Question

Compute the fractional change in the volume of a glass slab, when subjected to a hydraulic pressure of $10$$atm$.

Answer 1

Hint: As the glass slab is subjected to a pressure, the glass will be in compression mode. As a result, the volume of the glass slab will reduce due to compression. The fractional change in the volume is equal to $\dfrac{{\Delta V}}{V}$. Think of a quantity which involves this fractional change and then use the given data to find the fractional change in the volume of the glass slab.

Complete step by step answer:
Given the hydraulic pressure on the slab is $10$$atm$, it can be written in pascals as $1.013 \times {10^6}$$Pa$.
Now, the bulk modulus of the glass slab is given by $B = - V\dfrac{{\Delta P}}{{\Delta V}}$.
From the above equation, the fractional change in the volume of the glass slab can be written as $\dfrac{{\Delta V}}{V} = - \dfrac{{\Delta P}}{B}$. The value of the bulk modulus of glass is $3.7 \times {10^{10}}$$Pa$.
Therefore,
$
  \dfrac{{\Delta V}}{V} = - \dfrac{{1.013 \times {{10}^6}}}{{3.7 \times {{10}^{10}}}} \\
  \dfrac{{\Delta V}}{V} = - 0.2738 \times {10^{ - 4}} \\
 $
Hence, the fractional change in the volume of a glass slab, when subjected to a hydraulic pressure of $10$$atm$ is $\dfrac{{\Delta V}}{V} = - 0.2738 \times {10^{ - 4}}$.

Additional Information:
The bulk modulus is the measurement of resistance offered by a substance to compression. It is the ratio of change in pressure to the relative change in volume. For instantaneous changes, the equation for the bulk modulus becomes
$B = - \dfrac{{(dP)}}{{\left( {\dfrac{{dV}}{V}} \right)}}$.
The compressibility of a substance is given by the reciprocal of the bulk modulus, that is $\dfrac{1}{B}$. The bulk modulus is also referred to as incompressibility.
The bulk modulus is actually a thermodynamic quantity. For an adiabatic and reversible process, we have $P{V^\gamma } = c$, where $c$ is a constant
Differentiating both sides,
$
  P(\gamma ){(V)^{\gamma - 1}}dV + {(V)^\gamma }dP = 0 \\
  \gamma P = - \dfrac{{VdP}}{{dV}} \\
  \gamma P = B \\
 $
Therefore, we have the bulk modulus as the product of the pressure and the ratio of specific heats at constant pressure and volume.

Note:
Remember the value of the bulk modulus given in terms of the fractional change in volume and change in pressure. Remember that the bulk modulus has a negative in its formula. Reason being, the volume due to compression decreases. The value of the bulk modulus varies in between $35 - 55$ $GPa$.