Chapter 5: Q. 5.50 (page 185)

The compression factor of a fluid is defined as the ratio PV/NkT; the deviation of this quantity from 1 is a measure of how much the fluid differs from an ideal gas. Calculate the compression factor of a Van der Waals fluid at the critical point, and note that the value is independent of a and b. (Experimental values of compression factors at the critical point are generally lower than the Van der Waals prediction, for instance, 0.227 for H22O, 0.274 for CO22, and 0.305 for He.)

Short Answer

Expert verified

The compression factor of van der Waal fluid at critical point is $\frac{3}{8}$ .

Step by step solution

Given

The compression factor of a fluid is given by

$Z = \frac{P V}{N k T}$

where Z is compression factor.

van der Waal's equation

$P = \frac{N k T}{(V - N b)} - \frac{a N^{2}}{V^{2}}$

Partial differentiation of the equation w.r.t V

we get

role="math" localid="1646976952526" $\frac{δ P}{δ V} = - \frac{N k T}{{(V - N b)}^{2}} + \frac{2 a N^{2}}{V^{3}}$

Again differentiating we get

role="math" localid="1646977057779" $\frac{δ^{2} P}{δ^{2} V} = \frac{N k T}{{(V - N b)}^{3}} - \frac{6 a N^{2}}{V^{4}}$

At critical point

$\frac{δ P}{δ V} = 0 \frac{δ^{2} P}{δ^{2} V} = 0$

role="math" localid="1646977123022" $\Rightarrow \frac{N k T_{c}}{{(V_{c} - N b)}^{2}} = \frac{2 a N^{2}}{{V_{c}}^{3}} a n d \frac{N k T_{c}}{{(V_{c} - N b)}^{3}} = \frac{6 a N^{2}}{{V_{c}}^{4}}$

Finding Vc, Tc, Pc.

On equating the above equations we get

$V_{c} = 3 N b$

Substituting the equation in equation

role="math" localid="1646977903462" $\frac{N k T_{c}}{{(3 N b - N b)}^{2}} = \frac{2 a N^{2}}{{(3 N b)}^{3}} \Rightarrow k T_{c} = \frac{3}{27} \frac{a}{b}$

Substituting the above values in equation (2)

$P_{c} = \frac{N k (\frac{3 a}{27 b)}}{(3 N b - N b)} - \frac{a N^{2}}{{(3 N b)}^{2}} \Rightarrow P_{c} = \frac{1}{27} \frac{a}{b^{2}}$

Substituting the values in Equation (1)

$Z_{C} = \frac{P_{c} V_{c}}{N k T_{c}} = \frac{1}{27} \frac{a}{b^{2}} \times 3 N b \times \frac{1}{N} \times \frac{27}{3} \times \frac{b}{a} = \frac{3}{8}$

The value of the compression factor is less than 1

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Short Answer

Step by step solution

Given

van der Waal's equation

At critical point

Finding Vc, Tc, Pc.

Substituting the values in Equation (1)

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