Solution

Problem

$5.5.2.$ To measure the true volume of a granular material, it is placed in a cylinder that is hermetically sealed by a piston. Then, the air pressures $P_1$ and $P_2$ are measured at the same temperature for two positions of the piston, when the total volume of air and material is $V_1$ and $V_2$. What is the volume of the material according to these data?

Solution

Let $V_0$ be the true (intrinsic) volume of the solid particles of the granular material.
The air in the cylinder occupies the space above the material as well as the voids between its particles. Therefore, the actual volume occupied by the air is the total geometric volume under the piston minus the volume of the material itself:
$$V_{air} = V - V_0$$

Since the cylinder is hermetically sealed (the amount of air $\nu = \text{const}$) and the temperature remains the same ($T = \text{const}$), we can apply Boyle's Law for the two states of the air:

$$P_1(V_1 - V_0) = P_2(V_2 - V_0)$$

Let's solve this equation for $V_0$:

$$P_1V_1 - P_1V_0 = P_2V_2 - P_2V_0$$

$$V_0(P_2 - P_1) = P_2V_2 - P_1V_1$$

$$V_0 = \frac{P_1V_1 - P_2V_2}{P_1 - P_2}$$