Question

X rays have wavelengths on the order of \(1 \times 10^{-10} \mathrm{m} .\) Calcu- late the energy of \(1.0 \times 10^{-10} \mathrm{m} \mathrm{X}\) rays in units of kilojoules per mole of \(\mathrm{X}\) rays. AM radio waves have wavelengths on the order of \(1 \times 10^{4} \mathrm{m}\) . Calculate the energy of \(1.0 \times 10^{4} \mathrm{m}\) radio waves in units of kilojoules per mole of radio waves. Consider that the bond energy of a carbon-carbon single bond found in organic compounds is 347 \(\mathrm{kJ} / \mathrm{mol}\) . Would \(\mathrm{x}\) rays and/or radio waves be able to disrupt organic compounds by breaking carbon-carbon single bonds?

Short answer

The energy of X rays is \(119.4 \mathrm{kJ/mol}\), which is higher than the bond energy of carbon-carbon single bond (347 \(\mathrm{kJ/mol}\)), so X rays can disrupt organic compounds by breaking carbon-carbon single bonds. The energy of radio waves is \(1.195 \times 10^{-4} \mathrm{kJ/mol}\), which is much lower than the bond energy, so radio waves cannot break carbon-carbon single bonds.

Step-by-step solution

Identifying the given values

The given values are: - Wavelength of X rays: \(\lambda_{X}\) = \(1.0 \times 10^{-10} m\) - Wavelength of radio waves: \(\lambda_{radio}\) = \(1.0 \times 10^{4} m\) - Bond energy of carbon-carbon single bond: 347 \(kJ/mol\) - Calculate the energy of X rays and radio waves in \(kJ/mol\), and compare them to the bond energy of carbon-carbon single bond.

Calculate the energy of the X rays

To calculate the energy of the X rays, we can use the Planck's formula: \[E = hf = \frac{hc}{\lambda}\] where - E is the energy of the X rays - h is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{J s}\)) - c is the speed of light (\(3.0 \times 10^{8} \mathrm{m/s}\)) - f is the frequency of the X rays - \(\lambda\) is the wavelength of the X rays Substitute the given values into the Planck's formula: \(E_{X} = \frac{(6.626 \times 10^{-34} \mathrm{J s})(3.0 \times 10^{8} \mathrm{m/s})}{1.0 \times 10^{-10} \mathrm{m}}\)

Convert energy of X rays to \(kJ/mol\)

To convert the energy of X rays to kilojoules per mole, we can use the conversion factors: - Multiply the energy by \(10^3\) to convert it from joules to kilojoules. - Multiply the energy by Avogadro's number (\(6.022 \times 10^{23} \mathrm{mol}^{-1}\)) to convert it from energy per photon to energy per mole. \(E_{X} = (1.986 \times 10^{-25} \mathrm{J}) (10^3) (6.022 \times 10^{23} \mathrm{kJ/mol})\)

Calculate the energy of the radio waves

Just as in step 2, we can now calculate the energy \(E_{radio}\) of the radio waves using the Planck's formula: \(E_{radio} = \frac{(6.626 \times 10^{-34} \mathrm{J s})(3.0 \times 10^{8} \mathrm{m/s})}{1.0 \times 10^{4} \mathrm{m}}\)

Convert energy of radio waves to \(kJ/mol\)

Similar to step 3, we convert the energy of radio waves to kilojoules per mole: \(E_{radio} = (1.986 \times 10^{-29} \mathrm{J}) (10^3) (6.022 \times 10^{23} \mathrm{kJ/mol})\)

Compare the energies with the bond energy of carbon-carbon single bond

At this step, we compare the energy of X rays and radio waves in \(kJ/mol\) with the bond energy of a carbon-carbon single bond (347 \(kJ/mol\)): - Energy of X rays: \(E_{X} = 119.4 \mathrm{kJ/mol}\) - Energy of radio waves: \(E_{radio} = 1.195 \times 10^{-4} \mathrm{kJ/mol}\) Since the energy of X rays is higher than the bond energy of the carbon-carbon single bond, they can potentially disrupt organic compounds by breaking carbon-carbon single bonds. On the other hand, the energy of radio waves is much lower than the bond energy, so they cannot cause any breakage of the carbon-carbon single bonds.

Key concepts

Planck's constant

Planck's constant is a fundamental constant in physics denoted by the symbol \( h \). It is used to relate the energy of a photon to its frequency. The value of Planck's constant is approximately \( 6.626 \times 10^{-34} \mathrm{J\,s} \). This constant is a cornerstone of quantum mechanics, helping to bridge the gap between the wave and particle behavior of light.
In the context of calculating energies of electromagnetic waves, Planck's constant enables us to use the equation \( E = hf \), where \( E \) is the energy, \( h \) is Planck's constant, and \( f \) is the frequency. Alternatively, because frequency \( f \) can be expressed as \( \frac{c}{\lambda} \) where \( c \) is the speed of light and \( \lambda \) is the wavelength, we use \( E = \frac{hc}{\lambda} \).
This relation is crucial for understanding how energy is quantized within photons and serves as a foundational principle in much of modern physics and chemistry, particularly when dealing with atomic and molecular spectra.

Wavelength and Energy Relationship

The relationship between wavelength and energy is inverse, meaning that as the wavelength of light decreases, its energy increases. This can be understood through the equation \( E = \frac{hc}{\lambda} \). Here, \( E \) is the energy of the photon, \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the light.

• Shorter wavelengths like X-rays have high energy.
• Longer wavelengths like radio waves have lower energy.

In practical application, having the ability to calculate the energy of electromagnetic waves allows us to predict their potential effects. For instance, X-rays are energetic enough to break chemical bonds and thus have medical and industrial applications. In contrast, radio waves have much longer wavelengths, resulting in lower energy and are generally used for communication.
This inverse relationship is a fundamental aspect of wave-particle duality, a concept which is essential for explaining many phenomena in the natural sciences.

Bond Energy Comparison

Bond energy refers to the amount of energy required to break a chemical bond in a molecule. In organic chemistry, carbon-carbon single bonds often serve as a measure for bond stability, with typical bond energies around 347 \( \mathrm{kJ/mol} \).
Comparing this value to the energy of electromagnetic radiation helps us understand potential chemical interactions or disruptions. For example:

• The calculated energy of X-rays is approximately 119.4 \( \mathrm{kJ/mol} \).
• The energy of AM radio waves is about 1.195 \( \times 10^{-4} \) \( \mathrm{kJ/mol} \).

Given these comparisons, X-rays possess enough energy to potentially break carbon-carbon bonds, which can alter or destroy organic compounds. Conversely, radio waves have insufficient energy for this purpose. These insights are invaluable in fields such as molecular chemistry and radiation therapy, where understanding the interaction between radiation and matter is critical.