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

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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?

We know that:

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:

Therefore, I get

From a casual google search, I find that the force constant is given as

#k = mu omega^2#

where

Is it

#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 agreementwith 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"# ,with HyperPhysics.notin agreement

However, if I use

#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 agreementwith HyperPhysics. But this I think is actually correct.

But I just used two different definitions of

##### 1 Answer

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

#tildeomega_0 = omega/(2pic)# .

HyperPhysics seems to give

From NIST for

#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

#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

(Had we assumed

So now **we're in agreement** that we're looking at

#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