Calculating the ratio between heat capacities of a gas

[DoubleHelix]

Hi, my task is to calculate the ratio (gamma) between the specific heats of a gas (Cp and Cv). The only information I have is a table of data for the pressures and volumes of the gas at different temperatures. I don't know if its monatomic, diatomic etc. (it's a later task to determine this).

My attempt so far. I've been researching this for hours and I've discovered that the ratio I need to find is gamma = Cp / Cv. I've learned that Cp = Cv + R where R is the gas constant and that Cv = 3R/2 for a monatomic gas, 5R/2 for a diatomic gas and that Cp = 5R/2 for a monatomic gas, 7R/2 for a diatomic gas but have been unable to find where these numbers come from! I also found a useful formula that
gamma = 1 + (R/Cv) so I only really need to find either Cv or Cp using my data to solve this question.

Thanks for any help.

 

[Gunthi]

What kind of transformation is the gas doing?

If it is diabatic, then you can find $$\gamma$$ from $$PV^{\gamma}=K$$, and then compare that value with the theoretical values of $$\frac{C_p}{C_v}$$ and find out what kind of molecule the gas is made of.

Here's an example for a monoatomic gas: It has 3 translational degrees of freedom, no rotation and no vibration, so $$N=3$$, and by the energy equipartition theorem you have a total energy of
$$U=n(3+0+0)\frac{1}{2}RT$$
where n is the mole number.

In order to calculate $$C_v$$ you can prove that
$$C_v=\frac{1}{n}(\frac{dU}{dT})_{v=const}$$
and because
$$C_p=C_v+R$$ you know the theoretical $$\gamma$$.

 

[DoubleHelix]

After reading your reply I read through the question again and noticed that it mentions the the gas is undergoing adiabatic expansion in a later part of the question, I never read this far into the question before so I didn't realize. Would the relation you mentioned hold true for an adiabatic expansion? I'm going to do more research with this new found information!

Thanks a lot!

 

[Gunthi]

It holds for any adiabatic process,in other words, any process in which there is no heat exchange between the system and the surrounding universe.

 

[DoubleHelix]

What kind of transformation is the gas doing?

If it is diabatic, then you can find $$\gamma$$ from $$PV^{\gamma}=K$$, and then compare that value with the theoretical values of $$\frac{C_p}{C_v}$$ and find out what kind of molecule the gas is made of.

Here's an example for a monoatomic gas: It has 3 translational degrees of freedom, no rotation and no vibration, so $$N=3$$, and by the energy equipartition theorem you have a total energy of
$$U=n(3+0+0)\frac{1}{2}RT$$
where n is the mole number.

In order to calculate $$C_v$$ you can prove that
$$C_v=\frac{1}{n}(\frac{dU}{dT})_{v=const}$$
and because
$$C_p=C_v+R$$ you know the theoretical $$\gamma$$.

OK so I had a go at using the method you told me and, I did:
$$\frac{C_p}{C_v}$$ = $$\frac{C_v + R}{C_v}$$
= $$\frac{\frac{1}{n}(\frac{dU}{dT}) + R}{\frac{1}{n}(\frac{dU}{dT})}$$

And assumed (for now) that it is a monatomic gas to calculate a set of U values which I plotted against the T values and drew a trendline through these points to get the gradient. Now, I (wrongly) canceled the top and bottom n to get

$$\frac{C_p}{C_v}$$ = $$\frac{(\frac{dU}{dT}) + R}{(\frac{dU}{dT})}$$

and substituted the values into get 1.66 which relates to H or He. However, as I stated I wrongly canceled the ns I think, or am I allowed the cancel the ns? This means n must equal 1 BUT it's a later part of the question to calculate n which is why I'm panicing a little! Also I assumed monatomic gas, is there no way to do this for a general case because it's also a later part of the question to determine if its monatomic/diatomic etc. lol. I have all the answers but in the wrong order!
Thanks a lot anyway, you've been a great help. Do you have any other ideas?