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Hi guis, i need your help...

Evaluate the rotational and vibrational contributions to the heat capacity of a gas of DBr (D=deuterium, Br=mixture at 50% of

##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!

1. Homework Statement1. Homework Statement

Evaluate the rotational and vibrational contributions to the heat capacity of a gas of DBr (D=deuterium, Br=mixture at 50% of

^{79}Br and^{81}Br) at 380 K temperature, knowing that the bond distance is 1.41 Å and the vibration frequency of^{1}H^{79}Br 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!

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