Problem 30 - Chemistry The Central Science

Question

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of 242 \(\mathrm{kJ} / \mathrm{mol}\) is required to break the chlorine-chlorine bond in \(\mathrm{Cl}_{2} .\) What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

Step-by-Step Solution

Answer

The minimum energy to break the chlorine-chlorine bond is given as 242 kJ/mol. First, we convert this energy to J/molecule: \(E_\mathrm{molecule} = \frac{242 \times 10^3}{6.022 \times 10^{23}} \mathrm{J/molecule}\). Next, we use Planck's equation to find the frequency \(\nu = \frac{E}{h}\), where \(h = 6.626 \times 10^{-34} \mathrm{Js}\). Then, we use the speed of light equation to calculate the wavelength \(\lambda = \frac{c}{\nu}\), where \(c = 3.00 \times 10^8 \mathrm{m/s}\). Lastly, we compare the obtained wavelength with the electromagnetic spectrum ranges to determine the type of radiation.

Step 1: Convert energy to J/molecule

Given energy is 242 kJ/mol which needs to be converted to J/molecule, for easier calculations. \[1 \mathrm{kJ} = 1000 \mathrm{J} \] \[1 \mathrm{mol} \approx 6.022 \times 10^{23} \mathrm{molecules} \] Conversion equation: \[ E_\mathrm{molecule} = \frac{E_\mathrm{mol} \times 1000}{6.022 \times 10^{23}}\]

Step 2: Calculate the frequency using Planck's equation

Next, we will find the frequency (\(\nu\)) using Planck's equation. Planck's equation: \[E = h\nu\] Where \(E\) is the energy per molecule and \(h\) is Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)). Rearrange the equation to solve for frequency (\(\nu\)): \[\nu = \frac{E}{h}\]

Step 3: Calculate the wavelength using the speed of light equation

Now, calculate the wavelength using the speed of light equation relating frequency and wavelength. Speed of light equation: \[c = \nu\lambda\] Rearrange the equation to solve for wavelength (\(\lambda\)): \[\lambda = \frac{c}{\nu}\] Where \(c\) is the speed of light (\(3.00 \times 10^8 \mathrm{m/s}\)) and \(\nu\) is the frequency obtained in Step 2.

Step 4: Identify the type of electromagnetic radiation

With the wavelength calculated, we can identify the type of electromagnetic radiation. The electromagnetic spectrum is divided into different ranges based on the wavelength: - Radio waves: > 1 m - Microwaves: 1 m - 1 mm - Infrared: 1 mm - 700 nm - Visible light: 700 nm - 400 nm - Ultraviolet: 400 nm - 10 nm - X-rays: 10 nm - 0.01 nm - Gamma rays: < 0.01 nm Compare the calculated wavelength to the ranges above to find out the corresponding type of electromagnetic radiation.

Key Concepts

Chemical Bond Rupture

Chemical bond rupture involves breaking the bonds that hold atoms together within a molecule. For the chlorine-chlorine bond in chlorine gas (\( \mathrm{Cl}_2 \)), a specific amount of energy is required to overcome the attraction between the two chlorine atoms. This energy is referred to as bond dissociation energy. In this exercise, the bond rupture requires 242 kJ/mol. To break this bond using electromagnetic radiation, this energy must be provided in the form of electromagnetic waves.

Planck's Equation

Planck's equation is essential in understanding how energy and frequency are related in electromagnetic radiation. The equation is expressed as:\[E = hu\]Here:

• \(E\) is the energy of a single photon.
• \(h\) is Planck's constant, \(6.626 \times 10^{-34} \mathrm{Js}\).
• \(u\) is the frequency of the radiation.

Using this equation, if you know the energy required to break a bond, you can calculate the necessary frequency of light to achieve this. Rearrange the equation to find frequency: \(u = \frac{E}{h}\). This relationship is vital for translating energy requirements into specific electromagnetic properties.

Wavelength Calculation

To find the wavelength of electromagnetic radiation, you can use its relationship with frequency. The equation that relates these properties is:\[c = u\lambda\]Where:

• \(c\) is the speed of light, \(3.00 \times 10^8 \mathrm{m/s}\).
• \(u\) is the frequency.
• \(\lambda\) is the wavelength.

By rearranging the formula to \(\lambda = \frac{c}{u}\), you can calculate the wavelength once the frequency is known. This helps in identifying what type of electromagnetic radiation can provide the energy needed to break a specific chemical bond.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, categorized by wavelength. Different types include:

• Radio waves: > 1 m
• Microwaves: 1 m - 1 mm
• Infrared: 1 mm - 700 nm
• Visible light: 700 nm - 400 nm
• Ultraviolet: 400 nm - 10 nm
• X-rays: 10 nm - 0.01 nm
• Gamma rays: < 0.01 nm

Once the wavelength is calculated using the speed of light equation, it can be compared to these ranges to determine which type of electromagnetic radiation is involved. Knowing the type helps explain how different portions of the spectrum interact with matter, such as how ultraviolet light can break chemical bonds like those in \( \mathrm{Cl}_2 \).