Properties of Light: Videos & Practice Problems

The electromagnetic spectrum encompasses various types of radiation, from long radio waves to gamma rays. Visible light, a small segment, ranges from approximately 700 nm (red) to 380 nm (violet). Key relationships include energy (E), frequency (ν), and wavelength (λ), where E = hν and c = νλ, with c being the speed of light (2.998 × 10⁸ m/s). As frequency increases, wavelength decreases, and energy rises, affecting atomic and molecular transitions, such as electron excitation in UV-visible spectroscopy and core electron interactions with X-rays.

The Nature of Light

The electromagnetic spectrum encompasses a wide range of radiant energy, with visible light representing only a small segment. The spectrum is organized from long radio waves on the left to gamma rays on the right. As we move from left to right, both frequency (represented by \( f \)) and energy (represented by \( E \)) increase, while wavelength (denoted by \( \lambda \)) decreases. This relationship can be summarized by the equation:

\[ c = \lambda f \]

where \( c \) is the speed of light. The visible light spectrum, which can be remembered using the acronym ROYGBIV (red, orange, yellow, green, blue, indigo, violet), spans approximately from 700 nanometers (nm) to 380 nm.

Different regions of the electromagnetic spectrum interact with matter in unique ways. For instance, radio waves primarily affect the nuclear spin of atoms, while microwaves influence molecular rotations and electron spins, which are fundamental in techniques like Nuclear Magnetic Resonance (NMR). Infrared radiation is associated with molecular vibrations, allowing for the identification of functional groups through infrared spectroscopy.

In the visible and ultraviolet (UV) regions, the focus shifts to valence electrons, with UV-visible spectroscopy examining conjugated systems—compounds with alternating double and single bonds. This process involves electronic excitation, where electrons jump to higher energy states.

As we progress to x-rays, the interaction involves core electrons, leading to bond breaking and ionization due to their higher energy levels. Finally, gamma rays, the most energetic form of electromagnetic radiation, are associated with nuclear reactions.

In summary, the electromagnetic spectrum illustrates the diversity of radiant energy, with each type influencing atomic and molecular structures in distinct ways, highlighting the importance of understanding these interactions in fields such as chemistry and physics.

The Nature of Light

The electromagnetic spectrum encompasses three key variables: energy, frequency, and wavelength. In this context, frequency, denoted by the symbol \( \mu \), represents the number of waves that pass a given point per second, measured in hertz (Hz) or seconds inverse. Wavelength, represented by \( \Lambda \), is the distance between consecutive crests of a wave, typically expressed in meters.

When visualizing an electromagnetic wave, the crest-to-crest distance corresponds to the wavelength \( \Lambda \), while the number of waves occurring in one second indicates the frequency \( \mu \). The electromagnetic wave itself can be illustrated in a three-dimensional space with axes representing different components, where the magnetic field vectors extend along the y-axis.

Importantly, the energy of an electromagnetic wave is directly proportional to its frequency; thus, a higher frequency results in greater energy. Conversely, the amplitude of the wave, which is the height from the crest to the trough, does not directly affect the energy of the wave. Instead, it is the frequency that plays a crucial role in determining energy levels.

It is essential to understand the inverse relationship between frequency and wavelength: as frequency increases, wavelength decreases. This means that if there are many waves in a second (high frequency), the distance between those waves (wavelength) will be shorter. This relationship is fundamental in understanding the various types of radiation within the electromagnetic spectrum, as they are all interconnected through these variables.

Properties of Light

The speed of a typical wave is defined as the product of frequency (μ) and wavelength (λ). In a vacuum, all forms of electromagnetic radiation travel at approximately ${2.998}^8$ meters per second, known as the speed of light (c). This relationship can be expressed with the equation:

$c=\mu \times \lambda$

Physicists Max Planck and Albert Einstein proposed that light consists of small packets of electromagnetic energy called quanta. The energy (E) of a single photon can be calculated using the formula:

$E=h\times \mu$

Here, h represents Planck's constant, valued at ${6.626}^{\times }$ joules times seconds. Additionally, the energy of a single photon can also be expressed in terms of wavelength:

$E=h\times \lambda$

To relate frequency and wavelength, we can isolate frequency (μ) using the speed of light:

$\mu =c/\lambda$

Substituting this into the energy equation gives:

$E=h\times c/\lambda$

This provides two versions of the energy equation: one for when frequency is known and another for when wavelength is given. The inverse relationship between frequency and wavelength is crucial for understanding how to calculate the energy of a single photon. Remembering these formulas will aid in solving problems related to electromagnetic radiation.

To practice, try calculating the wavelength from the provided information in the example. Compare your results to the expected answer to reinforce your understanding.

