Question

Calculate (a) the zero point energy of a He atom in a one-dimensional box of length \(1.00 \mathrm{cm}\) and (b) the ratio of the zero point energy to \(k T\) at \(300 .\) K.

Short answer

(a) The zero point energy of a helium atom in a one-dimensional box of length 1.00 cm is approximately \(9.79 \times 10^{-18} \mathrm{J}\). (b) The ratio of the zero point energy to \(kT\) at 300 K is approximately 2.36.

Step-by-step solution

Determine the energy levels of a particle in a one-dimensional box

For a quantum particle in a one-dimensional box, the energy is given by the formula: \( E_n = \frac{n^2 h^2}{8mL^2} \), where \( E_n \) is the energy of the particle at quantum number n, \( h \) is Planck's constant, \( m \) is the mass of the particle, and L is the length of the box. In this exercise, we are asked to find the zero point energy of a helium atom in a box, which corresponds to quantum number n = 1. We are given the length of the box \( L = 1.00 \mathrm{cm} = 1.00\times10^{-2} \mathrm{m} \), and we will need the mass of a helium atom, which is approximately \( 4 \times 10^{-3} \mathrm{g/mol}\). We will convert the mass to kg/particle: \( m = \frac{4\times10^{-3} \mathrm{kg/mol}}{N_A} \), where \( N_A \) is Avogadro's number (\( 6.022 \times 10^{23} \mathrm{mol^{-1}} \)).

Calculate the zero point energy of the helium atom in the box

Now that we have all of the necessary information, we can plug the values into the equation for the energy in a one-dimensional box to find the zero point energy for n = 1: \( E_1 = \frac{1^2 h^2}{8mL^2} \). Calculate the value of m: \( m = \frac{4\times10^{-3} \mathrm{kg/mol}}{6.022 \times 10^{23} \mathrm{mol^{-1}}} \approx 6.64 \times 10^{-27} \mathrm{kg} \). Insert all the values: \( E_1 = \frac{(1^2)(6.626 \times 10^{-34} \mathrm{J \cdot s})^2}{8 (6.64 \times 10^{-27} \mathrm{kg}) (1.00 \times 10^{-2}\mathrm{m})^2} \approx 9.79 \times 10^{-18} \mathrm{J} \). So, the zero point energy of the helium atom in the box is \( \approx 9.79 \times 10^{-18} \mathrm{J} \).

Calculate the ratio of the zero point energy to kT

To find the ratio of the zero point energy to the product of Boltzmann's constant, k, and the temperature, T, we can use the following formula: \( \frac{E_1}{kT} \), where \( k = 1.381 \times 10^{-23} \mathrm{J \cdot K^{-1}} \) and \( T = 300 \mathrm{K} \). Plugging in the values: \( \frac{E_1}{kT} = \frac{9.79 \times 10^{-18} \mathrm{J}}{(1.381 \times 10^{-23} \mathrm{J \cdot K^{-1}}) (300\mathrm{K})} \approx 2.36 \). So, the ratio of the zero-point energy to kT is approximately 2.36.

Key concepts

Zero Point Energy

Zero point energy is a fascinating concept in quantum mechanics that illustrates that even at absolute zero, particles still possess a fundamental level of energy. This is a key distinction from classical mechanics, where it is assumed that an object at rest has no kinetic energy. In quantum systems, however, the uncertainty principle dictates that there must be a non-zero baseline energy level.

In the context of a particle in a one-dimensional box, the zero point energy (\(E_1\)) is the energy of the lowest state of the particle, where the quantum number (\(n\)) is equal to 1. The formula to calculate this is:

• \(E_1 = \frac{h^2}{8mL^2} \),

where \(h\) is Planck's constant, \(m\) is the mass of the particle, and \(L\) is the length of the box. When applied to the helium atom in our exercise, its zero point energy is approximately \(9.79 \times 10^{-18} \text{ J}\).

Understanding zero point energy is crucial because it gives insight into the behavior of particles at a quantum level. Even in the absence of external forces, particles retain energy, which impacts quantum phenomena significantly.

Particle in a Box

The model of a particle in a box is a fundamental concept in quantum mechanics. It describes a particle that is free to move within a confined space but cannot escape its boundaries. This model helps illustrate how quantum mechanics governs the behavior of particles at a microscopic level.

In this scenario, the walls of the box are considered to be infinitely high potential barriers. This assumption simplifies the math and allows us to derive the energy levels of the particle, which are quantized. Quantization means the particle can only occupy specific energy levels, dependent on the quantum number \(n\), given by:

• \( E_n = \frac{n^2h^2}{8mL^2} \)

where \(h\) represents Planck's constant, \(m\) is the particle's mass, and \(L\) is the length of the box.

The particle in the box model is more than just a theoretical tool. It provides essential groundwork for understanding real-world quantum systems like electrons in atoms or molecules. The model illustrates how confinement leads to quantized energy levels, which is a key principle in designing quantum devices.

Quantum Number

Quantum numbers are integral values that describe the quantized states of a quantum mechanical system. They help determine specific properties of particles in quantum systems, like energy levels within the particle in a box model.

The primary quantum number, \(n\), is crucial as it defines the energy level and indirectly influences other properties of the system. In a particle in a box model, the energy of the particle is directly proportional to the square of \(n\), according to the formula:

• \( E_n = \frac{n^2h^2}{8mL^2} \)

where \(h\) is Planck's constant, \(m\) is the particle's mass, and \(L\) is the length of the box.

Quantum numbers are not just a mathematical tool; they have physical significance. They help physicists understand configurations of electrons in atoms, permitting more detailed predictions of atomic behaviors and interactions. Understanding quantum numbers is fundamental in progressing from basic quantum problems to more complicated scenarios in atomic and molecular physics.