Question

The highly reactive OH radical (a species with an unpaired electron) is believed to be involved in some atmospheric processes. Table 8.6 lists the bond enthalpy for the oxygen-to-hydrogen bond in \(\mathrm{OH}\) as \(460 \mathrm{~kJ} / \mathrm{mol}\). What is the longest wavelength (in \(\mathrm{nm}\) ) of radiation that can bring about the following reaction? $$ \mathrm{OH}(g) \longrightarrow \mathrm{O}(g)+\mathrm{H}(g) $$

Short answer

The longest wavelength is approximately 260 nm.

Step-by-step solution

Understand the Concept

The exercise involves calculating the longest wavelength of radiation that can break the O-H bond in the hydroxyl radical, OH, into oxygen and hydrogen atoms. The energy required to break this bond is given by its bond enthalpy.

Relate Wavelength to Energy

The energy of a photon is related to its wavelength by the equation \( E = \frac{hc}{\lambda} \), where \( E \) is the energy of the photon, \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \), \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength. We need to solve this equation for \( \lambda \).

Calculate Photon Energy

Given that the bond enthalpy of OH is \(460 \text{ kJ/mol}\), this energy represents the energy per mole required to dissociate the bond. Convert this energy to joules per photon using \( 1 \text{ kJ/mol} = \frac{1,000}{6.022 \times 10^{23}} \text{ J/photon} \), where \(6.022 \times 10^{23}\) is Avogadro's number. So, \( E = 460,000 \text{ J/mol} \div 6.022 \times 10^{23} \text{ molecules/mol} \).

Solve for Wavelength

Use the photon energy calculated in Step 3 to find the longest wavelength. Rearrange the formula from Step 2 to \( \lambda = \frac{hc}{E} \). Substitute in Planck's constant, the speed of light, and the energy to solve for \( \lambda \).

Convert Wavelength to Nanometers

The result from Step 4 is the wavelength in meters. Convert it into nanometers by multiplying by \(10^9\) (since 1 meter = \(10^9\) nanometers).

Key concepts

Photon Energy

To understand photon energy, let's start with a quick dive into what it means. Photons are tiny particles of light, and each photon carries energy. This energy is not constant but varies depending on the photon's wavelength or frequency.

The formula to calculate the energy of a photon is given by:

• \( E = \frac{hc}{\lambda} \)

Here, - \( E \) represents the energy of a photon in Joules. - \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ J} \cdot \text{s} \)).- \( c \) is the speed of light (\( 3.00 \times 10^8 \text{ m/s} \)).- \( \lambda \) is the wavelength of the photon in meters.

When photons interact with matter, such as molecules or atoms, they can transfer their energy, causing chemical reactions. In our exercise, we are interested in photons that have enough energy to break the bond between the oxygen (O) and hydrogen (H) atoms in the OH radical.

Understanding photon energy helps us determine the kind of radiation required to initiate specific reactions in atmospheric chemistry.

Wavelength Calculation

Now, to calculate the wavelength of a photon given its energy, we rearrange the photon energy formula to solve for wavelength \( \lambda \).

The formula becomes:

• \( \lambda = \frac{hc}{E} \)

This equation tells us the relationship between a photon's energy and its wavelength. When the energy is high, the wavelength is shorter and vice versa.

Since the bond enthalpy of the OH radical is \( 460 ext{ kJ/mol} \), we first need to convert this value to an equivalent energy per photon in joules:

• \( E = \frac{460,000 ext{ J/mol}}{6.022 \times 10^{23} ext{ molecules/mol}} \)

This operation changes the units from energy per mole to energy per individual photon.

Once we have \( E \), we plug it into the wavelength formula to find \( \lambda \). Finally, to express wavelength in nanometers (nm), we multiply the result by \(10^9\), because 1 meter equals \(10^9\) nanometers. This conversion helps us understand the photon energy in a more intuitive unit for wavelengths.

OH Radical Dissociation

The dissociation of the OH radical is a significant reaction in atmospheric chemistry. The OH radical, also known as the hydroxyl radical, plays a crucial role in processing pollutants and stabilizing the atmosphere.

Dissociation refers to a molecule breaking into two or more parts. In our context, the reaction is:

• \( \text{OH}(g) \longrightarrow \text{O}(g) + \text{H}(g) \)

Here, the O-H bond in the OH radical is broken, resulting in free oxygen and hydrogen atoms. The bond enthalpy provided (\(460 ext{ kJ/mol} \)) indicates the energy needed to break this bond.

For the dissociation to occur, the absorbed photon's energy must equal or exceed this bond enthalpy. Understanding this process helps illustrate how energy absorption can drive chemical changes, especially in the atmosphere where such radicals are active.

By calculating the precise wavelength of light needed for dissociation, researchers can better understand and predict the behavior of atmospheric phenomena, thereby contributing to environmental and climate studies.