Question

What is the energy (in kJmol \(^{-1}\) ) of X-ray photons with a wavelength of \(100 \mathrm{pm} ?\) (Section 3.2 ).

Short answer

The energy of the X-ray photons is approximately 1196 kJmol \(^{-1}\).

Step-by-step solution

Convert units for wavelength

The given wavelength is in picometers (pm). We need to convert it to meters for our calculations. Recall that 1 picometer = \(10^{-12}\) meters. Thus, \(100\, \mathrm{pm} = 100 \times 10^{-12}\, \mathrm{m} = 1 \times 10^{-10}\, \mathrm{m} \).

Use Planck's equation to find energy per photon

Planck's equation relates energy and wavelength: \[ E = \frac{hc}{\lambda} \] where \( h = 6.626 \times 10^{-34} \, \mathrm{J\cdot s} \) is Planck's constant, \( c = 3 \times 10^8 \, \mathrm{m/s} \) is the speed of light, and \( \lambda \) is the wavelength in meters. Substitute \( \lambda = 1 \times 10^{-10} \) m:\[E = \frac{(6.626 \times 10^{-34} \, \mathrm{J\cdot s}) \times (3 \times 10^8 \, \mathrm{m/s})}{1 \times 10^{-10} \, \mathrm{m}}\]Solving this gives us \( E = 1.986 \times 10^{-15} \, \mathrm{J} \) per photon.

Convert energy from Joules to kJ/mol

First, convert the energy from Joules to kilojoules. Since \( 1\, \, \mathrm{kJ} = 10^3 \, \mathrm{J} \), we get:\[ \text{Energy per photon} = 1.986 \times 10^{-15} \, \mathrm{J} \times 10^{-3} \frac{\mathrm{kJ}}{\mathrm{J}} = 1.986 \times 10^{-18} \, \mathrm{kJ} \] Next, find the energy per mole of photons by multiplying by Avogadro's number, \( 6.022 \times 10^{23} \, \mathrm{mol^{-1}} \):\[ \text{Energy per mole} = 1.986 \times 10^{-18} \, \mathrm{kJ} \times 6.022 \times 10^{23} \, \mathrm{mol^{-1}} = 1195.7 \, \mathrm{kJ\cdot mol^{-1}} \]

Round the result

Round the energy per mole of photons to a reasonable number of significant figures based on given data. Here we use three significant figures:\[ \text{Energy} = 1196 \, \mathrm{kJ\cdot mol^{-1}} \]

Key concepts

Wavelength Conversion

To find the energy of X-ray photons, it is essential to first understand wavelength conversion. Wavelength tells us the size of the light wave and is often provided in picometers (pm) when dealing with X-rays. However, for calculations involving energy, we need the wavelength in meters because this is the standard unit in equations like Planck's equation.
Here's how you convert from picometers to meters:

• 1 picometer = \(10^{-12}\) meters.
• To convert 100 pm to meters, multiply by \(10^{-12}\).
• Thus, 100 pm = \(1 \times 10^{-10}\) meters.

Understanding this conversion is crucial because incorrect conversions can lead to vastly different results in energy calculations.

Planck's Equation

Planck's equation is a powerful tool in physics that connects the energy of a photon with its wavelength. This relation is crucial when calculating the energy of photons across different spectra. Planck's equation is formulated as:

• \(E = \frac{hc}{\lambda}\)
• Where \(h = 6.626 \times 10^{-34} \, \mathrm{J\cdot s}\)
• \(c = 3 \times 10^8 \, \mathrm{m/s}\)
• \(\lambda\) is the wavelength in meters.

When you insert the wavelength of the X-ray, which is \(1 \times 10^{-10}\) meters, and multiply it with Planck’s constant \(h\) and the speed of light \(c\), you find the energy per photon. This equation emphasizes the inverse relationship between wavelength and energy—the shorter the wavelength, the higher the energy.

Energy Conversion

Once the energy in Joules per photon is calculated using Planck's equation, we need to convert it into kilojoules since it is a more practical unit, especially when dealing with large quantities like a mole of photons.
The conversion is quite straightforward:

• 1 kilojoule (kJ) = \(10^3\) Joules (J).
• Convert our result from joules: \(1.986 \times 10^{-15}\) J is converted to \(1.986 \times 10^{-18}\) kJ.

This is just the energy per single photon. Remember, X-ray explorations often require knowing the energy for a mole of photons for practical applications.

Avogadro's Number

Avogadro's number is a foundational concept in chemistry and physics, particularly useful when working with particles like atoms and photons. It tells us the number of units (like photons) in one mole of a substance. This number is defined as:

• \(6.022 \times 10^{23} \, \mathrm{mol^{-1}}\).

By multiplying the energy per photon (in kJ) by Avogadro's number, we convert this into an energy measurement applicable across larger scales, like chemistry experiments or X-ray interactions with materials.

• Energy per mole = energy per photon \(\times\) Avogadro's number.
• For our calculation, \(1.986 \times 10^{-18} \, \mathrm{kJ/photon} \times 6.022 \times 10^{23} \, \mathrm{mol^{-1}} = 1195.7 \, \mathrm{kJ/mol}\).

The use of Avogadro's number allows us to scale our calculations to real-world scenarios, giving a more comprehensive energy picture.