## COMPRESSIBILITY AND BULK MODULUS

**Introduction: **Bulk modulus, E_{v} is defined as the ratio of the change in pressure to the rate of change of volume due to the change in pressure. It can also be expressed in terms of change of density.

E_{v}= – dp/ (dv/v) = dp/ (d p/p)

Where dp is the change in pressure causing a change in volume dv when the original volume was v. The unit is the same as that of pressure, obviously. Note that dv/v = – dρ/ρ.

The negative sign indicates that if dp is positive then dv is negative and vice versa, so that the bulk modulus is always positive (N/m2). The symbol used in this text for bulk modulus is E_{v} (K is more popularly used).

This definition can be applied to liquids as such, without any modifications. In the case of gases, the value of compressibility will depend on the process law for the change of volume and will be different for different processes.

The bulk modulus for liquids depends on both pressure and temperature. The value increases with pressure as dv will be lower at higher pressures for the same value of dp. With temperature the bulk modulus of liquids generally increases, reaches a maximum and then decreases. For water the maximum is at about 50°C. The value is in the range of 2000 MN/m2 or 2000 × 10 ^{6} N/m2 or about 20,000 atm. Bulk modulus influences the velocity of sound in the medium, which equals (g _{o} × E_{v} /ρ)^{0.5.}

**Expressions for the Compressibility of Gases:** The expression for compressibility of gases for different processes can be obtained using the definition, namely, compressibility = – dp/ (dv/v). In the case of gases the variation of volume, dv, with variation in pressure, dp, will depend on the process used. The relationship between these can be obtained using the characteristic gas equation and the equation describing the process.

Process equation for gases can be written in the following general form

**Pv ^{n }= constant**

Where n can take values from 0 to ∞. If n = 0, then P = constant or the process is a constantpressure process. If n = ∞, then v = constant and the process is constant volume process. Theseare not of immediate interest in calculating compressibility. If dp = 0, compressibility is zeroand if dv = 0, compressibility is infinite.The processes of practical interest are for values of n = 1 to n = cp/cv(the ratio of specificheats, denoted as k). The value n = 1 means Pv = constant or isothermal process and n = cp/cv= k means isentropic process

Using the equation Pvn = constant and differentiating the same

**nPv ^{(n–1})dv v^{n}dp = 0**

Re arranging and using the definition of E_{v},** Ev = – dp/ (dv/v) = n × P**

Hence compressibility of gas varies as the product n × P.

For isothermal process, n = 1, compressibility = P.

For isentropic process, compressibility = k × P.

For constant pressure and constant volume processes compressibility values are zero and ∞ respectively

In the case of gases the velocity of propagation of sound is assumed to be isentropic. From the definition of velocity of sound as [g _{o }× Ev/ρ]^{0.5} it can be shown that

**c = [go × k P/ρ] ^{0.5} = [go × k × R × T]^{0.5}**

It may be noted that for a given gas the velocity of sound depends only on the temperature. As an exercise the velocity of sound at 27°C for air, oxygen, nitrogen and hydrogen may be calculated as 347.6 m/s, 330.3 m/s, 353.1 m/s and 1321.3 m/s.