Question

The radius r of a hydrogen atom (which consists of a proton and a comparatively lightweight electron) can only depend on the (equal) magnitudes e of the proton's and electron's charge, the Coulomb constant 1/4πϵ0​ that characterizes the strength of the electrostatic attraction between the proton and the electron, the mass m of the bound electron, and (because quantum mechanics is likely involved) Planck's constant in the form h/2π≡ℏ. (Since the proton remains essentially at rest, its mass turns out to be irrelevant. (a) Assuming that r depends only on the stated quantities, use dimensional analysis and the information from the inside cover of the [Unit C book] to find a plausible formula for the radius of a hydrogen atom. (b) Calculate a numerical estimate for the radius. (c) Explain why this is only an estimate. Some Physical Constants Standard Metric Prefixes (for powers of 10) Useful Conversion Factors

Answer 1

Step-by-step explanation:

(a) To find a plausible formula for the radius of a hydrogen atom using dimensional analysis, we need to consider the relevant quantities and their dimensions.

The given quantities are:

Magnitude of the charges, e

Coulomb constant, 1/4πϵ₀

Mass of the electron, m

Planck's constant, ℏ

We can assign dimensions to these quantities as follows:

[e] = C (Coulombs, the unit of charge)

[1/4πϵ₀] = C²/N·m² (Coulomb squared per Newton per square meter, the unit of the electric constant)

[m] = kg (kilograms, the unit of mass)

[ℏ] = J·s (Joule seconds, the unit of Planck's constant)

The formula for the radius, r, would then be a combination of these quantities with appropriate exponents to ensure dimensional consistency. By inspection, we can propose the following formula:

r = k(e²/ϵ₀)^(a)(ℏ^b)(m^c)

Here, k is a dimensionless constant, and a, b, and c are exponents that we need to determine.

(b) To calculate a numerical estimate for the radius of a hydrogen atom, we need the values of the relevant constants. Let's consider the following approximate values:

e ≈ 1.602 x 10^(-19) C (approximate charge of an electron or proton)

ϵ₀ ≈ 8.854 x 10^(-12) C²/N·m² (permittivity of free space)

ℏ ≈ 1.055 x 10^(-34) J·s (reduced Planck's constant)

m ≈ 9.109 x 10^(-31) kg (mass of an electron)

Plugging these values into the proposed formula, we can obtain a numerical estimate for the radius.

(c) This numerical estimate is only an approximation because the proposed formula is based on dimensional analysis and does not consider the specific details and complexities of the hydrogen atom, which involve quantum mechanics. The actual behavior of electrons in atoms is described by quantum mechanical wave functions and orbital models. To obtain a more precise value for the radius, a detailed quantum mechanical analysis is required, which goes beyond the scope of dimensional analysis.