# The pressure differential in the bulk modulus expression for liquids at isothermal condition

I was studying Pascal's Law which states(According to Wikipedia):-

Pascal's law is a principle in fluid mechanics that states that a pressure change occurring anywhere in a confined "incompressible" fluid is transmitted throughout the fluid such that the same change occurs everywhere.

This principle is stated mathematically as:

${\displaystyle \Delta P=\rho g(\Delta h)\,}$

This made me think what would the pressure be if the fluid was compressible. Now as I did study(but do not have a good command on) about elasticity of materials in the chapter before fluids, so I thought Bulk modulus must come into play here, and knowing the following about the Bulk modulus I set a task for myself to calculate the pressure at a depth in a compressible liquid.

According to Wikipedia(again):-

The bulk modulus ( ${\displaystyle K}$ or ${\displaystyle B}$) of a substance is a measure of how incompressible/resistant to compressibility that substance is. It is defined as the ratio of the "infinitesimal pressure increase" to the resulting relative decrease of the volume.

The infinitesimal pressure increase(or presure differential) part is posing a problem for me as you will see ahead in the post.

Consider a cuboid(made of water) of differential height $dh$ and cross sectional area $A(=lb)$ at a depth of $h$ below the free surface of water as shown in the diagram below:-

And as stated in Wikipedia:-

Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules.

So, consider the height $dh$ of the cuboid so small such that we can consider the pressure to be constant for that height.

If $\rho(h)$ denotes the density of the liquid at a height $h$ then the force exerted on all the faces of the cuboid is given by

$$\int_{F_0}^{F}{dF}=\int_{0}^{y}{\rho(y)Ag\cdot{dy}}$$

As, cross-sectional area for all such cuboids will be same so we can say that the pressure exerted on the faces of all such cuboids will be

$$\int _{P_0}^{P}{dP}=\int_{0}^{y}{\rho(y)g\cdot{dy}}\tag{1}$$ (where $P_0$ is the atmospheric pressure at the surface of the fluid)

Now, as we know that the bulk modulus(isothermal) is given as:-

$$B=-V\dfrac{dP}{dV}$$

Now, going according to the Wikipedia statement, what I make out of the statement "infinitesimal change in the pressure that results in the infinitesimal decrease in volume" is that that the pressure that acts to compress the element is $\displaystyle\int_{P_0}^{P}{dP}$ (summing all the pressure differences) which produces a net compression in the volume of element, given by $\Delta V$. So we have the relation

$$\int_{P_0}^{P}{dP}=-B\dfrac{\Delta V}{V}\\ \implies \int_{0}^{y}{\rho(y)g\cdot{dy}}=-B\dfrac{\Delta V }{V}$$

Now, after arriving at this erroneous equality for which I see no way to proceed further, I thought of searching for the same(hoping that it doesn't deal with something which leaves me empty handed) and figuring out where I erred. But, all I could find was the expression for barometric equation which I think, is based on statistical mechanics while what I am dealing with is continuum mechanincs.