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castrodisastro
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Homework Statement
Two copper cylinders, immersed in a water tank at 30.3 °C , contain helium and nitrogen, respectively. The helium-filled cylinder has a volume twice as large as the nitrogen-filled cylinder.
a) Calculate the average translational kinetic energy of a helium molecule and the average translational kinetic energy of a nitrogen molecule.
b) Determine the molar specific heat at constant volume (CV) and at constant pressure (Cp) for the two gases.
c) Find γ for the two gases.
Given/Known information
Cp = Cv+R
CV = [itex]\frac{3}{2}[/itex]R = 12.5 J/(mol*K) monatomic gas
Cp = [itex]\frac{5}{2}[/itex]R = 20.8 J/(mol*K) monatomic gas
CV = 20.7 J/(mol*K) Nitrogen gas
Cp = 29.1 J/(mol*K) Nitrogen gas
T = 30.3 °C = 303.3 K
Homework Equations
Kave = [itex]\frac{3}{2}[/itex]kBT
γ = Cp/CV
The Attempt at a Solution
Part a)
Helium is a monatomic molecule.
The average energy per degree of freedom is given by [itex]\frac{1}{2}[/itex]kB for each gas molecule. I was asked for the average translational kinetic energy, since a monatomic molecule has 3 translational degrees of freedom, and Nitrogen has 3 translational degrees of freedom and 2 rotational degrees, then the average kinetic energy is equal to,
Helium
Kave=[itex]\frac{3}{2}[/itex]kBT
Kave=[itex]\frac{3}{2}[/itex]([itex]1.38\ \times\ 10^{-23}\ J\ K^{-1}[/itex])(303.3 K)
Kave=[itex]6.278\ \times\ 10^{-21}\ J[/itex] ←INCORRECT
Nitrogen
Kave=[itex]\frac{3}{2}[/itex]([itex]1.38\ \times\ 10^{-23}\ J\ K^{-1}[/itex])(303.3 K)
Kave=[itex]6.278\ \times\ 10^{-21}\ J[/itex] ←INCORRECT
I don't know if the two gases being inside of copper cylinders affects the calculations somehow since there is no temperature change, no reaction going on, or anything that will change the equation. I know that for a polyatomic molecule you need to consider the additional degrees of freedom available, but for both He and N2, there are 3 translational degrees of freedom, giving us the [itex]\frac{3}{2}[/itex]. I don't know why that one is wrong.
Part b)
Values were obtained from a table in the book on Molar Specific Heats at constant volume and constant pressure. Also on this table was a column for the values of γ=Cp/CV for some gases.
CV, He = 12.5 J/(mol*K) ←CORRECT
Cp, He = 20.8 J/(mol*K) ←CORRECT
CV, N2 = 20.7 J/(mol*K) ←CORRECT
Cp, N2 = 29.1 J/(mol*K) ←CORRECT
Part c)
γ is stated to be the ratio Cp/CV
γHe = Cp, He/CV, He
γHe = (20.8 J/(mol*K))/(12.5 J/(mol*K))
γHe = 1.664 ≈ 1.67 ←INCORRECT
γN2 = Cp, N2/CV, N2
γN2 = (29.1 J/(mol*K))/(20.7 J/(mol*K))
γN2 = 1.4058 ≈ 1.4 ←CORRECT
I really don't know what other value γHe could be if we are told γHe=Cp/CV. Using values that I verified are correct for part b, taking Cp/CV for He should give me the correct answer, which the table in the book also verifies that it is a value of 1.67. Even more confusing is how doing that calculation for N2 gives me the correct answer for γN2.
All help is greatly appreciated.
Thanks in advance
Please let me know if something is confusing or if something looks like a typo, this took me a little while to proofread and edit and I went back numerous times to make it easier to read so I may have missed something.