Solved Chemistry Worksheet - Wavelength, frequency, & | Chegg.com - Free Printable

Let's solve each problem step by step using the relevant equations:

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Key Equations:

1. $ c = \lambda \nu $
→ $ \lambda = \frac{c}{\nu} $, $ \nu = \frac{c}{\lambda} $

2. $ E = h\nu $
→ $ \nu = \frac{E}{h} $, $ E = h\nu $

3. Constants:
- $ c = 3.00 \times 10^8 \, \text{m/s} $
- $ h = 6.626 \times 10^{-34} \, \text{J·s} $
- $ 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} $

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We'll go through each question with calculations, units, significant figures, and identify the type of radiation based on the electromagnetic spectrum.

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1. Wavelength of a wave with frequency $ 3.76 \times 10^{14} \, \text{s}^{-1} $?

$$
\lambda = \frac{c}{\nu} = \frac{3.00 \times 10^8 \, \text{m/s}}{3.76 \times 10^{14} \, \text{Hz}} = 7.98 \times 10^{-7} \, \text{m}
$$

Convert to nanometers: $ 7.98 \times 10^{-7} \, \text{m} = 798 \, \text{nm} $

- Answer: $ \boxed{7.98 \times 10^{-7} \, \text{m}} $ or $ 798 \, \text{nm} $
- Type: Visible light (red region)

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2. Frequency of a $ 6.9 \times 10^{-13} \, \text{m} $ wave?

$$
\nu = \frac{c}{\lambda} = \frac{3.00 \times 10^8}{6.9 \times 10^{-13}} = 4.35 \times 10^{20} \, \text{Hz}
$$

- Answer: $ \boxed{4.35 \times 10^{20} \, \text{Hz}} $
- Type: Gamma rays

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3. Wavelength of a 2.99 Hz wave?

$$
\lambda = \frac{c}{\nu} = \frac{3.00 \times 10^8}{2.99} = 1.00 \times 10^8 \, \text{m}
$$

- Answer: $ \boxed{1.00 \times 10^8 \, \text{m}} $
- Type: Radio waves (very long wavelength)

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4. Wavelength of a $ 1.28 \times 10^{17} \, \text{Hz} $ wave?

$$
\lambda = \frac{3.00 \times 10^8}{1.28 \times 10^{17}} = 2.34 \times 10^{-9} \, \text{m}
$$

- Answer: $ \boxed{2.34 \times 10^{-9} \, \text{m}} $ or $ 2.34 \, \text{nm} $
- Type: X-rays

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5. Frequency of a $ 7.43 \times 10^{-5} \, \text{m} $ wave?

$$
\nu = \frac{3.00 \times 10^8}{7.43 \times 10^{-5}} = 4.04 \times 10^{12} \, \text{Hz}
$$

- Answer: $ \boxed{4.04 \times 10^{12} \, \text{Hz}} $
- Type: Infrared (far IR)

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6. Frequency of a 2.600 cm wave?

Convert cm to m: $ 2.600 \, \text{cm} = 0.02600 \, \text{m} $

$$
\nu = \frac{3.00 \times 10^8}{0.02600} = 1.154 \times 10^{10} \, \text{Hz}
$$

- Answer: $ \boxed{1.154 \times 10^{10} \, \text{Hz}} $
- Type: Microwave

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7. Wavelength of a $ 4.34 \times 10^{15} \, \text{Hz} $ wave?

$$
\lambda = \frac{3.00 \times 10^8}{4.34 \times 10^{15}} = 6.94 \times 10^{-8} \, \text{m} = 69.4 \, \text{nm}
$$

- Answer: $ \boxed{6.94 \times 10^{-8} \, \text{m}} $
- Type: Ultraviolet (UV)

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8. Wavelength of a 109.6 MHz wave?

Convert MHz to Hz: $ 109.6 \, \text{MHz} = 1.096 \times 10^8 \, \text{Hz} $

$$
\lambda = \frac{3.00 \times 10^8}{1.096 \times 10^8} = 2.737 \, \text{m}
$$

- Answer: $ \boxed{2.74 \, \text{m}} $ (3 sig figs)
- Type: Radio waves (FM radio band)

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9. Energy of a $ 7.66 \times 10^{14} \, \text{Hz} $ wave?

$$
E = h\nu = (6.626 \times 10^{-34})(7.66 \times 10^{14}) = 5.077 \times 10^{-19} \, \text{J}
$$

- Answer: $ \boxed{5.08 \times 10^{-19} \, \text{J}} $ (3 sig figs)
- Convert to eV: $ \frac{5.08 \times 10^{-19}}{1.602 \times 10^{-19}} = 3.17 \, \text{eV} $
- Type: UV light

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10. Frequency of a wave with energy $ 8.35 \times 10^{-18} \, \text{J} $?

$$
\nu = \frac{E}{h} = \frac{8.35 \times 10^{-18}}{6.626 \times 10^{-34}} = 1.260 \times 10^{16} \, \text{Hz}
$$

- Answer: $ \boxed{1.26 \times 10^{16} \, \text{Hz}} $
- Type: X-rays

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11. Frequency of a $ 1.78 \times 10^{-15} \, \text{J} $ wave?

$$
\nu = \frac{1.78 \times 10^{-15}}{6.626 \times 10^{-34}} = 2.686 \times 10^{18} \, \text{Hz}
$$

- Answer: $ \boxed{2.69 \times 10^{18} \, \text{Hz}} $
- Type: Gamma rays

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12. Energy of a $ 3.12 \times 10^{18} \, \text{Hz} $ wave?

$$
E = h\nu = (6.626 \times 10^{-34})(3.12 \times 10^{18}) = 2.069 \times 10^{-15} \, \text{J}
$$

- Answer: $ \boxed{2.07 \times 10^{-15} \, \text{J}} $
- Type: Gamma rays

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13. Frequency of a $ 1.31 \times 10^{-22} \, \text{J} $ wave? What is its wavelength?

