# [PoM] Rotational and vibrational heat capacity

• BRN
In summary, the rotational constant and energy levels for the 1H79Br and 2H79Br isotopomers are the same.
BRN
Hi guis, i need your help...

1. Homework Statement

Evaluate the rotational and vibrational contributions to the heat capacity of a gas of DBr (D=deuterium, Br=mixture at 50% of 79Br and 81Br) at 380 K temperature, knowing that the bond distance is 1.41 Å and the vibration frequency of 1H79Br is ##\nu_0=2650cm^{-1}##

## The Attempt at a Solution

##R_M##=1.41 Å=##1.41*10^{-10}[m]##
## \nu_0=2650[cm^{-1}]= \nu_0=265000[m^{-1}]=7.9235*10^{13}[Hz] ##

Two isotopes have the same binding distance with inertia momentum:
$$I_1= \mu_1R_M^2= \frac{79}{80} \frac{10^{-3}}{N_A}R_M^2=3.2598*10^{-47}[Kgm^2]$$
$$I_2= \mu_2R_M^2= \frac{81}{82} \frac{10^{-3}}{N_A}R_M^2=3.2608*10^{-47}[Kgm^2]$$
$$I_{tot}=I_1+I_2=6.5206*10^{-47}[Kgm^2]$$

The characteristic rotational temperature is:
$$\Theta_{rot}= \frac{ \hbar^2}{2I_{tot}k_B}=6.1760[K]$$
I'm in ##T >> \Theta_{rot}## case, then:
$$C_{v,rot}=k_B=1.3806-10^{-23}[J/K]$$
and
$$C_{v,vib}= \frac{k_B( \beta \hbar \omega)^2e^{- \beta \hbar \omega}}{(1-e^{- \beta \hbar \omega})^2}$$
with
## \beta= \frac{1}{k_BT}## and ## \omega=2 \pi \nu_0=4.9784*10^{14}[rad/s] ##
$$\Rightarrow C_{v,vib}=6.2365*10^{-26}[J/K]$$

##C_{v,vib}## is wrong, why?

SOLUTIONS:##C_{v,rot}=k_B=1.3806-10^{-23}[J/K]; C_{v,vib}=5.597*10^{-25}[J/K]##

Thanks at all!

Last edited:
Have you taken into account that the vibrational frequencies depend on the reduced masses? Note that the given frequency is for 1H79Br, not D79Br.

Hi TSny and Happy New Year!

TSny said:
Note that the given frequency is for 1H79Br, not D79Br.
Yes, I know... but I don't know how to convert that frequency for D79Br.

The only relationship I know that involves frequency and reduced mass is this:

##\omega=\sqrt{\frac{k}{\mu}}=2\pi\nu##

Happy New Year!
BRN said:
##\omega=\sqrt{\frac{k}{\mu}}=2\pi\nu##
Use this equation to express the ratio of two frequencies in terms of the ratios of the masses.

BRN
OK, then:
for 1H79Br ##\omega=\sqrt{\frac{k}{\mu_1}}=2\pi\nu_0##
for 2H79Br ##\omega=\sqrt{\frac{k}{\mu_2}}=2\pi\nu_2##
and making the ratio, I get:

##\nu_2=\sqrt{\frac{\mu_1}{\mu_2}}\nu_0=5.6376*10^{13}[Hz]##

Now, ##\omega=2\pi\nu_2=3.5422*10^{14}[rad/s]## and ##C_{v,vib}=5.6708*10^{-25}[J/K]##

But the DBr gas is composed by 50% of 79Br and 50% of 81Br. I repeated as above for 2H81Br, getting ##C_{v,vib}=5.6799*10^{-25}[J/K]##

I think serves the average of the two results. it's correct?

Yes, I think that's the correct method. The accuracy of your answer will depend, of course, on the accuracy of the numbers you use for h, c, kB, and the atomic masses.

BRN
Thanks a lot! As usual you've been patient, timely and accurate.

For the rotation, you also calculated the moments of inertia for the 1H isotopomers instead of D. For some reason you added these together to give Itot, which is wrong. However in this case it didn't matter, as you say, T >> Θrot so Crot = kB.

Oh yes! I know! I forgot to correct. Sorry...

The exact moment of inertia are:

##I_1= \mu_1R_M^2= \frac{158}{81} \frac{10^{-3}}{N_A}R_M^2=6.4396*10^{-47}[Kgm^2]##
##I_2= \mu_2R_M^2= \frac{162}{83} \frac{10^{-3}}{N_A}R_M^2=6.4396*10^{-47}[Kgm^2]##

then, ## I_{tot}=1.2883*10^{-46}[kgm^2]##

Thanks!

Why do you add them together? That's meaningless. Do (if it was necessary) what you did for vibration - calculate C for both isotopomers, and take the average.

Hi mjc123,
I add them together because I think to particles system case where the inertia moments are added. It's wrong?

I don't know what "particles system case" you are referring to, but here you are not looking for a total moment of inertia of a system of particles (in which case just adding two would be inadequate!) but for the molecular moment of inertia. Assuming your calculations are correct, I(D79Br) = 6.4396 x 10-47 kg m2. (Why are they both the same? I get 6.4419 and 6.4458 for the two isotopomers.) From that you work out the rotational constant and energy levels, and hence Cv, for that species. Do the same for D81Br. Take an average for Cv of the mixture.

As T >> Θrot so Crot = kB, it doesn't matter in this case, but it's as well to get the principles right.

BRN
mjc123 said:
(Why are they both the same? I get 6.4419 and 6.4458 for the two isotopomers.)
Ops! I'm sorry! I made a copy / paste and did not correct the second value.

Ok, then I consider wrong the system of particles approach and I do the average of the two values.

Thanks!

## 1. What is rotational and vibrational heat capacity?

Rotational and vibrational heat capacity refers to the amount of heat that is needed to raise the temperature of a substance by one degree while keeping its rotational and vibrational energy constant.

## 2. How is rotational and vibrational heat capacity different from specific heat capacity?

Specific heat capacity is the amount of heat needed to raise the temperature of a substance by one degree while keeping its internal energy constant. Rotational and vibrational heat capacity takes into account the specific rotations and vibrations of molecules in a substance.

## 3. What factors influence the rotational and vibrational heat capacity of a substance?

The rotational and vibrational heat capacity of a substance is influenced by its molecular structure, molecular weight, and temperature. Substances with more complex molecular structures and higher molecular weights typically have higher rotational and vibrational heat capacities. Additionally, as temperature increases, so does the rotational and vibrational energy of molecules, leading to an increase in heat capacity.

## 4. How do scientists measure rotational and vibrational heat capacity?

Scientists use techniques such as calorimetry, which involves measuring the heat gained or lost by a substance during a chemical or physical process, to determine the rotational and vibrational heat capacity of a substance. They may also use spectroscopy, which involves studying the absorption and emission of light by a substance, to measure the vibrational and rotational energy levels of molecules.

## 5. What are some real-world applications of understanding rotational and vibrational heat capacity?

Understanding rotational and vibrational heat capacity is important in fields such as thermodynamics, material science, and chemical engineering. It can help in the design and optimization of industrial processes, as well as in the development of new materials with specific thermal properties. It also plays a key role in understanding the behavior of gases at different temperatures and pressures.

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