A specimen of oil having an initial volume of 600 cm3 is subjected to a pressure increase of 3.6×10^6 Pa, and the volume is found to decrease by 0.45 cm^3. What is the bulk modulus of the material and the compressibility?

Verified step by step guidance

1. Identify the given values: initial volume $V_0 = 600 \, \text{cm}^3$, change in volume $\Delta V = -0.45 \, \text{cm}^3$, and change in pressure $\Delta P = 3.6 \times 10^6 \, \text{Pa}$.
2. Recall the formula for bulk modulus, $B = -\frac{\Delta P}{(\Delta V / V_0)}$. The negative sign indicates that an increase in pressure results in a decrease in volume.
3. Substitute the given values into the bulk modulus formula to find $B$.
4. Calculate the compressibility, which is the reciprocal of the bulk modulus, $\kappa = \frac{1}{B}$.
5. Interpret the results: A higher bulk modulus indicates that the material is less compressible, while a higher compressibility indicates that the material is more easily compressed.

Key Concepts

Bulk Modulus

The bulk modulus is a measure of a material's resistance to uniform compression. It is defined as the ratio of the change in pressure to the relative change in volume. Mathematically, it is expressed as K = -ΔP / (ΔV/V₀), where ΔP is the change in pressure, ΔV is the change in volume, and V₀ is the initial volume. A higher bulk modulus indicates that the material is less compressible.

Compressibility

Compressibility is the measure of how much a material can be compressed under pressure. It is the reciprocal of the bulk modulus, defined as β = 1/K. This means that a material with high compressibility will experience a larger volume change for a given pressure increase. Compressibility is particularly important in fluid mechanics and material science, as it affects how substances behave under varying pressure conditions.

Pressure and Volume Relationship

The relationship between pressure and volume in a material is governed by the principles of fluid mechanics and thermodynamics. According to the principle of conservation of mass and the ideal gas law, an increase in pressure typically results in a decrease in volume for compressible materials. This relationship is crucial for calculating changes in volume and pressure, especially in scenarios involving liquids and gases under varying conditions.