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.
The mathematical representation of bulk modulus is given as follows:
For bulk modulus K > 0;
\(K=-V\frac{dP}{dV}\)
Where,
P: pressure
V: volume
\(\frac{dP}{dV}\) : derivative of pressure with respect to volume
Bulk modulus for unit mass;
\(K=\rho \frac{dP}{d\rho }\)
Where,
⍴: density
\(\frac{dP}{dV}\) : derivative of pressure with respect to density
Bulk modulus in thermodynamics with constant temperature and constant entropy is given as follows:
Isentropic bulk modulus Ks;
\(K_{s}=\gamma p\)
Isothermal bulk modulus KT;
\(K_{T}= p\)
Where,
p: pressure
γ: heat capacity ratio
Following is the table of bulk modulus of a few common materials:
|
Material |
Bulk modulus in GPa |
|
Diamond |
443 |
|
Rubber |
1.5 to 2 |
|
Steel |
160 |
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:
\(\frac{-\delta V}{V}=\frac{\delta p}{K}\) (negative sign to indicate that with increase in pressure, volume decreases)
Where,
δV: change in volume
δp: change in pressure
V: actual volume
K: bulk modulus
\(K=-V\frac{dp}{dV}\) (as δp tends to 0) (eq.1)
\(V=\frac{1}{\rho }\) (unit mass of the substance) (eq.2)
\(Vd\rho +\rho dV=0\) (after differentiation)
\(dV=-(\frac{V}{\rho })d\rho\) (eq.3)
\(K=\frac{-Vdp}{-(\frac{V}{\rho })d\rho }\) (after substituting eq.3 in eq.1)
\(∴ K=\rho \frac{dP}{d\rho }\)
Thus, above is the derivation of bulk modulus which is mainly applicable to liquids as gases are highly compressible which makes K vary.
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