Q.Compute the bulk modulus of water from the following data: Initial volume = 100.0 litre, Pressure increase = 100.0 atm ($1\ \text{atm} = 1.013 \times 10^{5}\ \text{Pa}$), Final volume = 100.5 litre. Compare the bulk modulus of water with that of air (at constant temperature). Explain in simple terms why the ratio is so large.

Concept: Bulk Modulus Application

Bulk modulus $B$ measures a material’s resistance to uniform compression:

Step 1 – Convert data to SI units

Initial volume $V = 100.0\ \text{L} = 0.100\ \text{m}^3$

Pressure increase $\Delta P = 100.0\ \text{atm} = 100.0 \times 1.013 \times 10^{5} = 1.013 \times 10^{7}\ \text{Pa}$

Volume increase $\Delta V = 100.5 - 100.0 = 0.5\ \text{L} = 5.0 \times 10^{-4}\ \text{m}^3$

Step 2 – Compute bulk modulus of water

The negative sign indicates volume increases with pressure decrease; magnitude is $2.03 \times 10^{9}\ \text{Pa}$.

Step 3 – Compare with air (isothermal)

For an ideal gas at constant temperature, $B_{\text{air}} = P$ (since $PV = \text{constant}$). At 1 atm, $B_{\text{air}} \approx 1.013 \times 10^{5}\ \text{Pa}$. …

Concept and Intuition

Bulk modulus ($B$) is the ratio of the volumetric stress (pressure change $\Delta P$) to the volumetric strain (fractional change in volume $-\Delta V / V_0$). The negative sign ensures $B$ is positive — when pressure increases, volume decreases.

Think of it as the "stiffness" of a material against compression. For a gas, molecules are far apart — you can push them closer easily, so $B$ is small. For a liquid like water, molecules are already nearly touching; pushing them even a little closer requires enormous force, so $B$ is huge. That's why water feels "incompressible" in everyday life, though it does compress slightly under extreme pressure.

Step-by-Step Calculation

1. Identify the given data

• Initial volume: $V_0 = 100.0\ \text{litre}$
• Final volume: $V_f = 100.5\ \text{litre}$
• Pressure increase: $\Delta P = 100.0\ \text{atm}$

So the actual volume change is:

2. Convert units to SI

Bulk modulus is usually expressed in pascals (Pa). Convert pressure and volume:

• Pressure: $1\ \text{atm} = 1.013 \times 10^{5}\ \text{Pa}$, so

• Volume: $1\ \text{litre} = 10^{-3}\ \text{m}^3$, so

3. Compute volumetric strain

Volumetric strain is the fractional change in volume:

This is a 0.5% decrease in volume — tiny for such a large pressure.

4. Apply the bulk modulus formula

5. Bulk modulus of air at constant temperature

For an ideal gas at constant temperature (isothermal process), Boyle's law applies: $PV = \text{constant}$. Differentiating gives $P dV + V dP = 0$, so:

Thus the isothermal bulk modulus of an ideal gas equals its pressure.

At normal atmospheric pressure:

$B_{\text{air}} = P_{\text{atm}} = 1.013 \times 10^{5}\ \text{Pa}$ …

Step 1: deltaP=100atm=1.013e7 Pa, V0=0.100 m^3, deltaV=-5.0e-4 m^3 (physically must decrease). Step 2: volumetric strain=-5.0e-3. Step 3: B=-deltaP/(deltaV/V0)=2.03e9 Pa. Step 4: for gas, B_air=P~=1.013e5 …