Respuesta :
Answer:
Vibrational Energy Of HCl in the lowest state :
[tex]2.86 \times 10^{-20} J[/tex]
Classical Limit for stretching of HCl bond from its equilibrium length :
[tex]Q_{0} = 0.0109 nm[/tex]
Percent of equilibrium Bond Length :
8.58 %
Explanation:
H-Cl bond Length = 0.127 nm =[tex] 1.27 \times 10^{-10}m[/tex]
Frequency from v = 0 to v = 1 is 2886 [tex]cm^{-1}[/tex]
[tex]\nu(Hz) = c\times \nu (cm^{-1})[/tex]
[tex]\nu(Hz) = 3 \times 10^{10} \times 2886 [/tex]
[tex]\nu = 8.568 \times 10^{13} Hz[/tex]
Reduced mass
[tex]\mu =\frac{m_{1}m_{2}}{m_{1}+m_{2}}[/tex]
[tex]\mu =\frac{1 \times 35.5}{1+35.5}[/tex]
[tex]\mu =\frac{1 \times 35.5}{36.5}[/tex]
but this has units in amu , to convert it in Kg divide it by
[tex]6.022\times 10^{23}[/tex] and 1000(to convert gram itno Kg)
[tex]\mu = 1.61 \times 10^{-27} Kg[/tex]
Calculation of Force constant :
[tex]\nu =\frac{1}{2\Pi }\sqrt{\frac{k}{\mu }}[/tex]
Here,
[tex]\nu = frequency[/tex]
k = force constant
[tex]\mu = Reduced mass[/tex]
Put the value of frequency , reduced mass and calculate for force constant
[tex]2(3.14)\times 8.658\times 10^{13} = \sqrt{\frac{k}{1.61\times 10^{-27}}}[/tex]
Solve the left hand side and square it. Then multiply it with reduced mass
k = 475.97 N/m
[tex]\omega = 2\Pi \nu[/tex]
[tex]\omega = 2(3.14)(8.568 \times 10^{13} Hz)[/tex]
[tex]\omega = 5.43 \times 10^{14}[/tex]
Calculation of lowest energy
[tex]E_{0} = \frac{h\omega }{2\Pi }[/tex]
h = planck's constant = [tex]6.626 \times 10^{-34}[/tex]
[tex]E_{0} = \frac{(6.626 \times 10^{-34})(5.43 \times 10^{14})}{2\Pi }[/tex]
On solving ,
[tex]E_{0} = 2.86 \times 10^{-20}J[/tex]
Calculation of Stretching of HCl bond:
Use formula :
[tex]\frac{1}{4\Pi }h\omega =\frac{1}{2}kQ_{0}[/tex]
here Q = stretching bond length
Put , k = 475.97 N/m ,[tex]\omega = 5.43 \times 10^{14}[/tex] and solve for Q
[tex]Q_{0}^{2} = 1.204 \times 10^{-22}[/tex]
take square root
[tex]Q_{0} = 1.097 \times 10^{-11}[/tex]
Calculation of Percentage extension:
Percentage[tex] =\frac{Q_{0}}{X_{eq}}\times 100[/tex]
[tex]\frac{1.097 \times 10^{-11}}{1.27 \times 10^{-10}}[/tex]
Percentage = 8.58 %
