Bulk Modulus Formula Derivation

Bulk Modulus

Bulk modulus of a substance is defined as the ratio of infinitesimal pressure increase to a decrease of the volume. Bulk modulus is meaningful only for a fluid. It is denoted as either K or B. The reciprocal of bulk modulus is compressibility of a substance and this is the relation between bulk modulus and compressibility.

The mathematical representation of bulk modulus is given as follows:

For bulk modulus K > 0;

Where,

P: pressure

V: volume

Bulk Modulus For Unit Mass

Where,

⍴: density

Isentropic bulk modulus Ks;

Isothermal bulk modulus K_T;

Where,

p: pressure

γ: heat capacity ratio

Following is the table of bulk modulus of a few common materials:

Bulk Modulus Formula Derivation

As we know the ratio between change in pressure to change in volumetric strain is dependent on bulk modulus of the material, following is the derivation showing the relationship:

Where,

δV: change in volume

δp: change in pressure

V: actual volume

K: bulk modulus

Thus, above is the derivation of bulk modulus which is mainly applicable to liquids as gases are highly compressible which makes K vary.

Solved Example

Q1. What is the bulk modulus of a body that experienced a change of pressure of 5 x 10⁴ N/m² and its volume goes from 4 cm³ to 3.9 cm³?

Solution: The bulk modulus is calculated using the formula,
B = ΔP /(ΔV/V)
B = (5 x 10⁴ N/m²)/((4 cm³ – 3.9 cm³)/4 cm³) = 0.125 x 10⁴ N/m²
B = 1.25 x 10⁴ N/m²

Q2. In the ammunition testing centre the pressure is found to be 255 MPa. Calculate the change in volume of the copper piece when subjected to this pressure in percentage. The bulk modulus of copper is 1.38 x 10¹¹ Pa.

Solution
The pressure in the testing centre is 255 MPa.
Bulk modulus, K = V(∆P) / ∆V
Substituting the values,
(ΔV / V) = 255 × 10⁶ / 1.38 × 10¹¹ × 100
Therefore, the change in volume percentage is 0.184 %

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