The equilibrium bond length of 1
H35Cl is 126 pm. What is wavelength of the required electromagnetic radiation to excite the rotational state from j=1 to j=2?
The masses of 1H and 35Cl are 1.0078 amu and 34.9689 amu, respectively)
Here is a site that spells that out.
Using m1 and m2 you can calculate the reduced mass,u, with u and R you can calculate I (moment of inertia), knowing I allows you to calculate B. Knowing J and B allows you to calculate E. Then E = hc/wavelength and wavelength can be determined.
http://hyperphysics.phy-astr.gsu.edu/%E2%80%8Chbase/molecule/rotrig.html#c3
To determine the wavelength of electromagnetic radiation required to excite the rotational state from j=1 to j=2 in an HCl molecule, we can use the formula:
\[ \Delta E = \frac{{h^2}}{{8 \pi^2 \mu r^{2}}}(j'(j'+1)-j(j+1)) \]
where:
- ΔE is the energy difference between the two rotational states
- h is Planck's constant (6.626 x 10^-34 J s)
- π is a mathematical constant (approximately 3.14159)
- μ is the reduced mass of the molecule, given by:
μ = (mH * mCl) / (mH + mCl)
where mH and mCl are the masses of hydrogen and chlorine respectively
- r is the equilibrium bond length of the molecule
First, let's calculate the reduced mass of the HCl molecule:
μ = (1.0078 amu * 34.9689 amu) / (1.0078 amu + 34.9689 amu)
μ ≈ 0.9991 amu
Next, we need to convert the reduced mass into kilograms to match the units of Planck's constant:
μ ≈ 0.9991 amu * (1.66054 x 10^-27 kg/1 amu)
μ ≈ 1.65972 x 10^-27 kg
Now, we can calculate the energy difference between the two rotational states:
ΔE = (6.626 x 10^-34 J s)^2 / (8 * (3.14159)^2 * 1.65972 x 10^-27 kg * (126 pm * 10^-12 m)^2) * (2(2+1)-1(1+1))
Calculating this, we get:
ΔE ≈ 1.659 x 10^-20 J
Finally, we can use the relationship between energy and wavelength for electromagnetic radiation:
ΔE = hc / λ
Where:
- ΔE is the energy difference in joules
- h is Planck's constant (6.626 x 10^-34 J s)
- c is the speed of light (approximately 3.00 x 10^8 m/s)
- λ is the wavelength of radiation in meters
Rearranging the equation to solve for λ:
λ = hc / ΔE
Substituting the values:
λ = (6.626 x 10^-34 J s * 3.00 x 10^8 m/s) / (1.659 x 10^-20 J)
Calculating this, we find:
λ ≈ 1.20 x 10^-6 meters or 1200 nm
Therefore, the wavelength of the required electromagnetic radiation to excite the rotational state from j=1 to j=2 in an HCl molecule is approximately 1200 nanometers.