Kinetic Energy of an Alpha Particle with a Wavelength of 12pm

To calculate the kinetic energy of an alpha particle with a wavelength of 12 pm, we need to use the wave-particle duality concept and the de Broglie wavelength equation.

Wave-Particle Duality
According to wave-particle duality, particles such as electrons, protons, and alpha particles exhibit both wave-like and particle-like properties. This means that they can be described both as particles with mass and as waves with a wavelength.

de Broglie Wavelength Equation
The de Broglie wavelength equation relates the wavelength of a particle to its momentum. It is given by the formula:

λ = h / p

Where:
λ is the wavelength of the particle
h is the Planck's constant (6.626 x 10^-34 J·s)
p is the momentum of the particle

To calculate the kinetic energy of the alpha particle, we need to determine its momentum first. The momentum of a particle can be calculated using the equation:

p = mv

Where:
p is the momentum
m is the mass of the particle
v is the velocity of the particle

Mass and Velocity of an Alpha Particle
An alpha particle consists of two protons and two neutrons, giving it a mass of approximately 4 atomic mass units (u) or 6.64 x 10^-27 kg. The velocity of an alpha particle can be determined using the equation:

v = λf

Where:
v is the velocity of the alpha particle
λ is the wavelength (12 pm or 1.2 x 10^-11 m)
f is the frequency of the alpha particle

Calculating the Momentum
Using the given wavelength of 12 pm, we can calculate the velocity of the alpha particle:

v = λf
= (1.2 x 10^-11 m)(c)
= 3.6 x 10^16 m/s

Where c is the speed of light (3 x 10^8 m/s).

Using the velocity and mass of the alpha particle, we can calculate its momentum:

p = mv
= (6.64 x 10^-27 kg)(3.6 x 10^16 m/s)
= 2.38 x 10^-10 kg·m/s

Calculating the Kinetic Energy
Now that we have the momentum of the alpha particle, we can use the de Broglie wavelength equation to calculate its kinetic energy. Rearranging the equation, we have:

λ = h / p
p = h / λ

Substituting the values:

p = (6.626 x 10^-34 J·s) / (1.2 x 10^-11 m)
= 5.522 x 10^-24 kg·m/s

The kinetic energy of the alpha particle can be calculated using the equation:

KE = (1/2) mv^2

Substituting the values:

KE = (1/2)(6.64 x 10^-27 kg)(3.6 x 10^16 m/s)^2
= 4.79 x 10^-11 J

Therefore, the kinetic energy of an alpha particle with a wavelength of 12 pm is approximately 4