Incompatibility between ideal gas equations of state

[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.

 

[Chestermiller]

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.

 

[Portuga]

Thank u very much!