Question

One reason carbon monoxide (CO) is toxic is that it binds to the blood protein hemoglobin more strongly than oxygen does. The bond between hemoglobin and CO absorbs radiation of \(1953 \mathrm{~cm}^{-1}\). (The unit is the reciprocal of the wavelength in centimeters.) Calculate the wavelength (in \(\mathrm{nm}\) and \(\dot{\mathrm{A}}\) ) and the frequency (in \(\mathrm{Hz}\) ) of the absorbed radiation.

Short answer

The wavelength is 5120 nm or 5.12 \times 10^4 \dot{\mathrm{A}}, and the frequency is 5.85 \times 10^{13} Hz.

Step-by-step solution

- Understand the given data

The problem provides that the absorption of radiation by the bond between hemoglobin and CO occurs at a wavenumber of \text{wavelength in reciprocal centimeters} (in \text{cm}^{-1}). The given wavenumber is \(\tilde{u} = 1953 \text{ cm}^{-1}\). We need to find the wavelength in \(\mathrm{nm}\) and \(\dot{\mathrm{A}}\), as well as the frequency in \(\mathrm{Hz}\).

- Convert wavenumber to wavelength in cm

The relationship between the wavenumber (\(\tilde{u}\)) and wavelength (\(\lambda\)) is given by: \[ \tilde{u} = \frac{1}{\lambda} \] This means: \[ \lambda = \frac{1}{\tilde{u}} \] Substituting the wavenumber: \[ \lambda = \frac{1}{1953 \text{ cm}^{-1}} \approx 5.12 \times 10^{-4} \text{ cm} \]

- Convert wavelength from cm to nm

To convert the wavelength from cm to nm, use the conversion factor \(1 \text{ cm} = 10^7 \text{ nm}\): \[ \lambda = 5.12 \times 10^{-4} \text{ cm} \times 10^7 \frac{\text{nm}}{\text{cm}} = 5120 \text{ nm} \]

- Convert wavelength from cm to angstroms

To convert the wavelength from cm to angstroms (\(\dot{\mathrm{A}}\)), use the conversion factor \(1 \text{ cm} = 10^{8} \dot{\mathrm{A}}\): \[ \lambda = 5.12 \times 10^{-4} \text{ cm} \times 10^{8} \frac{\dot{\mathrm{A}}}{\text{cm}} = 5.12 \times 10^{4} \dot{\mathrm{A}} \]

- Calculate frequency from wavenumber

Frequency (\(u\)) can be calculated using the speed of light \(c\) and the wavenumber (\(\tilde{u}\)): \[ u = c \tilde{u} \] where \(c = 2.998 \times 10^{10} \text{ cm/s}\): \[ u = 2.998 \times 10^{10} \text{ cm/s} \times 1953 \text{ cm}^{-1} = 5.85 \times 10^{13} \text{ Hz} \]

Key concepts

wavenumber to wavelength conversion

Understanding how to convert wavenumber to wavelength is essential in this problem. The wavenumber is simply the reciprocal of the wavelength in centimeters. If we know the wavenumber (\(\tilde{u}\)) of a radiation, we can find its wavelength (\(\lambda\)) using the formula: \[ \tilde{u} = \frac{1}{\lambda} \]. This can be rearranged to find the wavelength: \[ \lambda = \frac{1}{\tilde{u}} \]. By substituting the given wavenumber value, \(\tilde{u} = 1953 \text{ cm}^{-1}\), we get: \[ \lambda = \frac{1}{1953 \text{ cm}^{-1}} \approx 5.12 \times 10^{-4} \text{ cm} \]. This shows the conversion process in a simple way, highlighting the direct relationship between the wavenumber and wavelength.

wavelength in different units

Once we have the wavelength in centimeters, it's often useful to convert it into other units such as nanometers (nm) or angstroms (\( \dot{A} \)). Understanding the conversion factors is the key. For centimeters to nanometers, we use the conversion factor: \[ 1 \text{ cm} = 10^7 \text{ nm} \]. Therefore: \(\lambda = 5.12 \times 10^{-4} \text{ cm} \times 10^7 \frac{\m}{\text{cm}} \approx 5120 \text{ nm} \). Next, to convert from centimeters to angstroms, we use: \[ 1 \text{ cm} = 10^{8} \dot{A} \]. Substituting in the centimeter value: \(\lambda = 5.12 \times 10^{-4} \text{ cm} \times 10^{8} \frac{\dot{A}}{\text{cm}} \approx 5.12 \times 10^{4} \dot{A} \). These simple conversions make it easy to understand and compare wavelengths in different units.

frequency calculation

The frequency of radiation can be calculated if we know the wavenumber and the speed of light. The speed of light (\(c\)) is approximately \[ c = 2.998 \times 10^{10} \text{ cm/s} \]. The frequency (\( u \)) can be determined using the formula: \[ u = c \tilde{u} \]. Substituting in our given values: \(\tilde{u} = 1953 \text{ cm}^{-1} \text{ and } \ c = 2.998 \times 10^{10} \text{ cm/s} \), we find: \[ u = 2.998 \times 10^{10} \text{ cm/s} \times 1953 \text{ cm}^{-1} \approx 5.85 \times 10^{13} \text{ Hz} \]. This process shows how using simple formulas and given constants allows for calculating the frequency from the wavenumber.