Problem 62 - Chemistry Structure and Dynamics

Question

Use the relationship between the energy and the frequency of a photon to calculate the energy in kilojoules per mole of a photon of blue light that has a frequency of \(6.5 \times 10^{14} \mathrm{~s}^{-1}\). Compare the results of this calculation with the ionization energy of water \((1216 \mathrm{~kJ} / \mathrm{mol})\).

Step-by-Step Solution

Answer

The energy is 25948 kJ/mol, which is much greater than water's ionization energy of 1216 kJ/mol.

Step 1: Recall the Energy-Frequency Formula

The energy of a photon can be calculated using the formula \(E = h u\), where \(E\) is the energy of the photon, \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J s}\), and \(u\) is the frequency of the photon.

Step 2: Calculate the Energy of One Photon

Substitute the given frequency of the photon \((u = 6.5 \times 10^{14} \text{ s}^{-1})\) into the formula: \[E = (6.626 \times 10^{-34} \text{ J s}) \times (6.5 \times 10^{14} \text{ s}^{-1}) = 4.307 \times 10^{-19} \text{ J}\].

Step 3: Convert Energy to Kilojoules per Mole

To find the energy in kilojoules per mole, first convert the energy of a single photon from joules to kilojoules \((1 ext{ J} = 10^{-3} ext{ kJ})\): \[4.307 \times 10^{-19} ext{ J/photon} = 4.307 \times 10^{-22} ext{ kJ/photon}\]. Then multiply by Avogadro's number \((6.022 \times 10^{23} ext{ photons/mol})\): \[4.307 \times 10^{-22} ext{ kJ/photon} \times 6.022 \times 10^{23} ext{ photons/mol} = 25948 ext{ kJ/mol}\].

Step 4: Compare with Ionization Energy

The calculated energy of a photon \(25948 ext{ kJ/mol}\) greatly exceeds the ionization energy of water \(1216 ext{ kJ/mol}\). This implies that the energy associated with a photon of this frequency is much larger than the energy required to ionize water.

Key Concepts

Planck's Constant

When you're diving into the world of quantum mechanics, Planck's constant is your trusty companion. This constant is essential in the formula to calculate the energy of a photon. Remember the equation, \( E = h u \), where \( E \) is the energy of the photon and \( u \) is the frequency. Here, \( h \), Planck's constant, is valued at \( 6.626 \times 10^{-34} \,\text{J s} \).

This tiny constant bridges the gap between the energy a photon carries and its frequency. It's what helps us unravel mysteries at atomic and subatomic levels.

Planck's constant shows us that energy isn't as smooth and continuous as everyday life suggests but comes in discrete packets known as quanta. Remembering this will give you a deeper understanding of how light and other electromagnetic waves function at microscopic scales.

Ionization Energy

Ionization energy is a key concept when studying photons and their interaction with matter. It is the amount of energy required to remove an electron from a neutral atom in its gaseous phase.

In our problem, you compared the energy of a photon with the ionization energy of water, which is \( 1216 \,\text{kJ/mol} \). This energy amount tells us how tightly an electron is bound within an atom.

The photon's calculated energy was significantly larger than this ionization energy. This excessive energy indicates that the photon has enough power to not only ionize water but also to engage in further energetic reactions.

Understanding ionization energy is crucial, especially in fields like chemistry and physics, as it helps predict how atoms will behave in different environments.

Avogadro's Number

Avogadro's Number is a principal figure in chemistry that connects microscopic and macroscopic worlds. It's valued at \( 6.022 \times 10^{23} \), representing the number of entities, like atoms or molecules, in one mole of a substance.

In the problem, you used Avogadro's Number to convert the energy of a single photon into energy per mole, which is a more practical and usable form in chemistry.

• This allows for a direct comparison with quantities used routinely in laboratory settings, like kilojoules per mole.
• It's like a bridge that turns those small, challenging to handle numbers into something we can work with daily.


• In calculations involving moles and molecules, Avogadro's Number is indispensable, helping us scale to quantities we can measure and observe in the lab.

Avogadro's Number not only tells us how many particles there are in a mole but also aids in calculating significant energy levels, like that one solved in the exercise. Truly understanding this concept helps visualize the vastness of the molecular world.