Question

Why is there a discrepancy in the force constants of `"HBr"` vs. `"HCl"`?

We know that:

`tildeomega_0 = tildeomega_e - 2tildeomega_e chi_e`

where `tildeomega_0` is the forbidden frequency that is in the middle of the gap for `upsilon = 0 -> 1` in the infrared spectrum, `tildeomega_e` is the equilibrium fundamental frequency, and `tildeomega_e chi_e` is the anharmonicity constant.

NIST states:

`tildeomega_e ("HBr") = "2648.97 cm"^(-1)`
`tildeomega_e chi_e("HBr") = "45.2175 cm"^(-1)`
`tildeomega_e ("HCl") = "2990.95 cm"^(-1)`
`tildeomega_e chi_e("HCl") = "52.8186 cm"^(-1)`

Therefore, I get `tildeomega_0 = "2558.5 cm"^(-1)` for `"HBr"` and `"2885.3 cm"^(-1)` for `"HCl"`.

From a casual google search, I find that the force constant is given as `k = color(green)"410 N/m"` for `"HBr"` and `k = color(purple)"480 N/m"` for `"HCl"`. Here's the problem...

`k = mu omega^2`

where `mu = (m_1m_2)/(m_1 + m_2) cdot ("1 kg")/"1000 g" cdot "1 mol"/(6.0221413 xx 10^(23) "kg"` is the reduced mass (with `m_i` in `"g/mol"`), and `omega` is the angular frequency in `"rad/s"`. My concern?

Is it `tildeomega_e = omega/(2pic)`, or `tildeomega_0 = omega/(2pic)`?

`k("HBr") = mu(2pic tildeomega_e)^2`

`= 0.9951 cdot 1/1000 cdot 1/(6.0221413 xx 10^(23)) "kg" xx (2pi cdot 2.998 xx 10^(10) "cm/s" cdot "2648.97 cm"^(-1))^2`

`=` `color(green)"411.4 N/m"`, in agreement with HyperPhysics. But my gut says this is not right!

`k("HCl") = mu(2pic tildeomega_e)^2`

`= 0.9796 cdot 1/1000 cdot 1/(6.0221413 xx 10^(23)) "kg" xx (2pi cdot 2.998 xx 10^(10) "cm/s" cdot "2990.95 cm"^(-1))^2`

`=` `color(red)"516.3 N/m"`, not in agreement with HyperPhysics.

However, if I use `tildeomega_0`...

`k("HCl") stackrel(?" ")(=) mu(2pic tildeomega_0)^2`

`= 0.9796 cdot 1/1000 cdot 1/(6.0221413 xx 10^(23)) "kg" xx (2pi cdot 2.998 xx 10^(10) "cm/s" cdot "2885.3 cm"^(-1))^2`

`=` `color(purple)"480.5 N/m"`, easily in agreement with HyperPhysics. But this I think is actually correct.

But I just used two different definitions of `omega` to get there. Contradiction much?

Answer 1

After some experimentation, what it looks like to me is that

`tildeomega_0 = omega/(2pic)`.

HyperPhysics seems to give `nu = omega/(2pi)` for the table of frequencies vs. force constants, and it certainly is the case for `"HBr"`:

![)

From NIST for `"HBr"`,

![)

`tildeomega_e = "2648.975 cm"^(-1)`
`tildeomega_e chi_e = "45.2175 cm"^(-1)`

so as we said,

`tildeomega_0 = tildeomega_e - 2tildeomega_e chi_e`

`= 2648.975 - 2 cdot 45.2175` `"cm"^(-1) = "2558.2 cm"^(-1)`

If we say that `tildeomega_0 = omega/(2pi c) = nu/c`, then

`nu = tildeomega_0 cdot c`

`= ("2558.2 cm"^(-1))(2.998 xx 10^10 "cm/s")`

`= ul(7.67 xx 10^(13) "s"^(-1))`

And on HyperPhysics, their table has a frequency listed of `nu = ul(7.68 xx 10^13 "s"^(-1))`, and definitely agrees with what we just got.

(Had we assumed `nu = tildeomega_e cdot c`, we would get `7.94 xx 10^13 "s"^(-1)` instead.)

So now we're in agreement that we're looking at `bb(tildeomega_0 = omega/(2pi c))`. Let's just use it to get `omega`:

`omega = 2pi nu`

`= 2pi` `"rad/rev"cdot 7.68 xx 10^13 "s"^(-1)`

`= 4.83 xx 10^14 "rad/s"`

Then for the force constant, I get:

`color(blue)(k("HBr")) = mu omega^2`

`= overbrace([(1.007825*78.9183371)/(1.007825 + 78.9183371) "g"/"mol" cdot "1 kg"/"1000 g" cdot ("1 mol")/(6.0221413 xx 10^23)])^(mu) cdot (4.83 xx 10^14 "rad/s")^2`

`=` `color(blue)("385 N/m")`

and not `color(red)"411 N/m"`.

That seems to do it. Upon checking the rest of them, the following force constants are incorrect:

• `"HBr"` - Should be `"385 N/m"`, not `"410 N/m"`.
• `"HI"` - Should be `"293 N/m"`, not `"320 N/m"`.

These look good:

• `"HF"` - I got `"966 N/m"` compared to `"970 N/m"`.
• `"HCl"` - I see `"481 N/m"` compared to `"480 N/m"`.
• `"CO"` - I got `"1857 N/m"` compared to `"1860 N/m"`.
• `"NO"` - I got `"1551 N/m"` compared to `"1530 N/m"`.

So it's just `"HBr"` and `"HI"` that were using `tildeomega_e` instead of `tildeomega_0`.