#### Frequency:
$$
\nu = \frac{1.31 \times 10^{-22}}{6.626 \times 10^{-34}} = 1.975 \times 10^{11} \, \text{Hz}
$$

- Frequency: $ \boxed{1.98 \times 10^{11} \, \text{Hz}} $

#### Wavelength:
$$
\lambda = \frac{c}{\nu} = \frac{3.00 \times 10^8}{1.975 \times 10^{11}} = 1.518 \times 10^{-3} \, \text{m} = 1.52 \, \text{mm}
$$

- Wavelength: $ \boxed{1.52 \, \text{mm}} $
- Type: Microwave

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14. Wavelength of a $ 7.65 \times 10^{-17} \, \text{J} $ wave?

First find frequency:
$$
\nu = \frac{7.65 \times 10^{-17}}{6.626 \times 10^{-34}} = 1.154 \times 10^{17} \, \text{Hz}
$$

Then:
$$
\lambda = \frac{3.00 \times 10^8}{1.154 \times 10^{17}} = 2.599 \times 10^{-9} \, \text{m} = 2.60 \, \text{nm}
$$

- Answer: $ \boxed{2.60 \times 10^{-9} \, \text{m}} $
- Type: X-rays

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15. Energy of a 9,330 cm wave?

Convert cm to m: $ 9,330 \, \text{cm} = 93.30 \, \text{m} $

$$
\nu = \frac{3.00 \times 10^8}{93.30} = 3.214 \times 10^6 \, \text{Hz}
$$

$$
E = h\nu = (6.626 \times 10^{-34})(3.214 \times 10^6) = 2.128 \times 10^{-27} \, \text{J}
$$

- Answer: $ \boxed{2.13 \times 10^{-27} \, \text{J}} $
- Type: Radio waves

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16. Wavelength of a $ 1.32 \times 10^{-6} \, \text{eV} $ wave?

Convert eV to J:
$$
E = (1.32 \times 10^{-6})(1.602 \times 10^{-19}) = 2.114 \times 10^{-25} \, \text{J}
$$

Find frequency:
$$
\nu = \frac{2.114 \times 10^{-25}}{6.626 \times 10^{-34}} = 3.190 \times 10^8 \, \text{Hz}
$$

Now wavelength:
$$
\lambda = \frac{3.00 \times 10^8}{3.190 \times 10^8} = 0.940 \, \text{m}
$$

- Answer: $ \boxed{0.940 \, \text{m}} $
- Type: Radio waves

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17. Wavelength of a $ 1.528 \times 10^{-13} \, \text{J} $ wave?

$$
\nu = \frac{1.528 \times 10^{-13}}{6.626 \times 10^{-34}} = 2.305 \times 10^{20} \, \text{Hz}
$$

$$
\lambda = \frac{3.00 \times 10^8}{2.305 \times 10^{20}} = 1.301 \times 10^{-12} \, \text{m}
$$

- Answer: $ \boxed{1.30 \times 10^{-12} \, \text{m}} $
- Type: Gamma rays

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## ✔ Summary Table:

| # | Answer | Type |
|---|--------|------|
| 1 | $ 7.98 \times 10^{-7} \, \text{m} $ | Visible (Red) |
| 2 | $ 4.35 \times 10^{20} \, \text{Hz} $ | Gamma rays |
| 3 | $ 1.00 \times 10^8 \, \text{m} $ | Radio |
| 4 | $ 2.34 \times 10^{-9} \, \text{m} $ | X-rays |
| 5 | $ 4.04 \times 10^{12} \, \text{Hz} $ | Infrared |
| 6 | $ 1.154 \times 10^{10} \, \text{Hz} $ | Microwave |
| 7 | $ 6.94 \times 10^{-8} \, \text{m} $ | UV |
| 8 | $ 2.74 \, \text{m} $ | Radio |
| 9 | $ 5.08 \times 10^{-19} \, \text{J} $ | UV |
| 10 | $ 1.26 \times 10^{16} \, \text{Hz} $ | X-rays |
| 11 | $ 2.69 \times 10^{18} \, \text{Hz} $ | Gamma rays |
| 12 | $ 2.07 \times 10^{-15} \, \text{J} $ | Gamma rays |
| 13 | $ \nu = 1.98 \times 10^{11} \, \text{Hz}, \lambda = 1.52 \, \text{mm} $ | Microwave |
| 14 | $ 2.60 \times 10^{-9} \, \text{m} $ | X-rays |
| 15 | $ 2.13 \times 10^{-27} \, \text{J} $ | Radio |
| 16 | $ 0.940 \, \text{m} $ | Radio |
| 17 | $ 1.30 \times 10^{-12} \, \text{m} $ | Gamma rays |

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