Thermometric scale:
- Consider a system characterized by two independent variables, $x$ and $y$ for example. When the values ​​of $x$ and $y$ remain constant, as long as the external environment is not modified, the system is said to be in thermal equilibrium. The temperature $\theta$ is then assigned to the system, which obviously depends on the pairs $x$ , $y$ .

- A thermometer is a system where one of the variables is kept constant (y for example), the thermometer is brought into thermal equilibrium with the system whose temperature $\theta$ is sought. The equilibrium temperature $\theta$ only depends on the variable $x$ (which can be the volume of a fluid, the pressure, the resistance of a wire,...). The relation $\theta (x)$ defines the temperature scale.

-centisimal scale (with two fixed points):

The linear centisimal scale is defined by the thermometric function: $\theta(x)=ax+b $ $;\;\;\;(a,b)$  constants.


From the equation with $v=cst$ we have the termometric scale :$$c_{T}=(v-b)p=\alpha p$$

If $p\rightarrow 0$, the equation becomes:$$pv=c_{T}$$ 

this is the ideal gas equation, so $$c_{T}=nRT=Nk_{B}T$$