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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

01

Understanding Photon Energy

To calculate the energy of a photon, we use the formula \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant \((6.626 \times 10^{-34} \text{ J s})\), \(c\) is the speed of light \((3 \times 10^8 \text{ m/s})\), and \(\lambda\) is the wavelength \((5.00 \text{ nm} = 5.00 \times 10^{-9} \text{ m})\).

02

Calculating Photon Energy

Substitute the known values into the formula: \[E = \frac{(6.626 \times 10^{-34} \text{ J s})(3 \times 10^8 \text{ m/s})}{5.00 \times 10^{-9} \text{ m}}\]Calculating this gives \(E \approx 3.977 \times 10^{-17} \text{ J}\). Convert this energy into electronvolts using \(1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}\).\[E(\text{eV}) = \frac{3.977 \times 10^{-17} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 248 \, \text{eV}\]

03

Understanding Electron Wavelength

The de Broglie wavelength equation \(\lambda = \frac{h}{p}\), where \(p\) is momentum, applies to matter waves such as electrons. Momentum is \(p = mv\), but can be expressed with kinetic energy \(KE = \frac{p^2}{2m}\). Rearrange for \(KE\): \[KE = \frac{h^2}{2m\lambda^2}\]Here, \(m\) for an electron is \(9.109 \times 10^{-31} \text{ kg}\).

04

Calculating Electron Kinetic Energy

Substitute the known values into the formula:\[KE = \frac{(6.626 \times 10^{-34} \text{ J s})^2}{2(9.109 \times 10^{-31} \text{ kg})(5.00 \times 10^{-9} \text{ m})^2}\]Calculate the kinetic energy, which results in approximately \(KE \approx 1.505 \times 10^{-20} \text{ J}\). Convert this to electronvolts:\[KE(\text{eV}) = \frac{1.505 \times 10^{-20} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 0.094 \, \text{eV}\]

05

Comparing Energies and Conclusion

The energy of a photon is approximately 248 eV, while the kinetic energy of an electron is approximately 0.094 eV. Using electrons is preferable because their much lower energy reduces the risk of damaging the virus.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electron Microscopy

Electron microscopy is a powerful technique used to observe the minute details of small objects at the nanoscale, such as viruses. Unlike traditional optical microscopes that use visible light, electron microscopes use a beam of electrons. This method is advantageous for studying viruses due to the wave-particle duality of electrons, which allows them to exhibit both particle and wave-like behavior.
The much shorter de Broglie wavelength of electrons compared to that of visible light enables them to resolve much smaller structures.

• Transmission Electron Microscopes (TEM) allow electrons to pass through thin samples, yielding high-resolution images.
• Scanning Electron Microscopes (SEM) bounce electrons off the surface of an object, providing detailed topographic images.

The ability to study structures at such high resolutions, without the high energy that can damage specimens, is what makes electron microscopy particularly suitable for biological samples like viruses.

Photon Energy

The energy carried by a photon is directly related to its wavelength. As the wavelength of light decreases, the energy of each photon increases according to the formula:\[ E = \frac{hc}{\lambda} \]Here, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength. For small wavelengths, like 5 nm, the photon energy becomes quite high.
This high energy can be problematic when studying delicate specimens like viruses because it can damage or destroy them.

Thus, while photons provide the high resolution required due to their small wavelengths, their accompanying energy can be a limitation in the study of biological samples.

• A photon with a 5 nm wavelength carries an energy of about 248 eV.
• This much energy can disrupt delicate molecular structures in viruses.

de Broglie Wavelength

The de Broglie wavelength is a key concept when discussing the wave-nature of particles, such as electrons. It is calculated using the formula: \[ \lambda = \frac{h}{p} \]Where \( \lambda \) is the wavelength, \( h \) is Planck’s constant, and \( p \) is the momentum of the particle. For electrons used in microscopy, this wavelength is substantially shorter than that of visible light.
This shorter wavelength enables better resolution without high-energy drawbacks. Since momentum \( p = mv \) (mass times velocity), for electrons, it can also be expressed in terms of kinetic energy, leading to:\[ KE = \frac{h^2}{2m\lambda^2} \]

• An electron with a wavelength of 5 nm has a kinetic energy of about 0.094 eV.
• This is much lower than the photon energy, making electrons less likely to cause damage to the specimen.

Thus, de Broglie's concept supports the use of electrons in microscopy for nanoscale study, like investigating virus structures.

Virus Structure Analysis

Analyzing the structure of a virus is crucial for understanding its biology and developing treatments or vaccines. To effectively analyze these small structures, instruments like electron microscopes are used due to their ability to provide clarity without destructive energy levels. The size of a typical virus is around 50 nm, making them difficult to study using light-based microscopes because their wavelength doesn’t allow for the necessary resolution.
Electron microscopy overcomes this by using electrons, which can have wavelengths much shorter than light, providing heightened resolution.

• Electron microscopy allows for observing viruses down to the molecular level, uncovering details about capsid arrangements and nucleic acid placement.
• The low energy of electrons minimizes potential sample damage, maintaining the integrity of the virus structure for analysis.

Ultimately, this advanced method supports researchers in crafting effective antiviral strategies and furthering our understanding of viral functionalities.