Problem 2 - Basic Principles of Organic Chemistry

Question

Blue light has \(\bar{\nu}=20,800 \mathrm{~cm}^{-1}\). Calculate \(\nu\) in \(\mathrm{Hz}\) and \(\lambda\) in \(\mathrm{nm}\).

Step-by-Step Solution

Answer

Frequency \( \nu \) is \( 6.23 \times 10^{14} \) Hz and wavelength \( \lambda \) is 481 nm.

Step 1: Conversion of Wavenumber to Frequency

The given wavenumber \( \bar{u} = 20,800 \, \mathrm{cm}^{-1} \) is the reciprocal of wavelength in centimeters. To find frequency \( u \) in Hertz (Hz), we use the formula \( u = \bar{u} \times c \), where \( c \) is the speed of light \( 2.998 \times 10^{10} \, \mathrm{cm/s} \).Calculate \( u \):\[u = 20,800 \, \mathrm{cm}^{-1} \times 2.998 \times 10^{10} \, \mathrm{cm/s} = 6.23 \times 10^{14} \, \mathrm{Hz}\]

Step 2: Conversion of Wavenumber to Wavelength in cm

To find the wavelength \( \lambda \) in centimeters, use the reciprocal of the wavenumber.Calculate \( \lambda \):\[\lambda = \frac{1}{\bar{u}} = \frac{1}{20,800 \, \mathrm{cm}^{-1}} = 4.81 \times 10^{-5} \, \mathrm{cm}\]

Step 3: Conversion of Wavelength from cm to nm

Since 1 cm is equal to \( 10^7 \) nanometers, convert the wavelength from centimeters to nanometers.Calculate \( \lambda \) in nanometers:\[\lambda = 4.81 \times 10^{-5} \, \mathrm{cm} \times 10^7 \, \mathrm{nm/cm} = 481 \, \mathrm{nm}\]

Key Concepts

Frequency Calculation

Calculating the frequency (\(u\)) from a given wavenumber is an essential skill in understanding wave behavior and properties. Wavenumber \(\bar{u}\) signifies the number of wave cycles in a given unit distance and is inversely proportional to wavelength. It is given in the unit \(\text{cm}^{-1}\) (reciprocal centimeters).
To convert this wavenumber to frequency, we utilize the speed of light (\(c\)), which is a constant necessary for bridging the gap between the spatial and temporal characteristics of waves.

• The formula to convert wavenumber to frequency:\(u = \bar{u} \times c\).
• Where \(c\), the speed of light, is approximately \(2.998 \times 10^{10} \, \text{cm/s}\).

Plugging our values into the formula, the frequency is calculated as follows:\[u = 20,800 \, \text{cm}^{-1} \times 2.998 \times 10^{10} \, \text{cm/s} = 6.23 \times 10^{14} \, \text{Hz}\].
This gives us the frequency of the wave in Hertz, revealing how many cycles occur per second.

Wavelength in Nanometers

Wavelength, an inherent property of waves, is commonly expressed in nanometers (nm) when dealing with light to provide a more convenient size metric. To find the wavelength in nanometers, we must first determine the wavelength in centimeters using the wavenumber.

• The relationship between wavenumber and wavelength is given by:\( \lambda = \frac{1}{\bar{u}}\).
• From the exercise, this results in a wavelength of \(\lambda = \frac{1}{20,800 \, \text{cm}^{-1}} = 4.81 \times 10^{-5} \, \text{cm}\).

Next, convert this wavelength from centimeters to nanometers. Remember, one centimeter equals \(10^7\) nanometers.

• The conversion formula is: \(\lambda_{\text{nm}} = \lambda_{\text{cm}} \times 10^7\).

Applying this conversion:\[\lambda = 4.81 \times 10^{-5} \, \text{cm} \times 10^7 \, \text{nm/cm} = 481 \, \text{nm}\].
Thus, the wavelength of blue light is 481 nanometers, enabling precise characterization of its color and energy properties.

Speed of Light

The speed of light (\(c\)) is a universal constant vital for calculations involving wave properties. It links frequency and wavelength through its constancy in a vacuum, providing a consistent basis for translating spatial intervals into time intervals.

• Value in air or vacuum: \(2.998 \times 10^{10} \, \text{cm/s}\) or \(3.00 \times 10^8 \, \text{m/s}\).
• Acts as the maximum speed limit in the universe, ensuring physical laws hold across all frames of reference.

In the process of our calculations, it facilitates the conversion between wavenumbers (\(\bar{u}\)) to frequency (\(u\)) through its multiplication with the given wavenumber value.
The reliability of the speed of light makes it a cornerstone of calculations in wave mechanics, optics, and electromagnetic theory, providing a bridge between different units of physical quantities.