Injected gas versus pressure rise

Very simple concept for some, but quite honestly I’m striving to grasp:

Premises:

• You have a $20 \text{ m}^3$ pressure vessel.
• You start filling it with gas.
• You see a rise in the gauge pressure of $1 \text{ bar}$.

I’m told that this gauge pressure rise means you have injected a total of precisely $20 \text{ m}^3$ of gas when the reader shows exactly $1 \text{ bar}$. And that, therefore, knowing the gas density (lets say for example $2 \text{ kg/m}^3$ at normal conditions, this meaning $0^\circ\text{C}$ and absolute pressure of $1,01325 \text{ bar}$) you deduce the total amount of mass injected in the pressure vessel:

$20 \text{ m}^3 \cdot 2 \text{ kg/m}^3 =40 \text{ kg}$

Doubt 1: Where may I read anything proving that direct relationship between pressure rise and tidal volume?

Doubt 2: how may the stated density apply if this happens inside the vessel, where you have relative pressure and the value is given for absolute pressure?

Many thanks in advance!

Using Ideal Gas Law:

$\frac{p_1V_1}{n_1RT_1} = \frac{p_2V_2}{n_2RT_2}$

The volume stays the same, so we cancel those out. R is obviously the same, so that goes too. If we assume the temperature stays the same throughout the process,

$\frac{p_1}{n_1} = \frac{p_2}{n_2}$ , or

$p_2 = p_1\frac{n_2}{n_1}$

Since at a constant temperature, all (ideal) gases have the same molar volume, if you add a volume of gas equal to the initial volume (add a number of moles equal to the original number), the pressure will double.

• what to do if the temperature is not constant? – Innovine Aug 2 at 12:00