Determining temperature of a diatomic gas

In summary, the conversation discusses the calculation of the temperature of a diatomic gas after it has reached equilibrium. The internal energy of the gas is given as 10kJ for 3 moles of the gas. Using Boltzmann's constant and the number of moles, the average kinetic energy per molecule is calculated to be (5/2)kB T. However, it is determined that the factor of (5/2) is incorrect and the correct answer is 115k. This is due to the degrees of freedom being three for translation, two for rotation, and two for vibration.
  • #1
Deoxygenation
11
0
[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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  • #2
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.
 
  • #3
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 :+)
 
  • #4
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.
 
  • #5
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 :+)
 

1. How do you determine the temperature of a diatomic gas?

The temperature of a diatomic gas can be determined using the ideal gas law, which states that the pressure, volume, and temperature of an ideal gas are directly proportional. By measuring the pressure and volume of the gas and plugging those values into the ideal gas law equation, the temperature can be calculated.

2. What is the ideal gas law?

The ideal gas law is a mathematical equation that describes the relationship between the pressure, volume, and temperature of an ideal gas. It is represented by the equation PV = nRT, where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

3. How does the ideal gas law apply to diatomic gases?

The ideal gas law applies to all ideal gases, including diatomic gases. Diatomic gases are gases composed of two atoms, such as oxygen (O2) and nitrogen (N2), and behave similarly to other ideal gases at low pressures and high temperatures. This means that the ideal gas law can be used to determine the temperature of a diatomic gas.

4. Are there any limitations to using the ideal gas law to determine the temperature of a diatomic gas?

While the ideal gas law is a useful tool for determining the temperature of a diatomic gas, it does have some limitations. It assumes that the gas is at a low pressure and high temperature, and that the molecules of the gas do not interact with each other. In reality, diatomic gases may deviate from ideal behavior at high pressures or low temperatures, so the ideal gas law may not always give an accurate temperature measurement.

5. How can the temperature of a diatomic gas affect its properties?

The temperature of a diatomic gas can affect its properties in various ways. For example, an increase in temperature can cause the gas molecules to move faster, increasing the pressure and volume of the gas. It can also affect the specific heat capacity and thermal conductivity of the gas, which can impact its ability to transfer heat. Furthermore, temperature changes can also cause phase transitions in diatomic gases, such as changing from a gas to a liquid or solid state.

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