Relationship between Cp, Cv and R

[PSN03]

Homework Statement

Cp and Cv are specific heats at constant pressure and constant volume respectively. It is observed that
Cp−Cv =a for hydrogen gas
C p−Cv=b for nitrogen gas
The correct relation between a and b is :

Relevant Equations

Cp-Cv=R where R is gas constant

According to me a=b cause what I have been learning is R is gas constant and hence it will be same for both. But the solution have says something else.
According to them Cp-Cv=R/M where M is the molecular mass of gas.
So is the above mentioned formula correct? Do we have to take that M term always and in all the problems?

 

[etotheipi]

Meyer's relation can be written like$$C_p - C_v = nR$$where $n$ is the number of moles of the gas. Often you might want to consider the specific heat capacities, $c_p = \frac{C_p}{M}$ and $c_v = \frac{C_v}{M}$, in which case$$c_p -c_v = \frac{nR}{M} = \frac{R}{\mu}$$where $\mu = \frac{M}{n}$ is the molar mass of the gas

(To make matters more confusing, sometimes people use the lower-case notation for molar heat capacities instead, i.e. defining $c_p = \frac{C_p}{n}$ and $c_v = \frac{C_v}{n}$, in which case Meyer's relation reads $c_p - c_v = R$)

 

[PSN03]

If that is the original problem statement, then it's a really poorly worded question. Meyer's relation can be written like$$C_p - C_v = nR$$where $n$ is the number of moles of the gas. Often you might want to consider the specific heat capacities, $c_p = \frac{C_p}{M}$ and $c_v = \frac{C_v}{M}$, in which case$$c_p -c_v = \frac{nR}{M} = \frac{R}{\mu}$$where $\mu = \frac{M}{n}$ is the molecular mass of the gas. To make matters more confusing, sometimes people use the lower-case notation for molar heat capacities, i.e. defining $c_p = \frac{C_p}{n}$ and $c_v = \frac{C_v}{n}$, in which case Meyer's relation reads$$c_p - c_v = R$$Really the question is unanswerable unless you know whether the heat capacities are specific or molar. Furthermore, are we to assume that the mass, or alternatively the number of moles, is fixed?

Thanks a lot for your help. By the way only that much information is provided.
can I conclude that Cp/n is molar heat capacity and Cp/M is specific heat capacity and Cp is just heat capacity, right?

 

[etotheipi]

Yes, by specific it means per unit mass, and the correct relation in this case will be $c_p - c_v = \frac{R}{\mu}$. More generally, it should be fairly clear from context which definition is being used, but it's something to be careful about!

 

[PSN03]

Yes, by specific it means per unit mass, and the correct relation in this case will be $c_p - c_v = \frac{R}{\mu}$. More generally, it should be fairly clear from context which definition is being used, but it's something to be careful about!

Thanks a lot for your help. I was beating my head about this problem for long but finally it's done. Good day and stay safe

 

[Chestermiller]

Meyer's relation can be written like$$C_p - C_v = nR$$where $n$ is the number of moles of the gas. Often you might want to consider the specific heat capacities, $c_p = \frac{C_p}{M}$ and $c_v = \frac{C_v}{M}$, in which case$$c_p -c_v = \frac{nR}{M} = \frac{R}{\mu}$$where $\mu = \frac{M}{n}$ is the molar mass of the gas

(To make matters more confusing, sometimes people use the lower-case notation for molar heat capacities instead, i.e. defining $c_p = \frac{C_p}{n}$ and $c_v = \frac{C_v}{n}$, in which case Meyer's relation reads $c_p - c_v = R$)

I don’t think so, if Cp and Cv are the molar heat capacities, then Cp -Cv= R (without the n).

 

[etotheipi]

I don’t think so, if Cp and Cv are the molar heat capacities, then Cp -Cv= R (without the n).

I used $C_v$ and $C_p$ for the extensive heat capacity, and $c_p$ and $c_v$ for the specific heat capacity. Then in the note at the end I mentioned that if $c_p$ and $c_v$ are instead molar heat capacities, the relation is $c_p - c_v = R$ like you say. I don't know what notation is most widely used, though!

 

[Chestermiller]

I used $C_v$ and $C_p$ for the extensive heat capacity, and $c_p$ and $c_v$ for the specific heat capacity. Then in the note at the end I mentioned that if $c_p$ and $c_v$ are instead molar heat capacities, the relation is $c_p - c_v = R$ like you say. I don't know what notation is most widely used, though!

It seemed to me that the OP in post #1 was asking about the molar heat capacities.