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ENGR5003 / Week 6
Old spec L3 exercises 1-2 - expansivity and compressibility of a van der Waals gasTutorialOld spec ENGR2172:115 min

ENGR217 Lecture 3 student exercises 1-2 (slide 36)

The volumetric expansivity and isothermal compressibility are defined as β=1v(∂v∂T)p,α=−1v(∂v∂p)T\beta = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_p, \qquad \alpha = -\frac{1}{v}\left(\frac{\partial v}{\partial p}\right)_T A substance has the equation of state p=RTv−b−av2p = \dfrac{RT}{v - b} - \dfrac{a}{v^2}, where aa and bb are constants.

Formulas you may need
  • Cyclic relation: (∂x∂y)z(∂y∂z)x(∂z∂x)y=−1\left(\dfrac{\partial x}{\partial y}\right)_z\left(\dfrac{\partial y}{\partial z}\right)_x\left(\dfrac{\partial z}{\partial x}\right)_y = -1 (on the chemical thermodynamics formula sheet)
  • Reciprocal relation: (∂z∂x)y(∂x∂z)y=1\left(\dfrac{\partial z}{\partial x}\right)_y\left(\dfrac{\partial x}{\partial z}\right)_y = 1 (on the chemical thermodynamics formula sheet)
  • Definitions of β\beta and α\alpha (given in the question; learn them)
  1. (a)
    Derive expressions for the volumetric expansivity and the isothermal compressibility of this substance.
  2. (b)
    Show that for this equation of state β=α(∂p∂T)v\beta = \alpha\left(\dfrac{\partial p}{\partial T}\right)_v.