Properties of Light

To calculate the wavelength of red light emitted by a neon sign, we start with the relationship between the speed of light, frequency, and wavelength, expressed by the equation:

c = μ × λ

Here, c represents the speed of light, μ is the frequency, and λ is the wavelength. To find the wavelength, we rearrange the equation:

λ = c / μ

The speed of light is approximately 2.998 × 10⁸ m/s. The frequency given is 4.16 × 10⁸ MHz, which we need to convert to Hertz (Hz) since 1 MHz equals 10⁶ Hz. Thus:

4.16 × 10⁸ MHz = 4.16 × 10¹⁴ Hz

Now, substituting the values into the wavelength equation:

λ = (2.998 × 10⁸ m/s) / (4.16 × 10¹⁴ Hz)

Calculating this gives:

λ = 7.21 × 10^-7 m

Since the question asks for the wavelength in nanometers, we convert meters to nanometers using the conversion factor where 1 nm = 10^-9 m. Therefore:

λ = 7.21 × 10^-7 m × (1 nm / 10^-9 m) = 721 nm

This result indicates that the red light emitted by the neon sign has a wavelength of 721 nanometers, which is consistent with the visible light spectrum, particularly in the red range.

Next, to find the energy of a mole of photons, we can start by calculating the energy of a single photon using the equation:

E = h × μ

Where E is the energy, h is Planck's constant (6.626 × 10^-34 J·s), and μ is the frequency in Hz. After determining the energy of one photon, multiplying by Avogadro's number (6.022 × 10²³ mol^-1) will yield the energy of a mole of photons.

Properties of Light

To determine the energy in joules of a mole of photons associated with visible light of wavelength 493 nanometers, we first need to calculate the energy of a single photon. The energy of a photon can be expressed using the formula:

E = \frac{hc}{\lambda}

where:

• E is the energy of the photon in joules,
• h is Planck's constant, approximately \(6.626 \times 10^{-34}\) joules seconds,
• c is the speed of light, approximately \(2.998 \times 10^{8}\) meters per second,
• \(\lambda\) is the wavelength in meters.

First, we need to convert the wavelength from nanometers to meters. Since 1 nanometer equals \(10^{-9}\) meters, we convert 493 nanometers to meters:

\(\lambda = 493 \, \text{nm} = 4.93 \times 10^{-7} \, \text{m}\)

Now, substituting the values into the energy formula:

E = \frac{(6.626 \times 10^{-34} \, \text{J s})(2.998 \times 10^{8} \, \text{m/s})}{4.93 \times 10^{-7} \, \text{m}} \approx 4.029 \times 10^{-19} \, \text{J/photon}

Next, to find the energy per mole of photons, we use Avogadro's number, which is approximately \(6.022 \times 10^{23}\) photons per mole. The energy in joules per mole can be calculated as follows:

E_{mole} = E \times N_A = (4.029 \times 10^{-19} \, \text{J/photon}) \times (6.022 \times 10^{23} \, \text{photons/mole}) \approx 2.43 \times 10^{5} \, \text{J/mole}

Thus, the energy of a mole of photons associated with visible light of wavelength 493 nanometers is approximately \(2.43 \times 10^{5}\) joules per mole. This calculation highlights the importance of using both Planck's constant and Avogadro's number to convert between the energy of individual photons and the energy of a mole of photons.

Properties of Light Calculations 1

In this scenario, we are tasked with determining the wavelength of light emitted by a single photon from a laser pulse that produces 1.242 kilojoules of energy, containing approximately \(3.50 \times 10^{22}\) photons. To find the wavelength, we utilize the relationship between energy and wavelength, expressed by the equation:

E = \frac{hc}{\lambda}

where E is the energy of a photon, h is Planck's constant (\(6.626 \times 10^{-34} \, \text{J s}\)), c is the speed of light (\(3.00 \times 10^8 \, \text{m/s}\)), and \(\lambda\) is the wavelength in meters. To isolate the wavelength, we rearrange the equation:

\lambda = \frac{hc}{E}

Next, we need to calculate the energy of a single photon. First, we convert the total energy from kilojoules to joules:

1.242 \, \text{kJ} = 1.242 \times 10^3 \, \text{J}

Now, we divide the total energy by the number of photons to find the energy of one photon:

E_{\text{photon}} = \frac{1.242 \times 10^3 \, \text{J}}{3.50 \times 10^{22}} \approx 3.55 \times 10^{-20} \, \text{J}

With the energy of a single photon calculated, we can substitute this value back into the rearranged equation for wavelength:

\lambda = \frac{(6.626 \times 10^{-34} \, \text{J s})(3.00 \times 10^8 \, \text{m/s})}{3.55 \times 10^{-20} \, \text{J}} \approx 5.60 \times 10^{-7} \, \text{m}

This result indicates that the wavelength of light emitted by one photon from the laser pulse is approximately \(5.60 \times 10^{-7}\) meters, or 560 nanometers, which falls within the visible spectrum. Understanding the connection between energy and wavelength is crucial for solving similar problems, as it allows for the manipulation of the fundamental constants and the energy values to derive the desired wavelength.

