Question

As the weapons officer aboard the Starship Chemistry, it is your duty to configure a photon torpedo to remove an electron from the outer hull of an enemy vessel. You know that the work function (the binding energy of the electron) of the hull of the enemy ship is \(7.52 \times 10^{-19} \mathrm{J}\) a. What wavelength does your photon torpedo need to be to eject an electron? b. You find an extra photon torpedo with a wavelength of \(259 \mathrm{nm}\) and fire it at the enemy vessel. Does this photon torpedo do any damage to the ship (does it eject an electron)? c. If the hull of the enemy vessel is made of the element with an electron configuration of \([\mathrm{Ar}] 4 s^{1} 3 d^{10},\) what metal is this?

Short answer

a. The required wavelength of the photon torpedo needed to eject an electron is approximately \(264 \mathrm{nm}\). b. The found photon torpedo has a wavelength of \(259 \mathrm{nm}\), which is less than the required wavelength, so it can eject an electron and cause damage to the enemy vessel. c. The enemy vessel's outer hull is made of Zinc (Zn).

Step-by-step solution

Calculate the required wavelength of the photon torpedo

Given the work function of the enemy's hull, we can use the photoelectric effect equation to find the wavelength needed to eject an electron. The photoelectric effect equation is given by: \[E = hf\] where \(E\) is the energy of the photon, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), and \(f\) is the frequency of the light. Since \(f = \frac{c}{\lambda}\), where \(c\) is the speed of light in a vacuum (\(3 \times 10^8 \mathrm{m/s}\)), we can rewrite the equation as: \[E = h\left(\frac{c}{\lambda}\right)\] To find the wavelength (\(\lambda\)), we have: \[\lambda = \frac{hc}{E}\] Now we can plug in the given work function (\(E = 7.52 \times 10^{-19} \mathrm{J}\)): \[\lambda = \frac{(6.626 \times 10^{-34} \mathrm{Js})(3 \times 10^8 \mathrm{m/s})}{7.52 \times 10^{-19} \mathrm{J}}\]

Find the wavelength of the photon torpedo

Now, we calculate the wavelength: \[\lambda = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{7.52 \times 10^{-19}} \approx 2.64 \times 10^{-7} \mathrm{m}\] For simplicity and convenience, we can convert the wavelength into nanometers (nm), by multiplying it by \(10^9\): \[\lambda = 2.64 \times 10^{-7} \times 10^9 = 264 \mathrm{nm}\] The required wavelength of the photon torpedo needed to eject an electron is approximately \(264 \mathrm{nm}\).

Compare the found photon torpedo's wavelength to the required wavelength

The found photon torpedo has a wavelength of \(259 \mathrm{nm}\), which is less than the required wavelength of \(264 \mathrm{nm}\). Since the photon energy is inversely proportional to the wavelength, the energy of the found photon torpedo is actually greater than required. Thus, the found photon torpedo can eject an electron and cause damage to the enemy vessel.

Identify the metal composition of the enemy hull

The electron configuration of the metal in the enemy's vessel is given as: \([\mathrm{Ar}] 4s^{1} 3d^{10}\) The electron configuration of Argon is \([Ne] 3s^2 3p^6\). Therefore, the complete electron configuration of the given metal is: \([Ne] 3s^2 3p^6 4s^1 3d^{10}\) After examining the periodic table, we find that the element with this electron configuration is Zinc (Zn), with atomic number 30. Thus, the enemy vessel's outer hull is made of Zinc.

Key concepts

Wavelength Calculation

The photoelectric effect is a fascinating phenomenon that helps us understand how light interacts with matter. When photons (light particles) hit a material, they can provide enough energy to remove electrons from the material's surface. This energy is related to the photon's wavelength.

To find the wavelength needed to eject an electron, we use the formula:

• \( E = hf \), where \( E \) is energy, \( h \) is the Planck's constant \( (6.626 \times 10^{-34} \mathrm{Js}) \), and \( f \) is the frequency.
• Since frequency \( f \) and wavelength \( \lambda \) are related by \( f = \frac{c}{\lambda} \), where \( c \) is the speed of light \( (3 \times 10^8 \mathrm{m/s}) \), we can rewrite the equation as \( E = \frac{hc}{\lambda} \).
• To find the wavelength \( \lambda \), rearrange to \( \lambda = \frac{hc}{E} \).

In our case, the work function \( E \) is \( 7.52 \times 10^{-19} \mathrm{J} \). Plugging in the numbers, we find that \( \lambda \approx 264 \mathrm{nm} \). This wavelength is what your photon torpedo needs to knock an electron off the surface.

Photon Energy

Understanding photon energy is crucial for analyzing light-matter interactions. Energy and wavelength are interconnected because energy increases as wavelength decreases.

Let's go further into this relationship.

• High-energy photons (shorter wavelengths) have more ability to dislodge electrons due to their increased energy.
• The inverse relationship means a photon with a wavelength shorter than the necessary \( 264 \mathrm{nm} \) will contain more energy than a photon of exactly \( 264 \mathrm{nm} \).
• If a photon torpedo has a wavelength of \( 259 \mathrm{nm} \), it's shorter than what's needed, thus it carries more energy.

In our scenario, the photon torpedo with \( 259 \mathrm{nm} \) successfully ejects an electron to damage the enemy ship since it's more energetic than the minimum requirement.

Electron Configuration

Electron configuration provides insight into the arrangement of electrons in an atom. It's like following a map to understand where each electron goes.

For any atom, electrons fill lower energy levels first, moving to higher levels as they go.

• Consider the configuration \([\mathrm{Ar}] 4s^1 3d^{10}\): This tells us which orbitals contain electrons for that element.
• The configuration starts with Argon \([\mathrm{Ar}]\), which stands for \([Ne] 3s^2 3p^6\).
• Then it adds \(4s^{1}\) and \(3d^{10}\), illustrating all the electrons beyond Argon's.

In our case, this configuration matches Zinc (Zn), an element widely used, hinting that the enemy ship's hull is made of Zinc. Knowing the electron configuration helps identify the elements and predict chemical behaviors.

Zinc

Zinc is a versatile metal with a distinct electron configuration: \([\mathrm{Ar}] 4s^1 3d^{10}\). Let's explore why Zinc might be chosen for an enemy hull.

Here are some characteristics of Zinc:

• Zinc is resistant to corrosion, making it ideal for protection.
• It's often used in alloys to enhance strength and durability.
• The electron configuration of Zinc suggests that its chemical and physical properties are balanced for resilience.

Understanding Zinc's role helps appreciate its practical applications in shielding and construction, similar to its theoretical use in the Starship Chemistry scenario.