Dissociation percentages of N2O4 at different pressures using Gibbs' free energy


> 
> \begin{align}
> \ce{N2O4(g)&<=>2NO2(g)} & 
> \Delta\_\mathrm{r}G^\circ &= \pu{4.7 kJ mol^-1}
> \end{align}
> 
> 
> Knowing that the standard values for pressure and temperature are $\pu{1 bar}$ and $\pu{298 K}$ respectively, find: 
> 
> 
> a) the dissociation grade (noted as $\alpha$) of $\ce{N2O4}$ at equilibrium, in standard conditions
> 
> 
> b) the dissociation grade of $\ce{N2O4}$ at equilibrium and at $\pu{298 K}$ / $\pu{10 bar}$
> 
> 
> 


First, we find $K$ from $\Delta G$


Then we write the partial pressures as the molar fraction of the compound times the total pressure of the system (at equilibrium)


After that, we write the relation for $K\_p$ using partial pressures


and I'm stuck...


\begin{align}
 \Delta\_\mathrm{r}G^\circ &= -RT \ln K &
 \Rightarrow K&=0.15 \mathrm{~(unitless)} 
 \\
 P\_{\ce{N2O4}} &= \frac{1-\alpha}{1+\alpha}\cdot P & 
 P\_{\ce{NO2}} &= \frac{2\alpha}{1+\alpha}\cdot P & 
 K\_p = \frac{(P\_{\ce{NO2}})^2}{P\_{\ce{N2O4}}} 
 \\
 \Rightarrow K\_p &= \frac{4\alpha^2}{1-\alpha^2}\cdot P 
\\
\end{align}


My question is: how can I convert between $K$ (which is unitless) and $K\_p$ (which is not) so that I can find the dissociation grade at $\pu{1 bar}$ and $\pu{10 bar}$?


