In this video I go over another derivatives application and show how isothermal compressibility can be described using the derivative as a rate of change of the volume with increasing applied pressure. I also go over an example in determining the compressibility of air at 25 degrees Celsius.

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Derivatives Application: Compressibility

Compressibility

One of the quantities of interest in thermodynamics is compressibility. If a given substance is kept at a constant temperature, then its volume V depends on its pressure P.

We can consider the rate of change of volume with respect to pressure – namely, the derivative dV/dP. As P increases, V decreases, so dV/dP < 0.

The compressibility is defined by introducing a minus sign and dividing this derivative by the volume V:

Thus, β measures how fast, per unit volume, the volume of a substance decreases as the pressure on it increases at constant temperature.

Example

If the volume V (in m³) of a sample of air at 25°C was found to be related to the pressure P (in kPa) by the equation:

Determine the Compressibility of the air at a pressure of 50 kPa.