Problem 49 - College Physics

Question

Structure of a virus. To investigate the structure of extremely small objects, such as viruses, the wavelength of the probing wave should be about one-tenth the size of the object for sharp images. But as the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. One alternative is to use electron matter waves instead of light. Viruses vary considerably in size, but \(50 \mathrm{nm}\) is not unusual. Suppose you want to study such a virus, using a wave of wavelength \(5.00 \mathrm{nm} .\) (a) If you use light of this wavelength, what would be the energy (in eV) of a single photon? (b) If you use an electron of this wavelength, what would be its kinetic energy (in eV)? Is it now clear why matter waves (such as in the electron microscope) are often preferable to electromagnetic waves for studying microscopic objects?

Step-by-Step Solution

Answer

Photon energy: 248 eV; electron energy: 6 eV. Electron waves cause less damage than photons.

Step 1: Calculate Photon Energy with Light

The energy of a photon is given by the formula \[ E = \frac{hc}{\lambda} \]where \( h \) is Planck's constant \(6.626 \times 10^{-34} \text{ Js}\), \( c \) is the speed of light \(3.00 \times 10^8 \text{ m/s}\), and \( \lambda \) is the wavelength in meters.First, convert the wavelength from nanometers to meters:\[ 5.00 \text{ nm} = 5.00 \times 10^{-9} \text{ m} \]Substituting the values, we calculate:\[ E = \frac{(6.626 \times 10^{-34} \text{ Js})(3.00 \times 10^8 \text{ m/s})}{5.00 \times 10^{-9} \text{ m}} \]Simplified, \[ E \approx 3.98 \times 10^{-17} \text{ J} \]To convert energy from joules to electronvolts, use the conversion factor:\( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \).Thus, \[ E \approx \frac{3.98 \times 10^{-17} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 248 \text{ eV} \].

Step 2: Calculate Kinetic Energy of Electron

For an electron, the de Broglie wavelength is given by:\[ \lambda = \frac{h}{p} \]where \( p \) is the momentum of the electron. The kinetic energy \( KE \) is related to the momentum by:\[ KE = \frac{p^2}{2m} \]where \( m \) is the mass of the electron \(9.11 \times 10^{-31} \text{ kg}\).From the de Broglie equation, the momentum \( p \) is:\[ p = \frac{h}{\lambda} \approx \frac{6.626 \times 10^{-34} \text{ Js}}{5.00 \times 10^{-9} \text{ m}} \]\[ p \approx 1.325 \times 10^{-25} \text{ kg m/s} \]The kinetic energy then is:\[ KE = \frac{(1.325 \times 10^{-25} \text{ kg m/s})^2}{2 \times 9.11 \times 10^{-31} \text{ kg}} \]\[ KE \approx 9.62 \times 10^{-19} \text{ J} \]Convert this to eV:\[ KE \approx \frac{9.62 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 6.0 \text{ eV} \].

Step 3: Compare and Conclude

The energy of a photon with a 5 nm wavelength is approximately 248 eV, whereas the kinetic energy of an electron with the same wavelength is about 6.0 eV. This significant energy difference is why electron waves (matter waves) are preferable; they impart less energy to the object, reducing the potential for damage.

Key Concepts

Photon Energy

Photon energy plays a crucial role in understanding light interactions. A photon's energy is determined by its wavelength. The shorter the wavelength, the higher the energy the photon possesses. This energy can be calculated using the equation: \[ E = \frac{hc}{\lambda} \] where \( E \) is the photon energy, \( h \) is Planck's constant \( (6.626 \times 10^{-34} \text{ Js}) \), \( c \) is the speed of light \( (3.00 \times 10^8 \text{ m/s}) \), and \( \lambda \) is the wavelength.

• Photon energy is typically measured in electronvolts (eV) for convenience.
• This energy can potentially damage delicate samples like viruses, especially at shorter wavelengths which result in higher photon energy.

By understanding photon energy, scientists can make informed decisions about the imaging techniques that balance resolution and sample integrity.

de Broglie Wavelength

The concept of the de Broglie wavelength explains how particles such as electrons can exhibit wave-like properties. Proposed by physicist Louis de Broglie, any matter can have an associated wavelength, determined by its momentum. This wavelength is given by: \[ \lambda = \frac{h}{p} \] where \( \lambda \) is the de Broglie wavelength, \( h \) is Planck's constant, and \( p \) is the momentum of the particle.

• The de Broglie wavelength concept is critical in electron microscopy.
• It allows electrons to be considered as waves, probing materials with greater atomic level detail.
• This concept helps overcome limitations posed by traditional light-based microscopy.

Understanding the de Broglie wavelength gives us deeper insights into the wave nature of particles.

Kinetic Energy

Kinetic energy refers to the energy that a particle possesses due to its motion. For an electron, its kinetic energy can be derived once its momentum is known using the de Broglie relationship as follows: \[ KE = \frac{p^2}{2m} \] where \( KE \) is the kinetic energy, \( p \) is momentum, and \( m \) is the electron's mass.

• Kinetic energy is crucial in electron microscopy as it relates to the speed and penetrating power of electrons.
• Lower kinetic energies are preferable to minimize damage when probing delicate structures like viruses.
• Electron wave's lower kinetic energy compared to photon energy helps maintain the integrity of sensitive samples.

By controlling kinetic energy, researchers achieve precise imaging without compromising the sample.

Virus Structure

Investigating virus structures requires a careful choice of imaging techniques due to their small size and fragile nature. These structures can often be on the nanometer scale, making electron microscopy a preferred method for studying them.

• Typically, viruses are around 20 to 300 nanometers in size, with a common study scale of 50 nanometers provided in this context.
• The requirement of a probing wave one-tenth the size of the virus calls for sub-nanometer wavelengths.
• Electron waves are advantageous as their energy levels can be moderated, reducing the risk of damaging the virus during imaging.

Understanding virus structure through appropriate techniques is vital for advancing medical research and virology.

Matter Waves

Matter waves describe the wave-like behavior of particles. Fundamental to quantum mechanics, this concept allows particles such as electrons to be described in terms of waves.

• Matter waves are intrinsic to techniques such as electron microscopy.
• They enable high-resolution imaging of structures at atomic scales by utilizing electron's wave characteristics.
• The use of matter waves in electron microscopes provides a solution to high-energy photon difficulties, allowing detailed observation of sensitive materials like viruses.

Matter waves highlight the dual nature of particles, showcasing their particle and wave properties, which is essential in physics and various applications.