Bulk Modulus for Gases

The bulk modulus, also known as incompressibility, measures a substance’s capacity to sustain changes in volume when compressed on all sides. It’s calculated by dividing the compressive force by the relative deformation. The higher the bulk modulus of a substance, the greater the force required to deform it or change its volume. For example, steel would require a huge amount of force to deform it compared to plastic or wood. Hence, the bulk modulus of steel would be higher than the bulk modulus of either plastic or wood. Bulk modulus can be used not just for solids but also for liquids and Gases. Bulk modulus for gases is calculated by the formula B= ∆P/(∆V/V).

What is Bulk Modulus in Gases?

Bulk modulus for gases is the amount of resistance a volume of gas applies when under pressure. It can be calculated with the help of the ideal gas law formula. Gases behave differently when there is a change in temperature or heat. Thus, to cater to this situation, there are two types of bulk modulus for gases. Isothermal and adiabatic bulk modulus. The formula for isothermal bulk modulus is B=-∆P/(∆V/V) and for adiabatic Bulk modulus is Badiabatic= γP

The formula for Bulk Modulus:

Bulk modulus is just another type of modulus (example: young’s modulus) we measure for solids, but this one can also be used in the case of liquids and Gases. Bulk modulus is associated with a “volume strain”. To make it simple, let’s take an object and increase the pressure by applying equal amounts of force on it from all directions. This would induce a reduction in the volume of the object. From the above understanding, we can define bulk modulus “B” as 

B= Pressure applied Fractional /change in volume

Since the pressure is increased, it’s a positive change; hence the pressure is a positive number. The volume of the object is decreasing, that is, a negative change. Thus, the volume change is a negative number. Therefore, the final formula would be

B=- Pressure applied fractional /change in Volume

= –∆P/∆V/V

= -V.∆P/∆V

Some Values of Bulk Modulus

A few bulk modulus for gases examples we should know to make solving bulk modulus problems easy are listed here. Since the pressure required to deform or change the volume of an object becomes an astronomical number in terms of pascal, we write it in Gigapascal.

Note that a Giga pascal (GPa) is 109 times a pascal. 

Isothermic and Adiabatic Bulk Modulus

According to ideal gas law, PV=nRT  where P is pressure, V is volume, n is the amount of the substance, R is the ideal gas constant, and T is the temperature. We will be describing the bulk modulus for Gases concerning this equation.

Bulk modulus under isothermal conditions, that is with no change in temperature, will be depicted by the symbol Bisothermal. Let us solve the equation PV=nRT  under isothermal conditions.

PV=nRT  

T is constant in isothermal conditions. Therefore,

PV=constant

P.dV/dP+V.dP/dP=0

P= -V.dP/dV=B

Therefore, under isothermal conditions, the bulk modulus for gases is equal to the pressure of the gas.

Bulk modulus under adiabatic conditions, that is, with no heat transfer, will be depicted by the symbol Badiabatic. Let us solve the equation PV=nRT   under adiabatic conditions.

PV=constant

Applying product rule again:

P(γ.Vγ-1 .  dVdP)+Vγ=0

P(γ.Vγ-1 .  dVdP)= – Vγ

→γP=- V .  dP/dV=B

Therefore, adiabatic bulk modulus Badiabatic= γP where γ=CpCv.

Conclusion

Bulk Modulus for Gases is the constant that describes the resistance, and gas applies against deforming or external pressures. It can be easily found out by dividing volumetric stress with volumetric strain or, in simple words, dividing the change in pressure applied on the gas with the change in volume of the gas divided by the original volume of gas. Bulk modulus is usually represented with the symbol “K”. The SI unit of the Bulk modulus is the pascal, and the dimensional formula is  M1L-1T-2. The formula for Bulk modulus is given as B= ∆P/(∆V/V). Adiabatic means “no transfer of heat in a particular process”. Bulk modulus, which is calculated under conditions that prevent heat transfer, is known as adiabatic bulk modulus and is denoted by “Badiabatic”. The formula for adiabatic bulk modulus is given as Badiabatic= ∆P/(∆V/V)= γP .