Determine the value of compressibility factor at critical point (Zcp) for the Van der Waals’ gas.

Question

Determine the value of compressibility factor at critical point [latex]\left(Z_{c p}\right)[/latex] for then Van der Waals’ gas.

Accepted Answer

Refer Fig. 8.13.

From the isotherms plotted on p-v diagram in Fig. 8.13 it can be seen that the critical isotherms has an inflection point, whose tangent is horizontal at the critical point.

[latex]\left(\frac{\partial p_c}{\partial v} ight)_{c p}=0 ext { and }\left|\frac{\partial^2 p}{\partial v^2} ight|_{c p}=0[/latex]

The Van der Waal’s equation at the critical point is

[latex]p_{c p}=\frac{R_0 T_{c p}}{\bar{v}_{c p}-b}-\frac{a}{\bar{v}_{c p}^2}[/latex] ...(i)

As [latex]T_{c p}[/latex] is constant

[latex]\left(\frac{\partial p_{c p}}{\partial v_{c p}} ight)=-\frac{R_0 T_{c p}}{\left(\bar{v}_{c p}-b ight)^2}+\frac{2 a}{\bar{v}_{c p}{ }^3}=0[/latex] ...(ii)

[latex]\left(\frac{\partial^2 p_{c p}}{\partial v_{c p}^2} ight)=\frac{2 R_0 T_{c p}}{\left(\bar{v}_{c p}-b ight)^3}-\frac{6 a}{\left(\bar{v}_{c p} ight)^4}=0[/latex] ...(iii)

[latex]\left(\frac{3}{\bar{v}_{c p}} ight) imes(i i)+(i i i)[/latex] gives

[latex]-\frac{3 R_0 T}{\bar{v}_{c p}\left(\bar{v}_{c p}-b ight)^2}+\frac{2 R_0 T}{\left(\bar{v}_{c p}-b ight)^3}=0[/latex]

or [latex]\frac{3}{\bar{v}_{c p}}=\frac{2}{\left(\bar{v}_{c p}-b ight)}[/latex]

or [latex]\bar{v}_{c p}=3 b[/latex]

Substituting for b in (ii), we get

[latex]-\frac{R_0 T_{c p}}{\left[\bar{v}_{c p}-(1 / 3) \bar{v}_{c p} ight]^2}+\frac{2 a}{\left(\bar{v}_{c p} ight)^3}=0[/latex]

∴ [latex]a=\frac{9}{8} R_0 T_{c p} \bar{v}_{c p}[/latex]

Substituting for a and b in (i), we get

[latex]p_{c p}=\frac{R_0 T_{c p}}{\bar{v}_{c p}-(1 / 3) \bar{v}_{c p}}-\frac{(9 / 8) R_0 T_{c p} \bar{v}_{c p}}{\bar{v}_{c p^2}}[/latex]

∴ [latex]\frac{p_{c p} \bar{v}_{c p}}{R_0 T_{c p}}=\frac{1}{(2 / 3)}-\frac{(9 / 8)}{1}[/latex]

But [latex]\frac{p_{c p} \bar{v}_{c p}}{R_0 T_{c p}}=Z_{c p}[/latex]

∴ [latex]Z _{ cp }=\frac{3}{2}-\frac{9}{8}=\frac{ 3 }{ 8 }[/latex].

Question 8.16: Determine the value of compressibility factor at critical po...

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