Question

The HCl equilibrium bond length is 0.127 nm and the v = 0 to v = 1 transition is observed in the infrared at 2886 cm-1.Compute the vibrational energy of HCl in its lowest state. Compute the classical limit for the stretching of the HCl bond from its equilibrium length in this state. What percent of the equilibrium bond length is this extension

Answer 1

Vibrational Energy Of HCl in the lowest state :

`2.86 * 10^(-20) J`

Classical Limit for stretching of HCl bond from its equilibrium length :

`Q_(0) = 0.0109 nm`

Percent of equilibrium Bond Length :

8.58 %

Step-by-step explanation:

H-Cl bond Length = 0.127 nm =
`1.27 * 10^(-10)m`

Frequency from v = 0 to v = 1 is 2886
`cm^(-1)`

`\\u(Hz) = c* \\u (cm^(-1))`

`\\u(Hz) = 3 * 10^(10) * 2886`

`\\u = 8.568 * 10^(13) Hz`

Reduced mass

`\mu =(m_(1)m_(2))/(m_(1)+m_(2))`

`\mu =(1 * 35.5)/(1+35.5)`

`\mu =(1 * 35.5)/(36.5)`

but this has units in amu , to convert it in Kg divide it by

`6.022* 10^(23)` and 1000(to convert gram itno Kg)

`\mu = 1.61 * 10^(-27) Kg`

Calculation of Force constant :

`\\u =(1)/(2\Pi )\sqrt{(k)/(\mu )}`

Here,

`\\u = frequency`

k = force constant

`\mu = Reduced mass`

Put the value of frequency , reduced mass and calculate for force constant

`2(3.14)* 8.658* 10^(13) = \sqrt{(k)/(1.61* 10^(-27))}`

Solve the left hand side and square it. Then multiply it with reduced mass

k = 475.97 N/m

`\omega = 2\Pi \\u`

`\omega = 2(3.14)(8.568 * 10^(13) Hz)`

`\omega = 5.43 * 10^(14)`

Calculation of lowest energy

`E_(0) = (h\omega )/(2\Pi )`

h = planck's constant =
`6.626 * 10^(-34)`

`E_(0) = ((6.626 * 10^(-34))(5.43 * 10^(14)))/(2\Pi )`

On solving ,

`E_(0) = 2.86 * 10^(-20)J`

Calculation of Stretching of HCl bond:

Use formula :

`(1)/(4\Pi )h\omega =(1)/(2)kQ_(0)`

here Q = stretching bond length

Put , k = 475.97 N/m ,
`\omega = 5.43 * 10^(14)` and solve for Q

`Q_(0)^(2) = 1.204 * 10^(-22)`

take square root

`Q_(0) = 1.097 * 10^(-11)`

Calculation of Percentage extension:

Percentage
`=(Q_(0))/(X_(eq))* 100`

`(1.097 * 10^(-11))/(1.27 * 10^(-10))`

Percentage = 8.58 %