How many electrons, protons, and neutrons are in a neutral 207 Pb atom? Enter your answers numerically separated by commas. N.. Np. N, = 82,82,125 Submit Previous Answers v Correct Part B The lead nucleus has a diameter of 14.2 fm. What is the electric field strength at the surface of a lead nucleus? Express your answer with the appropriate units. kg E = 2.305 • 1017 m’
The Correct Answer and Explanation is :
Part A: Electrons, Protons, and Neutrons in ( {}^{207}\text{Pb} )
The isotope ( {}^{207}\text{Pb} ) has the following characteristics:
- Protons ((N_p)): The atomic number of lead (Pb) is 82. This means ( {}^{207}\text{Pb} ) has 82 protons.
- Electrons ((N_e)): Since the atom is neutral, the number of electrons equals the number of protons. So, (N_e = 82).
- Neutrons ((N_n)): The mass number of ( {}^{207}\text{Pb} ) is 207. The number of neutrons can be calculated as:
[
N_n = \text{Mass Number} – \text{Protons} = 207 – 82 = 125
]
Thus, ( N_e, N_p, N_n = 82, 82, 125 ).
Part B: Electric Field Strength at the Surface of the Nucleus
The electric field strength ((E)) at the surface of a spherical nucleus is given by:
[
E = \frac{k_e Q}{r^2}
]
where:
- (k_e = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2) (Coulomb constant),
- (Q = +Ze) is the total charge, with (Z = 82) and (e = 1.602 \times 10^{-19} \, \text{C}) (charge of a proton),
- (r = \text{radius of the nucleus}).
The diameter of the nucleus is (14.2 \, \text{fm} = 14.2 \times 10^{-15} \, \text{m}), so the radius is:
[
r = \frac{\text{Diameter}}{2} = \frac{14.2}{2} \times 10^{-15} = 7.1 \times 10^{-15} \, \text{m}.
]
The total charge (Q) is:
[
Q = Z e = 82 \cdot 1.602 \times 10^{-19} = 1.31364 \times 10^{-17} \, \text{C}.
]
Substitute the values into the formula:
[
E = \frac{(8.99 \times 10^9)(1.31364 \times 10^{-17})}{(7.1 \times 10^{-15})^2}.
]
Simplify:
[
E = \frac{1.18012 \times 10^{-7}}{5.041 \times 10^{-29}} = 2.34 \times 10^{21} \, \text{N/C}.
]
Explanation
This calculation highlights the incredible strength of electric fields at the scale of atomic nuclei due to the small size and high charge concentration. This strong electric field contributes to nuclear stability and interactions in quantum scales.