Use the bohr model to calculate the radius and the energy of the b⁴⁺ ion in the n 3 state. how much energy would be required to remove the electrons from 1 mol of b⁴⁺ in this state? what frequency and wavelength of light would be emitted in a transition from the n 3 to the n 2 state of this ion? express all results in si units.

E =  3.6 x 10⁶ J/mol

f =   1.1 x 10 ¹⁶ s⁻¹

λ =  2.6 x 10⁻⁸ m

Explanation:

Rydberg´s equation for hydrogen-like atoms is:

1/λ = Z²Rh (1/n₁² - 1/n₂²)

where  λ = wavelength  

Z² = atomic number of hydrogen-like atom  

 Rh= Rydberg´s constatn                                       

n₁ = principal quantum number of initial state                        

 n₂ = principal quantum number of final state

We also know that E = h(c/ λ ) = hf, where f is frequency equal to c/λ,  so we have all the information needed to answer the questions.

a)  We are asked the energy to remove the electron from 1 mol of B⁴⁺ , that means the transition is from  n₁ = 3 to n₂ = ∞. The term 1/n₂ approaches zero in the infinity so:

Working in  SI units

1/λ =  5² x1.097 x 10⁷ m⁻¹ ( 1/3² - 0) = 3.0 x 10⁷ m⁻¹

E= h(c/ λ )= hc(1/ λ) = 6.626 x 10⁻³⁴ J/s x  3 x 10⁸ m/s x  (3.0 x 10⁷ m⁻¹)

= 6.0x 10⁻¹⁸ J

This is the energy per atom, so per mol of atoms  is

= 6.0x 10⁻¹⁸ J/atom x 6.022 x 10²³ atoms/mol = 3.6 x 10⁶ J/mol

b) f and λ from a transition n= 3 to n=2

1/ λ = 5² x1.097 x 10⁷ m⁻¹ x ( 1/2² - 1/3²) = 3.8 x 10⁷  m⁻¹    ⇒

λ = 1/ 3.8 x 10⁷  m⁻¹ = 2.6 x 10⁻⁸ m

f = 3 x 10⁸ m/s / 2.6 x 10⁻⁸ m = 1.1x 10 ¹⁶ s⁻¹

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