Properties of Light Calculations 1

To determine the total energy required to remove electrons from a mole of hydrogen atoms, we start with the ionization energy of a single hydrogen atom, which is given as \(2.178 \times 10^{-18}\) joules. This value represents the energy needed to ionize one atom, or equivalently, one photon.To convert this energy into microjoules per mole, we first recognize that 1 microjoule is equal to \(10^{-6}\) joules. Therefore, we can express the ionization energy in microjoules:\[2.178 \times 10^{-18} \text{ joules} \times \frac{1 \text{ microjoule}}{10^{-6} \text{ joules}} = 2.178 \times 10^{-12} \text{ microjoules per atom}\]Next, we utilize Avogadro's number, which is \(6.022 \times 10^{23}\) atoms per mole, to convert the energy from per atom to per mole:\[2.178 \times 10^{-12} \text{ microjoules per atom} \times 6.022 \times 10^{23} \text{ atoms per mole} = 1.31 \times 10^{12} \text{ microjoules per mole}\]Thus, the total energy required to ionize one mole of hydrogen atoms is \(1.31 \times 10^{12}\) microjoules. This calculation illustrates the process of converting energy units and applying Avogadro's number to transition from atomic to molar quantities, reinforcing the importance of unit conversion in physical chemistry.

A low-pressure mercury-vapor lamp has a characteristic emission line at 253 nm. Knowing that this lamp is putting out 11.8 watts of light energy, how many mercury atoms are emitted per second during operation?

• A. 7.86x10^-19 Hg atoms/s
• B. 4.73x10⁵ Hg atoms/s
• C. 1.50x10¹⁹ Hg atoms/s
• D. 1.50x10²⁸ Hg atoms/s
• E. 9.27x10^-18 Hg atoms/s

Answer: (C)

Here’s what students ask on this topic:

What is the electromagnetic spectrum and what are its different types of radiation?

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from long radio waves to gamma rays. The spectrum includes, in order of increasing frequency and energy: long radio waves, radio waves (AM and FM), microwaves, infrared, visible light, ultraviolet (UV), X-rays, and gamma rays. Visible light, which humans can see without instruments, ranges from approximately 700 nm (red) to 380 nm (violet). As we move from radio waves to gamma rays, the frequency and energy increase, while the wavelength decreases.

How are energy, frequency, and wavelength related in the electromagnetic spectrum?

Energy (E), frequency (ν), and wavelength (λ) are interrelated in the electromagnetic spectrum. The energy of a photon is given by the equation $E=h\nu$, where $h$ is Planck's constant (6.626 × 10^-34 J·s). The speed of light (c) is related to frequency and wavelength by the equation $c=\nu \lambda$, where $c$ is approximately 2.998 × 10⁸ m/s. As frequency increases, wavelength decreases, and energy increases.

What is the visible light spectrum and its range in nanometers?

The visible light spectrum is the portion of the electromagnetic spectrum that can be seen by the human eye without the aid of instruments. It ranges from approximately 700 nanometers (nm) to 380 nm. The colors in the visible spectrum can be remembered using the acronym ROYGBIV, which stands for Red, Orange, Yellow, Green, Blue, Indigo, and Violet. Red light has the longest wavelength (~700 nm), while violet light has the shortest (~380 nm).

What is the speed of light and how is it related to frequency and wavelength?

The speed of light (c) in a vacuum is approximately 2.998 × 10⁸ meters per second (m/s). It is related to frequency (ν) and wavelength (λ) by the equation $c=\nu \lambda$. This means that the speed of light is the product of the frequency and the wavelength of the electromagnetic wave. As frequency increases, wavelength decreases, and vice versa, to maintain the constant speed of light.

What is Planck's constant and how is it used to calculate the energy of a photon?

Planck's constant (h) is a fundamental constant in physics with a value of 6.626 × 10^-34 joules·seconds (J·s). It is used to calculate the energy (E) of a photon using the equation $E=h\nu$, where $\nu$ is the frequency of the photon. Alternatively, if the wavelength (λ) is known, the energy can be calculated using $E=\frac{hc}{\lambda }$, where $c$ is the speed of light.