The internuclear distance (bond length) of carbon monoxide molecule is 1.13 Å. Calculate the energy (in joules and eV) of this molecule in the first excited rotational level. Also calculate the angular velocity of the molecule. Given atomic masses of ^12C=1.99×10^-26 kg; ^16O = 2.66× 10^-26 kg.

Question

The internuclear distance (bond length) of carbon monoxide molecule is 1.13 Å. Calculate the energy (in joules and eV) of this molecule in the first excited rotational level. Also calculate the angular velocity of the molecule. Given atomic masses of [latex]^{12}C=1.99×10^{-26} kg; ^{16}O = 2.66× 10^{-26} kg[/latex].

Accepted Answer

[latex]\begin{array}{l}r=1.13 \mathrm{~A}^{\circ}=1.13 \times 10^{-10} \mathrm{~m} \ \\mu=\frac{m_{1} m_{2}}{m_{1}+m_{2}}=\frac{1.99 \times 10^{-26} \mathrm{~kg} \times 2.66 \times 10^{-26} \mathrm{~kg}}{1.99 \times 10^{-26} \mathrm{~kg}+2.66 \times 10^{-26} \mathrm{~kg}}=1.14 \times 10^{-26} \mathrm{~kg}\end{array}[/latex]

moment of inertia [latex]I=\mu r^{2}\[/latex]

[latex]\quad \quad \qquad I=1.14 \times 10^{-26} \mathrm{~kg} \times\left(1.13 \times 10^{-10} \mathrm{~m}\right)^{2}=1.46 \times 10^{-46} \mathrm{~kg} \mathrm{~m}^{2}[/latex]

Energy of a rigid diatomic molecule is given by

[latex]E_{J}=\frac{h^{2}}{8 \pi^{2} I} J(J+1) \text { Joule }[/latex]

For the first excited level, J=1; hence

[latex]\begin{array}{l}E_{1}=\frac{\left(6.626 \times 10^{-34}\right)^{2} \times 1(1+1)}{8 \times 3.14 \times 3.14 \times 1.46 \times 10^{-46} \mathrm{~kg} \mathrm{~m}^{2}} \ \ \quad=7.61 \times 10^{-23} \text { Joule }\\end{array}\[/latex]

since 1 eV = 1.602 × [latex]10^{-19}[/latex] J

[latex]E_1=\frac{7.61×10^{-23}}{1.602×10^{-19} J eV^{-1}}\\quad =4.76 \times 10^{-4} \mathrm{eV}\[/latex]

[latex]\text { angular velocity } \omega=\sqrt{\frac{2 E_{1}}{I}}=\sqrt{\frac{2 \times 7.61 \times 10^{-23} \mathrm{~J}}{1.46 \times 10^{-46} \mathrm{~kg} \mathrm{~m}^{2}}}=3.32 \times 10^{11} \mathrm{~radians}^{-1}[/latex]

Question 17.5.2: The internuclear distance (bond length) of carbon monoxide m......

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