Chapter 5: Q. 5.48 (page 185)

As you can see from Figure5.20,5.20,the critical point is the unique point on the original van der Walls isotherms (before the Maxwell construction) where both the first and second derivatives ofPPwith respect toVV(at fixedTT) are zero. Use this fact to show that
V_c=3Nb, P_c = $\frac{1}{27} \frac{a}{b^{2}}$ and kT_c= $\frac{8}{27} \frac{a}{b}$

Short Answer

Expert verified

V_c=3Nb, P_c= $\frac{1}{27} \frac{a}{b^{2}}$ and kT_c= $\frac{8}{27} \frac{a}{b}$

Step by step solution

van der Waal's equation

$P = \frac{N k T}{(V - N b)} - \frac{a N^{2}}{V^{2}}$ (1)

Partial differentiation of the equation w.r.t V

we get

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

Again differentiating we get

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

At critical point

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

$\frac{N k T_{c}}{{(V_{c} - N b)}^{2}} = \frac{2 a N^{2}}{{V_{c}}^{3}} a n d \frac{N k T_{c}}{{(V_{c} - N b)}^{3}} = \frac{6 a N^{2}}{{V_{c}}^{4}}$

Step 3: Finding Vc, Tc, Pc.

On equating the above equations we get

$V_{c} = 3 N b$ (4)

Substituting the equation (4) in equation (2)

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

Substituting the above values in equation (1)

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

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Short Answer

Step by step solution

van der Waal's equation

At critical point

Step 3: Finding Vc, Tc, Pc.

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As you can see from Figure5.20,5.20,the critical point is the unique point on the original van der Walls isotherms (before the Maxwell construction) where both the first and second derivatives ofPPwith respect toVV(at fixedTT) are zero. Use this fact to show thatVc=3Nb, Pc =127ab2 and kTc=827ab

Short Answer

Step by step solution

van der Waal's equation

At critical point

Step 3: Finding Vc, Tc, Pc.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Atoms and Radioactivity

Dynamics

Physics of Motion

Conservation of Energy and Momentum

Torque and Rotational Motion

Fields in Physics

Study anywhere. Anytime. Across all devices.