What Is the Difference Between PV = nRT and Pv = RT?

[askor]

There are two formula as below:

1. PV = nRT
2. Pv = RT (v is specific volume)

What is the difference between them?

 

[Tallus Bryne]

Both equations (if all the symbols represent the same quantity) imply that V = v*n, is this true?

One equation has a proportionality constant (R) written on a per mol basis, and another on a per unit mass basis, and so they will have different values if both equations are to be consistent with the ideal gas model.

 

[Chestermiller]

There are two formula as below:

1. PV = nRT
2. Pv = RT (v is specific volume)

What is the difference between them?

If v is the molar volume (volume per mole), then v = V/n. The gas constant R in both equations is the same.

 

[askor]

What about P = ρRT?

Isn't above equation shows that $P = \frac{1}{v}RT$ or Pv = RT?

So Why v = V/n, not v = 1/ρ?

If PV = nRT, Pv = RT, and P = ρRT (PV = mRT). What is the difference?

 

[Chestermiller]

What about P = ρRT?

Isn't above equation shows that $P = \frac{1}{v}RT$ or Pv = RT?

This is correct if $\rho$ is the molar density.

So Why v = V/n, not v = 1/ρ?

They're both the same.

If PV = nRT, Pv = RT, and P = ρRT (PV = mRT). What is the difference?

There is no difference. They all say the same thing (and give the same results).

 

[askor]

You said something about molar volume and molar density which is studied in Chemistry.

What I'm asking about is studied in fluid mechanics. In my fluid mechanics textbook, ρ in P = ρRT is ρ = m/V, not ρ = n/V.

So, which one is correct, ρ = m/V or ρ = n/V?

 

[jack action]

I'm surprised by the answers you got.

In both of your equations, $R$ is not the same. For clarity purposes, the equations are often written the following ways:
$$PV = n\bar{R}T$$
$$PV = mRT$$
where $\bar{R}$ is the universal gas constant and is equal to 8.3144598 J/mol/K, for any gas.

$R$ is the specific gas constant, and there is a value for each gas. The relationship between the two is $R = \frac{\bar{R}}{M}$, where $M$ is the molar mass of the gas (or $M = \frac{m}{n}$).

So:
$$PV = n\bar{R}T$$
$$PV = \frac{m}{m}n\bar{R}T$$
$$PV = m\frac{\bar{R}}{M}T$$
$$PV = mRT$$
Then, knowing that density $\rho = \frac{m}{V}$ and that specific volume $v = \frac{V}{m}$, then $P = \rho RT$ or $Pv = RT$.

Gas constant on Wikipedia

 

[Chestermiller]

We have a symbology issue here.

In some developments v is used to represent the volume per unit mass, and in others v is used to represent the volume per mole
In some developments V is used to represent the total volume, and in others V is used to represent the molar volume.
In some developments, $\rho$ is used to represent the mass density, and in others, $\rho$ is used to represent the molar density.
In Action Jack's experience, R is the gas constant for the specific gas, and $\bar{R}$ is the universal gas constant; in my experience, R is used for the universal gas constant, and it is also sometimes used for the gas constant for the specific gas; I have never seen $\bar{R}$ used for the universal gas constant.

To summarize, these symbols are used differently in different developments, and one needs to know the specific way it is being used (from the context) in the development one is reading. There is obviously not one correct way.