Problem 54 - General Chemistry: Principles and Modern Applications

Question

Calculate the wavelength of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.

Step-by-Step Solution

Answer

Using the above steps, one can calculate the wavelength (\(\lambda\)) of the electromagnetic radiation required to excite a proton from the ground state to the level with \(n=4\) in a one-dimensional box 50. pm long.

Step 1: Calculate the Energy Levels

First, calculate the energy levels for \(n=1\) and \(n=4\) using the formula \(E = n^2h^2/(8mL^2)\), where \(n\) is the energy level, \(h\) is the Planck's constant equal to \(6.63 x 10^{-34}\) Js, \(m\) is the mass of the proton, about \(1.672 x 10^{-27}\) kg, and \(L\) is the length of the box, \(50 x 10^{-12}\).

Step 2: Find the Difference in Energy Levels

Next, find the difference between the energy levels for \(n=4\) and \(n=1\) to get the energy of the radiation needed to excite the proton from the ground state to \(n=4\).

Step 3: Calculate the Wavelength

Finally, find the wavelength using Einstein's relation \(E=h\nu = hc/ \lambda\), where \(E\) is the energy of the radiation, \(h\) is the Planck's constant, \(c\) is the speed of light at \(3 x 10^{8}\) m/s, and \(\lambda\) is the wavelength. Solving for \(\lambda\) yields \(\lambda = hc/E\).

Key Concepts

Energy Levels Calculation

In quantum mechanics, energy levels in a one-dimensional box can be calculated to determine the specific energies that a particle can have inside the box. This is crucial for understanding how a particle, such as a proton, behaves in such confines. The key formula to calculate energy levels in a one-dimensional box is:\[ E_n = \frac{n^2h^2}{8mL^2} \]where:

• \( n \) is the principal quantum number, representing the energy level. In this case, it is an integer starting from 1.
• \( h \) is Planck's constant, approximately \( 6.626 \times 10^{-34} \) Js.
• \( m \) is the mass of the particle. For a proton, this is approximately \( 1.672 \times 10^{-27} \) kg.
• \( L \) is the length of the box, which is \( 50 \times 10^{-12} \) m for this problem.

To calculate the energy levels, you plug in the values for each energy state, such as \( n=1 \) and \( n=4 \) for this problem. This gives you the specific energy values that correspond to each level, providing a clear picture of the quantum state transitions possible within the system.Calculating energy levels allows us to explore how much energy is required for transitions, like moving from the ground state to an excited state, which is crucial for further calculations of physical phenomena such as wavelength.

Wavelength Calculation

Once the energy levels are calculated, the next step is to find out the wavelength of the electromagnetic radiation required for transitions between these levels. In quantum mechanics, the energy difference between two states determines the type of radiation required. For the transition from \( n=1 \) to \( n=4 \), we must first calculate the difference in energy, \( \Delta E \).\[ \Delta E = E_4 - E_1 \]This energy difference can then be used with Einstein's relation for electromagnetic radiation energy:\[ E = hu = \frac{hc}{\lambda} \]where:

• \( u \) is the frequency of the radiation.
• \( c \) is the speed of light, approximately \( 3 \times 10^8 \) m/s.
• \( \lambda \) is the wavelength of the radiation we are trying to find.

By rearranging the equation to solve for \( \lambda \), we get:\[ \lambda = \frac{hc}{\Delta E} \]This equation allows us to find the wavelength of the required radiation. Understanding this helps students bridge the gap between energy differences and the physical manifestation of this energy as electromagnetic waves, crucial in quantum transitions.

One-Dimensional Box Model

The one-dimensional box model is a fundamental concept in quantum mechanics, used to describe a particle that is free to move in a small space with infinitely high walls - essentially a simplified system to understand quantum behavior. This model allows us to visualize and calculate quantized energy levels.The assumptions of this model include:

• The particle is confined to move only along one dimension, the length \( L \) of the box.
• The walls are impenetrable, meaning no potential energy is considered outside the box.
• The particle exhibits wave-like properties, confined by boundary conditions that dictate that the wave function must be zero at the walls of the box.

In our problem, the box is \( 50 \times 10^{-12} \) m long, a typical size for quantum mechanical problems involving subatomic particles, such as protons. The mathematical description through quantized energy levels in the box model highlights the particle's discrete nature when confined at such small scales.Utilizing this model simplifies the calculations, making it an excellent starting point for students venturing into understanding complex quantum systems. It serves as a stepping stone to grasp more elaborate scenarios in quantum mechanics, providing a basic framework to explore wave-particle duality and the quantization of energy.