Determining temperature of a diatomic gas

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To determine the temperature of a diatomic gas with 3.0 moles and an internal energy of 10 kJ, the average kinetic energy formula (5/2) kB T was initially applied. However, the calculation yielded 160 K instead of the expected 115 K, prompting a review of the degrees of freedom for diatomic molecules. It was clarified that diatomic gases have three translational, two rotational, and two vibrational degrees of freedom, leading to a total of seven degrees of freedom. Each degree contributes (1/2)kT, which aligns with the book's answer. The initial misunderstanding stemmed from incorrectly applying the degrees of freedom in the energy equation.
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[SOLVED] Determining temperature of a diatomic gas

1. Assume 3.0 moles of a diatomic gas has an internal energy of 10kJ. Determine the temperature of the gas after it has reached equilibrium (assuming that molecules rotate and vibrate at that tmeperature).


2.Boltzmann's constant and # of moles 6.022x10^23



3.The average kinetic energy per diatomic gas molecule is (5/2) kB T where kB is Boltzmann's constant.
The number of molecules in 3 moles is 3 * 6.022 x 10^23, so I just solved for T:

10000 J = (5/2)* (1.3807 x 10^-23 J/K) * 3 * (6.022 x 10^23) * T


Which, I get 160k, but the answer is suppose to be 115k, which I have no idea what I did wrong, maybe the wrong constant perhaps? No clue. Thanks for any help given :D
 
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Everything looks fine. I just whipped out my calculator and got the same answer. You *might* be missing something about the internal energy, i.e. maybe you can't equate U = 5/2kT, but off the top of my head it looks fine.

It's possible that the book is wrong. Wouldn't be the first time.
 
Awesome, that's what I thought it could be a book error. I will go and ask my teacher about this whenever its possible, thanks for your help :+)
 
Hi Deoxygenation,

Deoxygenation said:
1. Assume 3.0 moles of a diatomic gas has an internal energy of 10kJ. Determine the temperature of the gas after it has reached equilibrium (assuming that molecules rotate and vibrate at that tmeperature).


2.Boltzmann's constant and # of moles 6.022x10^23



3.The average kinetic energy per diatomic gas molecule is (5/2) kB T where kB is Boltzmann's constant.
The number of molecules in 3 moles is 3 * 6.022 x 10^23, so I just solved for T:

10000 J = (5/2)* (1.3807 x 10^-23 J/K) * 3 * (6.022 x 10^23) * T


Which, I get 160k, but the answer is suppose to be 115k, which I have no idea what I did wrong, maybe the wrong constant perhaps? No clue. Thanks for any help given :D

I think the factor of (5/2) is incorrect here. These diatomic molecules are vibrating and rotating, so the degrees of freedome are:

three from translation (x,y,z)
two from rotation (the two axes perpendicular to the line joining the atoms)
two from vibration (kinetic and potential energy)

Each degree of freedom will give (1/2)kT of energy per molecule, and so I think the answer in your book is correct.
 
Oooo, wow, good thing I came back and check out the question. Ok, that's what must of been wrong. Thank you for finding the error :+)
 
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