**315nm (wavelength of ultraviolet light from the sun in the first band). Express your answer up to three significant figures.****0.0780nm (a wavelength used in medical X-rays). Express your answer up to three significant figures.****632.8nm (wavelength of red light from a helium-neon laser). Express your answer up to three significant figures.**

This question aims to determine the frequency of various electromagnetic radiations through their wavelengths. The wavelength of an electromagnetic wave refers to the distance between its consecutive crests or troughs. Whereas the frequency of an electromagnetic wave refers to the number of times a wavelength is repeated in a second.

The relation between wavelength and frequency is expressed through the following equation:

\[ c = \lambda \times v \]

Where $c$ refers to the speed of light ($3 x10^{8} m/s$), lambda refers to the wavelength, and v refers to the frequency.

In the question, three different wavelengths are mentioned. In part (1), the wavelength of ultraviolet light coming from the sun in the first band is given. In part (2), the wavelength of an X-ray is given, and similarly, in part (3) the wavelength of the red light from a helium-neon laser is given. The above equation can be used to determine the frequency of these wavelengths.

**Expert Solution**

- The wavelength given in this part is $315nm$ ($315 x 10^{-9}m$). In order to determine the frequency of this wavelength, the following equation will be used:

\[ c = \lambda \times v \]

Upon rearranging this equation, the following equation is obtained for determining the frequency:

\[ v = c / \lambda \]

Inserting all the values in the above equation:

\[ v = c / \lambda \]

\[ v = 3 x 10^{8} / 315 x 10^{-9} \]

**\[ v = 9.52 x 10^{14} Hz \]**

2. The wavelength given in this part is $0.0780nm$ ($0.0780 x 10^{-9}m$). In order to determine the frequency of this wavelength, the following equation will be used:

\[ c = \lambda \times v \]

Upon rearranging this equation, the following equation is obtained for determining the frequency:

\[ v = c / \lambda \]

Inserting all the values in the above equation:

\[ v = c / \lambda \]

\[ v = 3 x 10^{8} / 0.0780 x 10^{-9} \]

**\[ v = 3.85 x 10^{18} Hz \]**

3. The wavelength given in this part is $632.8nm$ ($632.8 x 10^{-9}m$). In order to determine the frequency of this wavelength, the following equation will be used:

\[ c = \lambda \times v \]

Upon rearranging this equation, the following equation is obtained for determining the frequency:

\[ v = c / \lambda \]

Inserting all the values in the above equation:

\[ v = c / \lambda \]

\[ v = 3 x 10^{8} / 632.8 x 10^{-9} \]

**\[ v = 4.74 x 10^{14} Hz \]**

**Alternate Solution**

For determining the frequency of the given wavelengths, the following formula will be used:

\[ v = c / \lambda \]

- $\lambda$ = $315nm$

\[ v = 3 x 10^{8} / 315 x 10^{-9} \]

**\[ v = 9.52 x 10^{14} Hz \]**

2. $\lambda$ = $0.0780nm$

\[ v = 3 x 10^{8} / 0.0780 x 10^{-9} \]

**\[ v = 3.85 x 10^{18} Hz \]**

3. $\lambda$ = $632.8nm$

\[ v = 3 x 10^{8} / 632.8 x 10^{-9} \]

**\[ v = 4.74 x 10^{14} Hz \]**

**Example**

The wavelength of blue light in the electromagnetic spectrum is $487nm$. Determine its frequency and express the answer in five significant figures.

The formula for determining the frequency of this wavelength is given below:

\[ c = \lambda \times v \]

\[ v = c / \lambda \]

Where c = $3 x 10^{8}m$.

Inserting the values in the formula:

\[ v = 3 x 10^{8} / 487 x 10^-{9} \]

**\[ v = 6.1602 x 10^{14} Hz \]**