Compressibility, τ, is defined as: τ = − 1 /ν *(dν/ dp) Show that the ratio...

Compressibility, τ, is defined as: τ = − 1 /ν *(dν/ dp) Show that the ratio of the isothermal compressibility to the isentropic compressibility is equal to the specific heat ratio for a perfect gas.

Solutions

Expert Solution

Here is the thermal expansion coefficient:

is the isothermal compressibility:

and is the isentropic compressibility:

A corresponding expression for the difference in specific heat capacities (intensive properties) at constant volume and constant pressure is:

where ρ is the density of the substance under the applicable conditions.

The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size-dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Thus:

The difference relation allows one to obtain the heat capacity for solids at constant volume which is not readily measured in terms of quantities that are more easily measured. The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio.

queen_honey_blossom answered 10 months ago

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