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Magazine Chapter 3: Problem 1
What is the energy (in kJmol \(^{-1}\) ) of X-ray photons with a wavelength of \(100 \mathrm{pm} ?\) (Section 3.2 ).
Short Answer
Expert verified
The energy of the X-ray photons is approximately 1196 kJmol \(^{-1}\).
Step by step solution
01
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, \(100\, \mathrm{pm} = 100 \times 10^{-12}\, \mathrm{m} = 1 \times 10^{-10}\, \mathrm{m} \).
02
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.
03
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, \( 6.022 \times 10^{23} \, \mathrm{mol^{-1}} \):\[ \text{Energy per mole} = 1.986 \times 10^{-18} \, \mathrm{kJ} \times 6.022 \times 10^{23} \, \mathrm{mol^{-1}} = 1195.7 \, \mathrm{kJ\cdot mol^{-1}} \]
04
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:\[ \text{Energy} = 1196 \, \mathrm{kJ\cdot mol^{-1}} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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.
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:
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.
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:
- \(6.022 \times 10^{23} \, \mathrm{mol^{-1}}\).
- Energy per mole = energy per photon \(\times\) Avogadro's number.
- For our calculation, \(1.986 \times 10^{-18} \, \mathrm{kJ/photon} \times 6.022 \times 10^{23} \, \mathrm{mol^{-1}} = 1195.7 \, \mathrm{kJ/mol}\).
