Question

Assuming that the equation of state for a gas can be written in the form P(Vm​−b(T))=RT, derive an expression for α=V1​(∂T∂V​)P​ and κT​=−V1​(∂P∂V​)T​. Show the steps of differentiation.

Answer 1

To calculate the isobaric expansion coefficient (\(\alpha\)) and the isothermal compressibility (\(\kappa_T\)), one must differentiate the modified gas equation of state with respect to volume and pressure respectively while maintaining the respective variables constant.

Step-by-step explanation:

The question involves deriving expressions for the isobaric expansion coefficient (\(\alpha\)) and the isothermal compressibility (\(\kappa_T\)) using a modified equation of state for a gas.

The given equation of state is P(V_m - b(T)) = RT,

where P stands for pressure,

V_m is the molar volume,

T is the temperature in Kelvin,

R is the gas constant,

and b(T) is a temperature-dependent volume correction term.

To derive \(\alpha\), one must take the derivative of the temperature with respect to volume at constant pressure and multiply by the inverse of the volume.

For \(\kappa_T\), take the partial derivative of pressure with respect to volume at constant temperature and then multiply by the negative inverse of volume.

During the calculations, it's crucial to remember to consistently use Kelvin for temperature.