Ask AI: Calculate the force constants of the C=O bonds with the values of carbon monoxide (1902 cm-1)

The force constant (k) of a bond is a measure of the bond's stiffness and can be calculated using the wavenumber of the vibrational mode. For carbon monoxide, the wavenumber (ν) is given as 1902 cm⁻¹. The relation between the wavenumber and the force constant is given by the formula: \[ \nu = \frac{1}{2\pi c} \sqrt{\frac{k}{\mu}} \] Where: - \(\nu\) is the wavenumber (in cm⁻¹) - \(c\) is the speed of light in vacuum (approximately \(3 \times 10^{10} \, \text{cm/s}\)) - \(k\) is the force constant (in dyn/cm or N/m; dyn/cm is preferred here because of the units of ν and c) - \(\mu\) is the reduced mass of the C-O system First, we need to calculate the reduced mass (\(\mu\)) of the C-O bond. The reduced mass is given by: \[ \mu = \frac{m_C m_O}{m_C + m_O} \] Where: - \(m_C\) is the mass of a carbon atom (approximately 12 atomic mass units, amu) - \(m_O\) is the mass of an oxygen atom (approximately 16 amu) Converting atomic masses to grams: \[ m_C = 12 \, \text{amu} \times 1.66 \times 10^{-24} \, \text{g/amu} = 1.99 \times 10^{-23} \, \text{g} \] \[ m_O = 16 \, \text{amu} \times 1.66 \times 10^{-24} \, \text{g/amu} = 2.66 \times 10^{-23} \, \text{g} \] Now calculate the reduced mass (\(\mu\)) in grams: \[ \mu = \frac{1.99 \times 10^{-23} \, \text{g} \times 2.66 \times 10^{-23} \, \text{g}}{1.99 \times 10^{-23} \, \text{g} + 2.66 \times 10^{-23} \, \text{g}} \] \[ \mu = \frac{5.29 \times 10^{-46} \, \text{g}^2}{4.65 \times 10^{-23} \, \text{g}} \] \[ \mu = 1.14 \times 10^{-23} \, \text{g} \] Convert the reduced mass to kg for consistency with SI units: \[ \mu = 1.14 \times 10^{-23} \, \text{g} \times 10^{-3} \, \text{kg/g} \] \[ \mu = 1.14 \times 10^{-26} \, \text{kg} \] Now we can rearrange the equation to solve for \(k\): \[ k = (2 \pi c \nu)^2 \mu \] Insert the values: \[ c = 3 \times 10^{10} \, \text{cm/s} \] \[ \nu = 1902 \, \text{cm}^{-1} \] Putting it all together: \[ k = \left(2 \pi \times 3 \times 10^{10} \, \text{cm/s} \times 1902 \, \text{cm}^{-1}\right)^2 \times 1.14 \times 10^{-26} \, \text{kg} \] \[ k = \left(2 \pi \times 3 \times 10^{10} \times 1902\right)^2 \times 1.14 \times 10^{-26} \] \[ k = \left(1.14164 \times 10^{15} \, \text{s}^{-1}\right)^2 \times 1.14 \times 10^{-26} \] \[ k = 1.30234 \times 10^{30} \, \text{s}^{-2} \times 1.14 \times 10^{-26} \] \[ k = 1.48 \times 10^4 \, \text{N/m} \] However, the original units of \(k\) in dyn/cm are preferred: \[ k_{\text{dyn/cm}} = 1.48 \times 10^4 \, \frac{N}{m} \times \frac{10^5 \, \text{dyn}}{1 \, \text{N}} \times \frac{1 \, \text{m}}{10^2 \, \text{cm}} \] \[ k_{\text{dyn/cm}} = 1.48 \times 10^7 \, \text{dyn/cm} \] Therefore, the force constant for the C=O bond in carbon monoxide is approximately \(1.48 \times 10^7 \, \text{dyn/cm}\).