(a) The reduced mass of the O2 is
μ = m o m o m o + m o = 16.00 × 16.00 16.00 + 16.00 = 8 u = 8 × 1.66 × 1 0 − 27 = 1.33 × 1 0 − 26 k g \mu= \frac{m_om_o}{m_o+m_o} \\
= \frac{16.00 \times 16.00}{16.00+16.00} = 8 \;u \\
= 8 \times 1.66 \times 10^{-27} = 1.33 \times 10^{-26} \; kg μ = m o + m o m o m o = 16.00 + 16.00 16.00 × 16.00 = 8 u = 8 × 1.66 × 1 0 − 27 = 1.33 × 1 0 − 26 k g
The momentum inertia is
I = μ r 2 = 1.33 × 1 0 26 × ( 1.20 × 1 0 − 10 ) 2 = 1.91 × 1 0 − 46 k g × m 2 I = \mu r^2 \\
= 1.33 \times 10^{26} \times (1.20 \times 10^{-10})^2 \\
= 1.91 \times 10^{-46} \; kg \times m^2 I = μ r 2 = 1.33 × 1 0 26 × ( 1.20 × 1 0 − 10 ) 2 = 1.91 × 1 0 − 46 k g × m 2
The rotational energies are
E r o t = h 2 2 l J ( J + 1 ) = ( 6.626 × 1 0 − 34 J s / 2 π ) 2 2 × 1.91 × 1 0 − 46 k g m 2 J ( J + 1 ) = ( 2.91 × 1 0 − 23 J ) J ( J + 1 ) E_{rot}= \frac{h^2}{2l}J(J+1) \\
= \frac{(6.626 \times 10^{-34 \;J \;s/2 \pi )^2}}{2 \times 1.91 \times 10^{-46} \;kg \; m^2}J(J+1) \\
= (2.91 \times 10^{-23}\;J) J(J+1) E ro t = 2 l h 2 J ( J + 1 ) = 2 × 1.91 × 1 0 − 46 k g m 2 ( 6.626 × 1 0 − 34 J s /2 π ) 2 J ( J + 1 ) = ( 2.91 × 1 0 − 23 J ) J ( J + 1 )
For J=0,1,2
E r o t = 0 , 3.63 × 1 0 − 4 e V , 1.09 × 1 0 − 3 e V E_{rot}= 0, 3.63 \times 10^{-4}\;eV, 1.09 \times 10^{-3}\;eV E ro t = 0 , 3.63 × 1 0 − 4 e V , 1.09 × 1 0 − 3 e V
(b) The vibrational energies are given by
E v i b = ( v + 1 2 ) h k μ = ( v + 1 2 ) ( 6.626 × 1 0 − 34 J s 2 π ) 1177 N / m 8 ( 1.66 × 1 0 − 27 k g ) = ( v + 1 2 ) ( 3.14 × 1 0 − 20 J ) ( 1 e V 1.602 × 1 0 − 19 J ) = ( v + 1 2 ) ( 0.196 e V ) E_{vib}=(v+\frac{1}{2})h \sqrt{\frac{k}{\mu}} \\
=(v+ \frac{1}{2})(\frac{6.626 \times 10^{-34}J \;s}{2 \pi}) \sqrt{ \frac{1177 \;N/m}{8(1.66 \times 10^{-27} \;kg)} } \\
= (v+ \frac{1}{2})(3.14 \times 10^{-20}\;J)( \frac{1 \;eV}{1.602 \times 10^{-19} \;J} ) \\
= (v+ \frac{1}{2})(0.196 \;eV) E v ib = ( v + 2 1 ) h μ k = ( v + 2 1 ) ( 2 π 6.626 × 1 0 − 34 J s ) 8 ( 1.66 × 1 0 − 27 k g ) 1177 N / m = ( v + 2 1 ) ( 3.14 × 1 0 − 20 J ) ( 1.602 × 1 0 − 19 J 1 e V ) = ( v + 2 1 ) ( 0.196 e V )
For v=0,1,2
E v i b = 0.098 e V , 0.294 e V , 0.490 e V E_{vib}=0.098 \;eV, 0.294 \;eV, 0.490 \;eV E v ib = 0.098 e V , 0.294 e V , 0.490 e V
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