Question

It takes 476 kJ to remove 1 mole of electrons from the atoms at the surface of a solid metal. What is the maximum wavelength of light that can remove a single electron from an atom at the surface of this solid metal?

Short answer

The maximum wavelength of light that can remove a single electron from an atom at the surface of this solid metal is \(\lambda = \frac{2.998 \times 10^{8} \text{ m/s} \times 6.626 \times 10^{-34} \text{Js}}{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}\), which can be calculated to get the value in meters.

Step-by-step solution

Convert energy in kJ/mol to J/mol

Given, the energy required to remove 1 mole of electrons is 476 kJ. We need to convert it to J/mol. 1 kJ = 1000 J So, 476 kJ = 476 * 1000 J = 476000 J/mol

Calculate the energy required to remove one electron

The energy required to remove 1 mole of electrons is 476000 J/mol. We need to find the energy required to remove one electron. Number of electrons in a mole = Avogadro's number = \(6.022 \times 10^{23}\) electrons/mol Energy per electron = \(\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}\)

Calculate the frequency of light

From the energy per electron, we can find the frequency of light using Planck's equation: \(E = h\nu \) Where E is the energy per electron, h is Planck's constant (\(6.626 \times 10^{-34} \text{ Js} \)), and \(\nu\) is the frequency. Energy per electron = \(\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}\) \(\nu = \frac{E}{h}\) \( \nu = \frac{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}{6.626 \times 10^{-34} \text{Js}}\)

Calculate the speed of light

Since the speed of light (c) is equal to the product of frequency (\(\nu\)) and wavelength (\(\lambda\)), we can find the wavelengths by dividing the speed of light by the frequency. \(c = 2.998 \times 10^{8} \text{m/s}\) \(\lambda = \frac{c}{\nu}\) \(\lambda = \frac{2.998 \times 10^{8} \text{m/s}}{\frac{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}{6.626 \times 10^{-34} \text{Js}}}\)

Solve for the maximum wavelength

Now, we can put all the values in the given formula and find the maximum wavelength. \(\lambda = \frac{2.998 \times 10^{8} \text{ m/s} \times 6.626 \times 10^{-34} \text{Js}}{\frac{476000\text{ J/mol}}{6.022\times 10^{23}\text{ electrons/mol}}}\) After solving the above expression, we get the maximum wavelength (in meters) that can remove a single electron from an atom at the surface of this solid metal.

Key concepts

Energy Conversion

Energy conversion is crucial when dealing with chemical and physical processes. In this exercise, we are converting energy from kilojoules per mole (kJ/mol) to joules per mole (J/mol). This conversion is necessary to make calculations compatible with physical constants that often use joules, such as Planck's constant.

• To convert from kJ to J, remember that 1 kJ equals 1000 J.
• By multiplying the given kJ value by 1000, you switch the units from kJ to J.

Understanding these conversions helps make sure the energy values are in a form that fits with other parts of the equation, like Planck's equation for energy calculation.
Making sure your units match is important for accurate results.

Planck's Equation

Planck's equation is used to relate the energy of a photon to its frequency. The equation is expressed as \(E = hu\). Here, \(E\) represents energy, \(h\) is Planck's constant, and \(u\) stands for frequency.
This equation is essential when studying the photoelectric effect, where light energy dislodges electrons from a material. Key points:

• In this exercise, you're asked to find the frequency of light capable of removing electrons.
• We use Planck's constant, \(6.626 \times 10^{-34} \text{ Js}\), to calculate this frequency.
• The result provides a direct connection between the energy of photons and the frequency of light.

Applying Planck's equation helps to establish foundational connections between energy and light properties.

Wavelength Calculation

Wavelength calculation plays a significant role in understanding how light interacts with matter. The wavelength is derived from the speed of light equation \(c = u\lambda\). Using this formula, if you know the speed of light and frequency, you can find the wavelength \(\lambda\).
Some important aspects include:

• The speed of light \(c\) is a constant \(2.998 \times 10^{8} \text{ m/s}\).
• The wavelength \(\lambda\) is inversely related to frequency \(u\), meaning higher frequencies lead to shorter wavelengths.
• The maximum wavelength corresponds to the minimum frequency required to remove a single electron.

This method offers insights into the light's specific properties, explaining how different wavelengths can affect electron behavior.

Avogadro's Number

Avogadro's number, denoted as \(6.022 \times 10^{23} \text{ mol}^{-1}\), is a fundamental constant in chemistry and physics. It tells us the number of atoms, ions, or molecules in one mole of a substance.
In this context:

• Avogadro's number helps calculate energy per electron by dividing the total energy for a mole by this constant.
• This conversion gives an individual electron's energy, crucial for understanding reactions at an atomic level.
• Applying Avogadro's number simplifies transitions between macroscopic measurements in kilojoules per mole and atomic-level energy measurements.

Thus, Avogadro's number interconnects the molar scale to individual atomic scales, which is vital for thorough scientific analysis and calculations.