Question

The spectrum shown here is for aspirin. The vertical axis is the amount of light absorbed, and the horizontal axis is the wavelength of incident light (in \(\mathrm{nm}\) ). (For more on spectroscopy, see pages \(183-199 .\) )What is the frequency of light with a wavelength of \(278 \mathrm{nm} ?\) What is the energy of one mole of photons with \(\lambda=278 \mathrm{nm} ?\) What region of the electromagnetic spectrum is covered by the spectrum above? Knowing that aspirin only absorbs light in the region depicted by this spectrum, what is the color of aspirin? (GRAPH CAN'T COPY)

Short answer

The frequency is approximately \(1.08 \times 10^{15} \mathrm{Hz}\), the energy of one mole of photons is \(4.31 \times 10^5 \mathrm{J/mol}\), and the spectrum is in the ultraviolet region. Therefore, aspirin appears colorless.

Step-by-step solution

Convert Wavelength to Frequency

To find the frequency of light, we use the equation: \[ c = \lambda u \] where \( c \) is the speed of light \( (3.00 \times 10^8 \mathrm{m/s}) \), \( \lambda \) is the wavelength \( (278 \mathrm{nm}) \), and \( u \) is the frequency. Convert the wavelength from nanometers to meters: \[ 278 \mathrm{nm} = 278 \times 10^{-9} \mathrm{m} \] Rearrange the equation to solve for frequency \( u \): \[ u = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \mathrm{m/s}}{278 \times 10^{-9} \mathrm{m}} \approx 1.08 \times 10^{15} \mathrm{Hz} \]

Calculate Energy of a Photon

The energy of a single photon is calculated using the equation: \[ E = h u \] where \( h \) is Planck's constant \( (6.626 \times 10^{-34} \mathrm{J \cdot s}) \) and \( u \) is the frequency from Step 1 \((1.08 \times 10^{15} \mathrm{Hz})\). Substitute and solve for \( E \): \[ E = (6.626 \times 10^{-34} \mathrm{J \cdot s})(1.08 \times 10^{15} \mathrm{Hz}) \approx 7.16 \times 10^{-19} \mathrm{J/photon} \]

Calculate Energy of One Mole of Photons

To find the energy for one mole of photons, multiply the energy per photon by Avogadro's number \((6.022 \times 10^{23} \mathrm{mol}^{-1})\): \[ E_{\text{mole}} = (7.16 \times 10^{-19} \mathrm{J/photon})(6.022 \times 10^{23} \mathrm{mol}^{-1}) = 4.31 \times 10^5 \mathrm{J/mol} \]

Identify the Spectral Region

The given wavelength \(278 \mathrm{nm}\) falls into the ultraviolet (\

Key concepts

Wavelength

Wavelength is the distance between consecutive peaks or troughs in a wave, typically measured in nanometers (nm) for the context of spectroscopy. It is a fundamental characteristic that determines many properties of light, including its color and energy. When dealing with the electromagnetic spectrum, the wavelength indicates which region of the spectrum the light belongs to, like visible light, ultraviolet, or infrared. Shorter wavelengths such as those in the ultraviolet region have higher energy and can affect materials more intensely. In the exercise, the wavelength of 278 nm is used to determine other properties of light like frequency and energy. To perform calculations, it’s often necessary to convert the wavelength into different units like meters. This is done by recognizing that 1 nm equals 10^-9 meters. Understanding wavelength helps to further explore how light interacts with materials, determining things like absorption and transmission in spectroscopy scenarios.

Frequency

Frequency describes how often waves pass a given point per second and is measured in hertz (Hz). It is directly related to the wavelength according to the equation: \[ c = \lambda u \] where \( c \) is the speed of light, \( \lambda \) is the wavelength, and \( u \) is the frequency. In simpler terms, frequency measures the number of wave cycles that occur each second. In the exercise, we converted the given wavelength of 278 nm into frequency to find this measurement for light. Once the conversion is made, we apply the formula to find that the frequency is approximately \( 1.08 \times 10^{15} \) Hz. Remember, frequency and wavelength are inversely proportional. As the wavelength gets shorter, the frequency increases, meaning more cycles pass in the same amount of time. This understanding helps in identifying radiation types and their properties.

Photon energy

Photon energy is the energy carried by a single photon, the smallest particle of light, and is a key concept in understanding spectroscopy. The energy of a photon is calculated using Planck's relationship: \[ E = h u \] where \( E \) is the energy, \( h \) is Planck's constant \((6.626 \times 10^{-34} \text{ J\cdot s})\), and \( u \) is the frequency. In our context, this was applied to find the energy of a photon with the previously calculated frequency. Discovering that the energy per photon for this wavelength of light is approximately \( 7.16 \times 10^{-19} \) joules (J). Moreover, by understanding that real-world applications often require conversions to larger scales, we multiply by Avogadro's number to determine the energy in a mole of photons, \( 4.31 \times 10^{5} \) J/mol. This conversion is essential in fields like chemistry and physics, where interactions at molecular levels are considered.

Ultraviolet spectrum

The ultraviolet (UV) spectrum is a part of the electromagnetic spectrum with wavelengths shorter than visible light but longer than X-rays. It generally includes wavelengths from about 10 nm to 400 nm. Ultraviolet light is well-known for its role in sunburn and other skin changes but is also crucial in scientific applications like spectroscopy, where it helps identify substances based on absorption patterns. In the exercise, the wavelength of 278 nm falls within the ultraviolet range, which suggests that aspirin absorbs light in this region of the spectrum. This property is often used to determine characteristics of various compounds by analyzing which wavelengths they absorb or emit, depicting their spectral "fingerprint.” Understanding where a substance falls in the electromagnetic spectrum aids in figuring out how it might interact with light and its potential applications or effects.