nu = \frac{E}{h}
$$

$$
\nu = \frac{2.30 \times 10^{-19}}{6.626 \times 10^{-34}} = 3.47 \times 10^{14} \, \text{Hz}
$$

✔ Answer: $ \boxed{3.47 \times 10^{14}} \, \text{Hz} $

---

Problem 8:

*Calculate the wavelength and frequency of photons with an energy of $ 1.4 \times 10^{-19} \, \text{J} $.*

#### Step 1: Find frequency
$$
\nu = \frac{E}{h} = \frac{1.4 \times 10^{-19}}{6.626 \times 10^{-34}} = 2.11 \times 10^{14} \, \text{Hz}
$$

#### Step 2: Find wavelength
$$
\lambda = \frac{c}{\nu} = \frac{3.00 \times 10^8}{2.11 \times 10^{14}} = 1.42 \times 10^{-6} \, \text{m}
$$

So:
- Frequency: $ \boxed{2.11 \times 10^{14}} \, \text{Hz} $
- Wavelength: $ \boxed{1.42 \times 10^{-6}} \, \text{m} $ or $ 1420 \, \text{nm} $

(Note: This is in the infrared region.)

---

✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ 1.99 \times 10^{-18} \, \text{J} $ |
| 2 | $ 1.99 \times 10^{-24} \, \text{J} $ |
| 3 | $ 3.38 \times 10^{-19} \, \text{J} $ |
| 4 | $ 2.68 \times 10^{-19} \, \text{J} $ |
| 5 | $ 2.92 \times 10^{-15} \, \text{J} $ |
| 6 | $ 2.87 \times 10^{-19} \, \text{J} $ |
| 7 | $ 3.47 \times 10^{14} \, \text{Hz} $ |
| 8 | $ \nu = 2.11 \times 10^{14} \, \text{Hz}, \, \lambda = 1.42 \times 10^{-6} \, \text{m} $ |

Let me know if you'd like these converted into other units (like eV), or explained further!