A Few Partial Derivatives Defined

It is useful to define a few special partial derivatives in terms of the state variables of gaseous systems, because the definitions either (a) can be considered as basic properties of the gas, or (b) will help simplify future equations. 

The expansion coefficient of a gas, labeled a, is defined as the change in volume as the temperature is varied at constant pressure. A 1/U multiplicative factor is included  

The isothermal compressibility of a gas, labeled k, is the change in volume as the pressure changes at constant temperature (the name of this coefficient is more descriptive). It too has a /V multiplicative factor, but it is negative  

Because dV/dp)j is negative for gases, the minus sign in equation 1.28 makes K a positive number. Again, for an ideal gas, it is easy to show that k = RT/ffiV. For both a and k, the /V term is included to make the quantities intensive (that is, independent of amount ). 

Because both of these definitions use p, V, and T, we can use the cyclic rule to show that, for example. 

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