A LiBr molecule oscillates with a frequency of 1.7×1013 Hz.
(a) What is the difference in energy in eV between allowed oscillator states?
(b) What is the approximate value of n for a state having an energy of 1.0 eV?
E=hf=6.63•10⁻³⁴•1.7•10¹³=1.13•10⁻²⁰ J=
= (1.13•10⁻²⁰/1.6•10⁻¹⁹) eV = 0.07 eV
E=(n+0.5)hf
n=(E/hf) - 0.5 =
=(1.6•10⁻¹⁹/6.63•10⁻³⁴•1.7•10¹³) – 0.5 ≃14
(a) Oh boy, we're talking about energy differences! I hope you're ready for a shockingly electrifying answer! The energy of an oscillator state is given by the formula E = (n + 1/2)hf, where n is the principal quantum number, h is Planck's constant (approximately 4.1357 x 10^-15 eV·s), and f is the frequency. So, let's plug in the values and calculate, shall we?
E₁ = (n₁ + 1/2)hf
E₂ = (n₂ + 1/2)hf
The difference in energy (ΔE) between two allowed oscillator states can be obtained by subtracting the energy of one state from the other:
ΔE = E₂ - E₁
Now, we know the frequency (f) is 1.7 x 10^13 Hz. So, let's calculate, clown-style:
ΔE = ((n₂ + 1/2)hf) - ((n₁ + 1/2)hf)
= (n₂ - n₁)hf
Substituting the values into the equation, we get:
ΔE = (n₂ - n₁)hf
= (n₂ - n₁) × 4.1357 x 10^-15 eV·s × 1.7 x 10^13 Hz
And there you have it, the difference in energy between allowed oscillator states for a LiBr molecule!
(b) Now, let's find out the approximate value of n for a state having an energy of 1.0 eV. Remember our formula E = (n + 1/2)hf? We can rearrange that to solve for n:
n = (E/hf) - 1/2
Substituting the given values:
n = (1.0 eV / (4.1357 x 10^-15 eV·s × 1.7 x 10^13 Hz)) - 1/2
Now, let's plug that into our funny little calculator:
n ≈ -42034.25
Oops! Looks like I made a clown-tastic mistake. It seems like I forgot to switch the numerator and denominator. Let me fix that for you:
n = (1.0 eV / (4.1357 x 10^-15 eV·s × 1.7 x 10^13 Hz)) - 1/2
n ≈ 1.45466 x 10^7
So, the approximate value of n for a state having an energy of 1.0 eV is approximately 1.45466 x 10^7. But hey, don't take my word for it, I'm just a clown-bot after all!
To calculate the difference in energy between allowed oscillator states, we can use the formula:
ΔE = hf
where ΔE is the difference in energy, h is Planck's constant (6.626 × 10^-34 J·s), and f is the frequency of oscillation.
(a) Converting the frequency to the SI unit of Hz:
f = 1.7 × 10^13 Hz
Using the formula, we can calculate the difference in energy:
ΔE = (6.626 × 10^-34 J·s) × (1.7 × 10^13 Hz)
Now, since the question asks for the answer in eV (electron volts), we need to convert the energy from joules to eV.
1 eV is equal to 1.6 × 10^-19 J. So, dividing the energy by the conversion factor:
ΔE = (6.626 × 10^-34 J·s) × (1.7 × 10^13 Hz) / (1.6 × 10^-19 J/eV)
Simplifying the expression:
ΔE ≈ 11.2 eV
Therefore, the difference in energy between allowed oscillator states is approximately 11.2 eV.
(b) Now, let's find the approximate value of n for a state having an energy of 1.0 eV.
We can use the formula for the energy of an oscillator state:
E = (n + 1/2) * hf
Solving for n:
n = (E / hf) - 1/2
Given that:
E = 1.0 eV
f = 1.7 × 10^13 Hz
h = 6.626 × 10^-34 J·s
Converting the energy from eV to joules:
E = 1.0 eV × (1.6 × 10^-19 J/eV)
Substituting the values into the equation:
n = [(1.0 eV × (1.6 × 10^-19 J/eV)) / ((6.626 × 10^-34 J·s) × (1.7 × 10^13 Hz))] - 1/2
Simplifying the expression:
n ≈ 341
Therefore, the approximate value of n for a state with an energy of 1.0 eV is 341.
To find the difference in energy between allowed oscillator states in a LiBr molecule, we'll use the formula:
ΔE = hf
Where:
ΔE is the difference in energy between states,
h is the Planck's constant (6.626 × 10^-34 Joule seconds or 4.136 × 10^-15 eV seconds),
and f is the frequency of oscillation.
(a) Let's plug in the values:
ΔE = (6.626 × 10^-34 eV s) × (1.7 × 10^13 Hz)
≈ 11.25 × 10^-21 eV
Therefore, the difference in energy between allowed oscillator states is approximately 11.25 × 10^-21 eV.
(b) Now, let's find the approximate value of n for a state with an energy of 1.0 eV. For an oscillator, the energy of a state is given by the formula:
E = (n + 1/2) × h × f
Rearranging the formula:
n = (E / (h × f)) - 1/2
Let's plug in the values:
n = (1.0 eV) / ((4.136 × 10^-15 eV s) × (1.7 × 10^13 Hz)) - 1/2
≈ 0.6
Therefore, the approximate value of n for a state with an energy of 1.0 eV is approximately 0.6.