Question # If ( Z ) is a compressibility factor, van der Waals' equation at low pressure can be written as

# If ( Z ) is a compressibility factor, van der Waals' equation at low pressure can be written as

(a) ( quad Z=1+frac{R T}{p b} )

(b) ( Z=1-frac{a}{V R T} )

(c) ( quad Z=1-frac{p b}{R T} )

(d) ( Z=1+frac{p b}{R T} )

Solution

Van der Waals equation is given by

[

left(rho+frac{a}{V^{2}}right)(V-b)=R T

]

At low pressure, ( V ) decreases. Thus, ( frac{a}{V^{2}} ) increases. However, ( V ) is still large enough is comparison to ( b ) hence ( b ) can be neglected. Thus, van der Waals' equation becomes

[

left(rho+frac{a}{V^{2}}right) V=R T

]

( Rightarrow quad rho V+frac{a}{V}=R T )

( Rightarrow quad rho V=R T-frac{a}{V} )

( Rightarrow quad frac{rho V}{R T}=1-frac{a}{R T V} )

( Z=1-frac{a}{R T V} quadleft[because frac{rho V}{R T}=Zright] )