Find molar mass and number of degrees of freedom from heat capacities.

[Junabes]

Hey having a bit of trouble with this one. I really don't know how to start. Think I just simply don't have the right equations or I'm using the wrong units. Hopefully you guys can help me.

Homework Statement

Find the molar mass and the number of degrees of freedom of molecules in a gas with known heat capacities: Cv = 650 J/(KgK) and Cp=910 J/(KgK).

Homework Equations

The equations that I can find in my book that I've been trying to figure out are :

1.Cv = (nRP)/2 where R = Gas constant and P = number of degrees of freedom. n= number of moles

2.Cp = nR(1+P/2)

3.Cp = Cv + R

The Attempt at a Solution

Well when I first started off I simply applied the Cv formula to the value of Cv and solve for P, the number of degrees of freedom. However when I did this, i got an answer of ~156 which I can immediately tell is not correct.

I am led to believe that I'm not using the correct units for Cv and Cp as they don't solve the last equation(3) correctly.

 

[Junabes]

Ok I just found the degrees of freedom.

Cp/Cv=1 + 2/P
7/5=1+2/P
P = 5 degrees of freedom.

Even though I'm still sure I have to change the units to find the Molar mass. I only found the degrees of freedom due to Cp/Cv is a ratio so units did not matter.

I have to somehow change the Cv and Cp units from J/(K*Kg) to J/(K*mol). My reasoning for this is that the gas constant R is J/(K*mol).

 

[Junabes]

I think I might be correct here: If anyone can confirm? Turns out I didn't really need to change any units just yet.

Cv = nRp/2
n=31.28moles/kg

Molecular mass = Mass/n = 1kg/31.28moles = 31.96g/mol.

I assume this would be Oxygen molecules.

 

[projjal]

Everything's OK.

 

[HalfLostAndSearching]

First, let's convert the given heat capacities from J/(KgK) to J/(molK) by dividing by the molar mass of the gas. This will give us:
Cv = 650 J/(KgK) = 650/(molar mass) J/(molK)
Cp = 910 J/(KgK) = 910/(molar mass) J/(molK)

Next, let's rearrange equation (1) to solve for the molar mass:
molar mass = (2Cv)/(nRP)

Substituting in the values for Cv and P from the given information, we get:
molar mass = (2*650)/(1*8.314*156) = 0.996 Kg/mol

Now, let's use equation (3) to solve for the number of degrees of freedom:
Cp = Cv + R
910/(molar mass) = 650/(molar mass) + 8.314
Solving for molar mass, we get:
molar mass = 0.996 Kg/mol
Substituting this back into the equation, we get:
P = (910-650)/8.314 = 31.2 degrees of freedom

So, the molar mass of the gas is 0.996 Kg/mol and the number of degrees of freedom is 31.2.