Question

Alpha particles are accelerated through a potential difference of $20.0\mathrm{kV}$. What is their de Broglie wavelength?

Short answer

Answer: To find the de Broglie wavelength of the alpha particles, follow the steps below: 1. Calculate the energy of the alpha particles using the equation E = eV, where E is the energy, e is the charge of an alpha particle (2e), and V is the potential difference (5.0 MV). 2. Find the speed of the alpha particles using the equation E = (1/2)mv², where m is the mass of an alpha particle (4m_p) and v is its speed. 3. Calculate the momentum of the alpha particles using the equation p = mv, where p is the momentum, m is the mass of an alpha particle, and v is its speed. 4. Finally, use the de Broglie wavelength equation λ = h/p, where λ is the de Broglie wavelength, h is Planck's constant, and p is the momentum of the alpha particles, to find the wavelength. After following these steps and plugging in the appropriate values, you will find the de Broglie wavelength of the alpha particles.

Step-by-step solution

Find the energy of the alpha particles using the given potential difference

We can find the energy of the alpha particles using the equation: $$E=eV$$ where $E$ is the energy, $e$ is the charge of an alpha particle (since an alpha particle consists of 2 protons and 2 neutrons, its charge is $2e$), and $V$ is the potential difference. Plug in the values given in the problem to find the energy.

Find the speed of alpha particles using the energy calculated in step 1

Next, we use the equation relating kinetic energy to the speed of the particles: $$E=\frac{1}{2}m{v}^2$$ where $m$ is the mass of an alpha particle and $v$ is its speed. The mass of an alpha particle is $4m_p$, where $m_p$ is the mass of a proton. Rearrange the equation and solve for $v$ using the energy found in step 1.

Find the momentum of alpha particles using the speed from step 2

Now we need to find the momentum of the alpha particles, which is given by: $$p=mv$$ where $p$ is the momentum, $m$ is the mass of an alpha particle, and $v$ is its speed found in step 2. Calculate the momentum using the given mass of an alpha particle and the speed from step 2.

Calculate the de Broglie wavelength of alpha particles using their momentum

Finally, we can use the de Broglie wavelength equation to find the wavelength of the alpha particles: $$\lambda =\frac{h}{p}$$ where $\lambda$ is the de Broglie wavelength, $h$ is Planck's constant, and $p$ is the momentum of the alpha particles calculated in step 3. Plug in the values and solve for the de Broglie wavelength.

Key concepts

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at the smallest scales, such as atoms and subatomic particles. Unlike classical physics, which uses deterministic laws and can predict the outcome of a physical system with great certainty, quantum mechanics is probabilistic, meaning it can only predict the likelihood of finding a particle in a particular state.

One of the key principles of quantum mechanics is wave-particle duality, which states that every particle exhibits both wave and particle properties. This dual nature is described by the de Broglie hypothesis, which asserts that any moving particle or object has an associated wave called the 'de Broglie wavelength'. The de Broglie wavelength is inversely proportional to the momentum of the particle; thus, a larger, heavier particle has a shorter wavelength, while a smaller, lighter particle has a longer one.

Understanding the wave-like behavior of particles is crucial for interpreting phenomena such as interference patterns and the behavior of electrons in atoms, which are essential to the modern understanding of chemical bonding and the function of electronic devices.

Alpha Particles

Alpha particles are a type of ionizing radiation ejected from the nucleus of an atom during radioactive decay. They consist of two protons and two neutrons bound together, which is essentially a helium-4 nucleus.

Due to their relatively large mass (compared to other subatomic particles like electrons) and the fact that they carry a double positive charge, alpha particles have a significant amount of kinetic energy. This energy can cause ionization when it interacts with matter, leading to chemical changes or damage to living cells. Alpha particles also have a very limited range in air and can be stopped by a sheet of paper or the outer layer of human skin, which makes them relatively safe, provided they are not ingested or inhaled.

In the context of quantum mechanics, when alpha particles are accelerated, as in the given exercise, their wave-particle duality becomes relevant, and calculating their de Broglie wavelength provides insight into their wave-like properties.

Potential Difference

The potential difference, often referred to as voltage, is a measure of the energy difference per charge between two points in an electric field. In simpler terms, it's the 'push' that causes charges to move in a particular direction and can accelerate charged particles from one point to another.

When charged particles such as alpha particles are accelerated through a potential difference, they gain kinetic energy, which can be calculated by the product of the charge and the potential difference. For an alpha particle, which has a charge of +2e (where e is the elementary charge), the potential difference determines how much energy the particle will acquire as it moves through an electric field. This energy transition is at the core of many technologies, from the cathode ray tubes in old television sets to particle accelerators that probe the fundamental particles of matter.

In the context of the original exercise, the potential difference of 20.0 kV applied to alpha particles is responsible for their acceleration and subsequent kinetic energy, which is necessary for determining their de Broglie wavelength.

Kinetic Energy

Kinetic energy is the energy of motion. Any object that has mass and is moving with a certain velocity has kinetic energy. In the classical sense, it's calculated using the formula $E_k=\frac{1}{2}m{v}^2$ where $E_k$ is the kinetic energy, $m$ is the mass of the object, and $v$ is its velocity.

For subatomic particles like alpha particles, knowing their kinetic energy is important for understanding their motion and interactions. When an alpha particle is accelerated through a potential difference, it gains kinetic energy. This energy is directly related to the speed that the particle will acquire and is crucial for calculations involving the particle's dynamics, such as its momentum and, as outlined by Louis de Broglie, its associated wavelength.

In the textbook solution provided, the kinetic energy gained by alpha particles when accelerated through a potential difference is used to calculate their speed, which is then used to determine their momentum. This information is essential for calculating their de Broglie wavelength, linking back to quantum mechanics and the wave-particle duality of the particles.