# Calculating the ratio between heat capacities of a gas

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 dont 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.

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 $$\gamma$$.

• gracy
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 realise. 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!

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

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) cancelled the top and bottom n to get

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

and substituted the values in to get 1.66 which relates to H or He. However, as I stated I wrongly cancelled 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?