Final answer:
The minimum kinetic energy of a proton confined in a 1-dimensional box can be determined using the uncertainty principle. To calculate the minimum kinetic energy, we need to find the uncertainty in position and momentum of the proton. The uncertainty principle states that the minimum uncertainty in position multiplied by the minimum uncertainty in momentum is greater than or equal to Planck's constant divided by 4π.
Step-by-step explanation:
The minimum kinetic energy of a proton confined in a 1-dimensional box can be determined using the uncertainty principle. According to the uncertainty principle, the minimum uncertainty in the position of the proton (Δx) multiplied by the minimum uncertainty in its momentum (Δp) is greater than or equal to Planck's constant divided by 4π. In this case, the proton is confined in a uranium nucleus of diameter 14.4x10^-15 m, so the length of the box is equal to the diameter of the nucleus.
The uncertainty in the position of the proton (Δx) is equal to half the length of the box, which is 7.2x10^-15 m. To find the minimum uncertainty in the momentum (Δp), we first need to find the wavelength of the proton. Since we know the mass of the proton (1.67x10^-27 kg) and the length of the box, we can use the de Broglie equation to find the wavelength (λ) of the proton.
Using the equation λ = h / p, where h is Planck's constant (6.63x10^-34 J·s) and p is the momentum of the proton, we can solve for p. Once we have the momentum, we can find the minimum uncertainty in the momentum (Δp).
Finally, using the equation E = p^2 / (2m), where E is the kinetic energy of the proton and m is the mass of the proton, we can calculate the minimum kinetic energy of the proton confined in the 1-dimensional box.