What is the relationship between the isothermal bulk modulus $E_\theta$ and the adiabatic bulk modulus $E_\phi$? (where $\gamma = C_p/C_v$)

Consider a fluid in a container. Let the density of the fluid at the surface and at depth $H$ be $\rho_0$ and $\rho$ respectively. The bulk modulus of the fluid is $B_0$. If $\rho = \frac{\rho_0}{1 - \frac{\rho g H}{B_0}}$, find the constant $\alpha$ in the expression $\rho = \frac{\rho_0}{1 + \alpha \rho g H}$ (Assume $\frac{\rho - \rho_0}{\rho_0} \ll 1$).

$A$ solid cube of copper of edge $10 \, cm$ is subjected to a hydraulic pressure of $7 \times 10^6 \, Pa$. If the Bulk modulus of copper is $140 \, GPa$,then the contraction in its volume will be ................ $m^3$.

The bulk modulus of copper is $1.4 \times 10^{11} \ Pa$ and the coefficient of linear expansion is $1.7 \times 10^{-5} (^{\circ}C)^{-1}$. What hydrostatic pressure is necessary to prevent a copper block from expanding when its temperature is increased from $20^{\circ}C$ to $30^{\circ}C$?

If a pressure of $8 \times 10^8 \ N \ m^{-2}$ is applied to a lead block,so that its volume reduces by $20 \%$. The Bulk modulus of the lead block is:

The increase in the pressure required to decrease the volume $(\Delta V)$ of water is $6.3 \times 10^7 \text{ N/m}^2$. The percentage decrease in the volume is . . . . . . . (Bulk modulus of water = $2.1 \times 10^9 \text{ N/m}^2$.) (in $\%$)