Closest Approach Estimate
- The Coulomb equation can be used to give an estimate for the radius of a nucleus
- When the alpha particle reaches its closest approach (to the nucleus), all of its kinetic energy has been converted to electric potential energy
- Initially, the alpha particles have kinetic energy equal to:
- The electric potential energy between the two charges is equal to:
- At this point, the kinetic energy Ek lost by the α particle approaching the nucleus is equal to the potential energy gain Ep, so:
- Where:
- m = mass of an α particle (kg)
- v = initial speed of the α particles (m s–1)
- q = charge of an α particle (C)
- Q = charge of the nucleus being investigated (C)
- r = the radius of closest approach (m)
- ε0 = permittivity of free space
Worked example
The first artificially produced isotope, phosphorus-30 (15P) was formed by bombarding an aluminium-27 isotope (13Al) with an α particle.
For the reaction to take place, the α particle must come within a distance, r, from the centre of the aluminium nucleus.
Calculate the distance, r, if the nuclear reaction occurs when the α particle is accelerated to a speed of at least 2.55 × 107 m s–1.
Step 1: List the known quantities
- Mass of an α particle, m = 4u = 4 × (1.66 × 10–27) kg
- Speed of the α particle, v = 2.55 × 107 m s–1
- Charge of an α particle, q = 2e = 2 × (1.6 × 10–19) C
- Charge of an aluminium nucleus, Q = 13e = 13 × (1.6 × 10–19) C
- Permittivity of free space, ε0 = 8.85 × 10–12 F m–1
Step 2: Write down the equations for kinetic energy and electric potential energy
Step 3: Rearrange for distance, r
Step 4: Calculate the distance, r
r = 2.77 × 10–15 m
Exam Tip
Make sure you're comfortable with the calculations involved with the alpha particle closest approach method, as this is a common exam question.You will be expected to remember that the charge of an α is the charge of 2 protons (2 × the charge of an electron)
