Question

An electron (restricted to one dimension) is trapped between two rigid walls 1.40 nm apart. The electron's energy is approximately 19 eV. (a) What is the quantum number n for the energy state that the electron occupies? (b) Based on the quantum number you found in part (a), calculate a more precise value for the electron's energy in eV), expressed to at least three significant figures. (Use any physical constants or unit conversions to at least four significant figures in your calculations.) _________ eV

Answer 1

a) n = 9.9 b) E₁₀ = 19.25 eV

Step-by-step explanation:

Solving the Scrodinger equation for the electronegative box we get

Eₙ = (h² / 8m L²2) n²

where l is the distance L = 1.40 nm = 1.40 10⁻⁹ m and n the quantum number

In this case En = 19 eV let us reduce to the SI system

En = 19 eV (1.6 10⁻¹⁹ J / 1 eV) = 30.4 10⁻¹⁹ J

n = √ (In 8 m L² / h²)

let's calculate

n = √ (8 9.1 10⁻³¹ (1.4 10⁻⁹)² 30.4 10⁻¹⁹ / (6.63 10⁻³⁴)²

n = √ (98) n = 9.9

since n must be an integer, we approximate them to 10

b) We substitute for the calculation of energy

In = (h² / 8mL2² n²

In = (6.63 10⁻³⁴) 2 / (8 9.1 10⁻³¹ (1.4 10⁻⁹)² 10²

E₁₀ = 3.08 10⁻¹⁸ J

we reduce eV

E₁₀ = 3.08 10⁻¹⁸ j (1ev / 1.6 10⁻¹⁹J)

E₁₀ = 1.925 101 eV

E₁₀ = 19.25 eV

the result with significant figures is

E₁₀ = 19.25 eV