# Incompatibility between ideal gas equations of state

• Portuga
In summary, to derive the equation of state for an ideal gas that relates pressure, density, and temperature, the equations PV = nRT and n = m/M were used, along with the consideration of density as n/V. Using these equations, it was found that the final equation should be P/ρ = RT, not P/ρ = RT/M. This is because ρ represents molar density, not mass density.

#### Portuga

Homework Statement
Derive the equation of state for an ideal gas that relates pressure, density, and temperature.
Relevant Equations
PV = nRT
To solve this problem I used two equations:
$$PV=nRT,$$
where ##P## is the pressure, ##V##the volume, ##R##the gas constant, ##T##for temperature and is##n##the number of moles, related to the
mass ##m## and molar mass ##M## by
$$n=\frac{m}{M}.$$
It will be also necessary consider the density ##\rho## as
$$\rho=\frac{m}{V}.$$

So,
\begin{align}
& PV=\frac{m}{M}RT\nonumber \\
\Rightarrow & \frac{P}{\frac{m}{V}}=\frac{RT}{M}\nonumber \\
\Rightarrow & \frac{P}{\rho}=\frac{RT}{M}.\nonumber
\end{align}
When I checked the answer, to my surprise I found
$$\frac{P}{\rho}=RT.$$
I am so confused because this is so simple and I have no idea about
what to do with the molar mass##M##to get the answer provided by the author.

Portuga said:
Homework Statement:: Derive the equation of state for an ideal gas that relates pressure, density, and temperature.
Relevant Equations:: PV = nRT

To solve this problem I used two equations:
$$PV=nRT,$$
where ##P## is the pressure, ##V##the volume, ##R##the gas constant, ##T##for temperature and is##n##the number of moles, related to the
mass ##m## and molar mass ##M## by
$$n=\frac{m}{M}.$$
It will be also necessary consider the density ##\rho## as
$$\rho=\frac{m}{V}.$$

So,
\begin{align}
& PV=\frac{m}{M}RT\nonumber \\
\Rightarrow & \frac{P}{\frac{m}{V}}=\frac{RT}{M}\nonumber \\
\Rightarrow & \frac{P}{\rho}=\frac{RT}{M}.\nonumber
\end{align}
When I checked the answer, to my surprise I found
$$\frac{P}{\rho}=RT.$$
I am so confused because this is so simple and I have no idea about
what to do with the molar mass##M##to get the answer provided by the author.
In that final equation, ##\rho## is the molar density n/V, not the mass density m/V.

MatinSAR and Portuga
Thank u very much!