The reduced mass (μ) of a diatomic molecule can be calculated using the formula:

μ = (mX * mY) / (mX + mY),

where mX and mY are the masses of atoms X and Y, respectively.

To calculate the reduced mass, we need to know the atomic masses of X and Y. Let's assume the atomic mass of X is MX and the atomic mass of Y is MY.

We can rearrange the formula for bond dissociation energy (D) to solve for the reduced mass:

D = μ * (v^2 / 2)

where v is the vibrational frequency. In this case, we can solve for v using the known values of D and the bond length (r):

D = μ * (v^2 / 2)
v^2 = (2 * D) / μ
v^2 = (2 * 400 kJ/mol) / μ
v^2 = 800 kJ/mol / μ

Next, we need to calculate the square of the vibrational frequency (v^2) in s^-2. Molecular vibrations are typically measured in wavenumbers (cm^-1). We can convert wavenumbers to s^-2 using the equation:

v^2 = (1 / λ^2) * c^2

where λ is the wavelength in cm and c is the speed of light (2.998 × 10^10 cm/s).

From the bond length (r) provided, we can calculate the wavelength (λ) using the equation:

λ = 2 * r

Now we can substitute the value of λ into the equation for v^2:

v^2 = (1 / (2r)^2) * c^2

Let's calculate the reduced mass:

First, convert the bond length from nm to cm:
r = 0.15 nm * (10 cm / 1 nm) = 1.5 cm

Next, calculate the wavelength:
λ = 2 * 1.5 cm = 3 cm

Then substitute the values into the equation for v^2:
v^2 = (1 / (2(1.5 cm))^2) * (2.998 × 10^10 cm/s)^2
v^2 = (1 / 4.5 cm^2) * (8.988 × 10^20 cm^2/s^2)
v^2 = (8.988 × 10^20 cm^2/s^2) / (4.5 cm^2)
v^2 = 1.996 × 10^20 s^-2

Now we can substitute the value of v^2 and D into the equation to solve for μ:
1.996 × 10^20 s^-2 = (800 kJ/mol) / μ
1.996 × 10^20 s^-2 * μ = 800 kJ/mol
μ = 800 kJ/mol / (1.996 × 10^20 s^-2)
μ = 4.01 × 10^-19 kJ/(mol.s^-2)

Finally, convert the reduced mass from kJ/(mol.s^-2) to kg by dividing by Avogadro's number (6.022 × 10^23 mol^-1):
μ = (4.01 × 10^-19 kJ/(mol.s^-2)) / (6.022 × 10^23 mol^-1)
μ = 6.66 × 10^-43 kg

Therefore, the reduced mass of the diatomic molecule XY is approximately 6.66 × 10^-43 kg.

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