With what velocity should an alpha particle travel towards the nucleus...
Calculating the Velocity of an Alpha Particle towards the Nucleus of a 'Cu' Atom
Introduction: An alpha particle is a helium nucleus consisting of two protons and two neutrons. To calculate the velocity of an alpha particle towards the nucleus of a 'Cu' atom, we need to consider the Coulomb force of interaction between the two charged particles.
Formula:The Coulomb force between two charged particles can be calculated using the following formula:
F = (k * q1 * q2) / r^2
Where F is the force of interaction, k is the Coulomb constant, q1 and q2 are the charges of the two particles, and r is the distance between them.
Calculating the Force of Interaction:In the case of an alpha particle and a 'Cu' atom, the charge of the alpha particle is +2e (where e is the charge of an electron) and the charge of the 'Cu' atom is +29e. The distance between them is given as 10^-13 m. Substituting these values in the Coulomb force formula, we get:
F = (9 × 10^9 Nm^2/C^2) * (2e) * (29e) / (10^-13 m)^2
F = 1.67 × 10^-11 N
Calculating the Velocity:The alpha particle is initially at rest and moves towards the 'Cu' atom. Therefore, the initial kinetic energy of the alpha particle is zero. The final kinetic energy of the alpha particle can be calculated using the work-energy principle:
W = ΔK.E
Where W is the work done by the Coulomb force, and ΔK.E is the change in kinetic energy of the alpha particle.
The work done by the Coulomb force can be calculated as:
W = F * d
Where d is the distance traveled by the alpha particle towards the 'Cu' atom. Since the alpha particle starts from rest, the distance traveled by it is equal to the distance between the alpha particle and the 'Cu' atom, which is 10^-13 m.
Substituting the values, we get:
W = (1.67 × 10^-11 N) * (10^-13 m)
W = 1.67 × 10^-24 J
The change in kinetic energy of the alpha particle can be calculated as:
ΔK.E = K.Ef - K.Ei
Where K.Ef is the final kinetic energy of the alpha particle, and K.Ei is the initial kinetic energy of the alpha particle.
Since the initial kinetic energy of the alpha particle is zero, we can write:
ΔK.E = K.Ef
Substituting the values, we get:
1.67 × 10^-24 J = (1/2) * m * v^2
Where m is the mass of the alpha particle, and v is the velocity of the alpha particle.
The mass of the alpha particle is 6.64 × 10^-27 kg. Substituting this value, we get:
1.67 × 10^-24 J = (1/2) * (6.64 × 10^-27 kg) * v^2
v = 2.69 × 10^7 m/s