The use of intensive variables in thermodynamics Using intensive variables simplify the analysis of thermodynamics . Also equations are divided by the total mass in order to use the "specific" version of a variable . But what's the difference if it were dependent on the mass? Why do they intend to get rid of the mass in the equation and how does it make it simpler ?
 A: In statistical mechanics, the most important behavior of a system (and the starting point for any more detailed analysis) is the bulk behavior in the limit where its size is macroscopic (i.e. $N\sim 10^{23}$). Statistical mechanical predictions of macroscopic thermodynamic properties (like total energy and entropy) can be decomposed into distinct sectors that grow at different rates with respect to the system size. For example, we could expand internal energy as $U(T,V)=Nu^{(1)}(T,v)+\sum_{\alpha_i<1}N^{\alpha_i}u^{(\alpha_i)}(T,v)+\log(N)u^{\ln}(T,v)+\dots$ (higher order terms with even slower increase with $N$). 
Contributions to extensive quantities such as $U$ with fractional powers of $N$ often represent boundary effects, like energy from surface tension in a two-phase mixture. These contributions are important, because they might reveal interesting physics that is obscured by the bulk behavior. However, in order to extract information about these smaller corrections, it is important to have a thorough understanding of the leading order "background" contributions so that these terms can be subtracted. 
The intensive quantity $\frac{U}{N}$ is therefore important because it can be directly measured by taking $N\rightarrow \infty$, and subtracted in order to reveal subextensive properties of the system. This line of reasoning can also be repeated iteratively, in order to extract the physics of each sector of $N$-dependent growth.
