How do you rewrite molarity to weight percentage or volume percentage?

Suppose I have a molar concentration $[C]$ in $mol/m^3$ in a continuous reactor, from which the differential equation is as follows:


$ \frac{\partial [C]}{\partial t} = k[A][B] + \phi C\_{,in} - \phi C\_{,out}$


Where the first term is the rate law with rate constant $k$, and with concentrations of reactants $A$ and $B$, as $A+B \longrightarrow C$.


The second and third terms are inflows and outflows of substance $C$ in a continuous reactor.


The outflow of $C$ is measured, and therefore I need to have each term correlate with each other by having the correct **dimensions.** I have the following question:


Suppose $C$ is a gaseous substance, and the measured output is in $vol\%$ (volume percentage), how would I convert $\phi C\_{,out}$ or $\phi C\_{,in}$, which is in $[Nm^3/s]$ (normal cubic meter per second), to $vol\%C$? I found the following:


$\hspace{30pt} \phi\_{C\_{,out}} = \dfrac{p\dfrac{vol\% C}{100\%}}{RT}\phi\_{out}\hspace{10pt}$ but this leads to a wrong dimension $[m^3/mol]$ for the $vol\%$




Relation of molarity and molar fraction ( = volume fraction for ideal gas approximation), using ideal gas state equation:


\begin{align}
c &= \frac{n\_1 }{ V} \tag{1}\\
pV &= nRT\tag{2}\\
c = \frac{n\_1 }{ \frac{(n1+n2)RT}{p}}&=\frac{p}{RT}\frac{n\_1}{n\_1+n\_2}\tag{3}
\end{align}


Computing volume fraction and percentage from molar concentration:


\begin{align}
\varphi&=c \cdot \frac{RT}{p}\tag{4}\\
vol\%&=\varphi \cdot 100\tag{5}
\end{align}


Dimension of $\frac{cRT}{p}$ is
$$\frac{\pu{mol m-3}\pu{J K-1 mol-1}\pu{K}}{\pu{N m-2}}= \frac{\pu{ m-3 J }}{\pu{(N m)(m-1 m-2)}}= \frac{\pu{ m-3 J }}{\pu{J m-3}}=1\tag{6}$$


Computing mass fraction and percentage from volume fraction:


\begin{align}
w\_1&=\frac{\varphi\_1 M\_1}{\varphi\_1 M\_1 + (1-\varphi\_1)M\_2}\tag{7}\\
mass\%&=w \cdot 100\tag{8}
\end{align}


Legend:


* $c$ - molar concentration $\pu{mol m-3}$
* $n\_1$ - analyte molar amount [$\pu{mol}$]
* $n\_2$ - molar amount of the mixture but the analyte[$\pu{mol}$]
* $V$ - volume of mixture [$\pu{m3}$]
* $R$ - gas constant [$\pu{8.314 J K-1 mol-1}$]
* $p$ - gas pressure [$\ce{Pa}$]
* $\varphi$ - volume fraction
* $\varphi\_1$ - the volume fraction of the analyte
* $w$ - mass fraction
* $w\_1$ - the mass fraction of the analyte
* $M\_1$ - molar mass of the analyte [$\pu{g/mol}$]
* $M\_2$ - molar mass of the major gas [$\pu{g/mol}$]  

( or the effective mean molar mass of used gas mixture, e.g. $\pu{28.8 g mol-1}$ for air)


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