Lecture 6 Virial Expansions and Critical Behavior September 11, 2008 Using the virial expansion in pressure: (1) at p = 0, Z = 1, perfect gas (2) at T , Z = 1 for all gasses Z (3) lim p 0 p T Z 1 a a a = p + ... 2b b + p RT (RT )3 RT T RT Z 1 a lim = b p RT p 0 T RT Page 1 of 8 If b&gt; a RT a RT a RT then b= then Z lim &gt; 0 (Repulsive interactions dominate) p 0 p T (note the initial slope is positive) Z lim = 0 (Boyle's condition) p 0 p T (note the initial slope is zero) Z lim &lt; 0 (Attractive interactions dominate) p 0 p T (note the initial slope is negative) b = a/RT can be rewritten as TB = a/Rb This is the Boyle Temperature!!! Perfect <a href="/keyword/gas-behavior/" >gas behavior</a> over a wide concentration range b&lt; then A 2-term virial expansion is only valid for describing the limit where p 0!!! As p increases, higher order terms take over!!! Page 2 of 8 Note how this is analogous to the expression for the virial expansion in terms of n/V for <a href="/keyword/van-der-waals/" >van <a href="/keyword/der-waals/" >der waals</a> </a> gases. The key differences are: Expand in p Expand in / Full expansion 2 Virial Coefficient (B' for empirical p expansions) 3rd Virial Coefficient (C' for empirical p expansions) nd b2 Nonetheless, they both describe the same thing and are only valid if we are talking about a <a href="/keyword/van-der-waals/" >van <a href="/keyword/der-waals/" >der waals</a> </a> gas. This is the reason that in engineering applications, B', C', etc. are empirical parameters to obtain the right results. Here we see the clear contribution of attractions and repulsions to the second and third term of the virial expansion. Page 3 of 8 Isotherms of <a href="/keyword/van-der-waals/" >van <a href="/keyword/der-waals/" >der waals</a> </a> Gasses 1 p = RT V Recall: (V is a molar volume) p= RT a (V b) V 2 T &gt; Tc T = Tc T &lt; Tc (V is a molar volume) Page 4 of 8 The van der Waal's loop is artificial, hence coexistent gas and liquid phases are connected by a tie line (horizontal line satisfying the equal areas construct. Along this line, gas is continually converted into liquid with decreasing molar volume until only liquid remains). Page 5 of 8 Page 6 of 8 van der Waal's Equation of State p= RT a V b V2 at the Critical Point where T = Tc, p = pc, V = Vc then RT a 1) p c = (Vc b) Vc2 , p 2) = 0, V Tc 2p =0, 3) V 2 Tc 1 1 1 V V = V = = 4) Tc = p V p T 0 c V Tc Singularity at the critical point, a fluid is infinitely compressible On the critical isotherm: p &lt; 0 for V &gt; Vc V Tc p = 0 for V = Vc V Tc p &lt; 0 for V &lt; Vc V Tc The first and second derivatives are: RT 2a p + , for T=Tc and V = Vc , = 2 V T (V b ) V3 p RTc 2a + =0 V = 2 Vc3 c Tc (Vc b ) Page 7 of 8 (Vc b ) RTc 2 = 2a Vc 3 2RTc = 6a Vc 4 2p = 2RT 6a = 0 or 2 3 V4 V T (V b ) Show the expressions, p c = Hence: 2 a = 3p c Vc , b = (Vc b ) 3 a 27b 2 , Vc = 3b , Tc = 8a 27Rb Vc pV 8p V 3 , R = c c , and Zc = c c = = 0.375 RTc 8 3 3Tc Reality for <a href="/keyword/van-der-waals/" >van <a href="/keyword/der-waals/" >der waals</a> </a> Eq. of State Although it is better than the perfect gas law, there is no experimental validation Only predicts phase behavior (gas to liquid transition), qualitatively Does predict Tc Fails quantitatively Hence, other equations of state like the virial expansions and those found in Table 1.7 do better jobs of predicting real <a href="/keyword/gas-behavior/" >gas behavior</a> . Page 8 of 8