Question

14-5. Using Eqs. (14-14) and (14-17), calculate the van der Waals constants a and b for nitrogen. For u(r), assume a Lennard-Jones 6-12 potential with ε and σ given in Table 12-3. Compare these calculated values to the experimentally determined values, a 1.39 x 106 cm atm/mole and b-39.1 cm/mole. Such poor agreement is quite typical, simply indi- cating the inadequacy of the van der Waals equation. 14-2 THE VAN DER WAALS EQUATION We start with Eq. (14-3), and take the reference system to be a hard-sphere fluid. We furthermore assume that UN İs pair-wise additive and to be of the form u(r) undr) + ud(r), where u (r) is some arbitrary attractive part (and so is negative). We then assume that BUN) small enough that we can write (14-11) Equation (14-11) follows since we are assuming that terms in β2 and higher can be neglected. From Eq. (14-10), 0 Certainly gus(r) was not available to van der Waals, and he effectively approximated (14-12) have 9Hs) by Its(r) (1 r > σ This form is correct for hard spheres only in the limit ρ--0. Using this gs(r), we (14-13) where a-2Tdr 14-14) THE VAN DER WAALS EQUATION 305 minus sign in the definition of a has been included to make a a positive number. The ember that )is negative. The specific form of u)) is not important here. Remem o far now, Eq. (14-3) has been reduced to (14-15) and so we can write p0) = ( ap (14-16) kT kT where po) is the pressure of the unperturbed system. The final approximation of the van der Waals theory is to assume that the hard- sphere configuration integral is of the form V, where the effective volume is deter- mined by assuming that the volume available to a molecule in the fluid has a volume 4 ơ3/3 excluded to it by each other molecule of the system. However, we have to divide this quantity by 2 since this factor of 4 ơ3/3 arises from a pair of molecules inter- acting, and only half the effect can be assigned to a given molecule. Therefore Ver V-2πNơ3/3, and we can write on 2nơ3 (14-17)- Substituting Eq. (14-17) into Eq. (14-16), then, we get the famous van der Waals equation, ESS (14-18) Usually one uses simple analytical expressions with adjustable pa that go asymptotically as r. These adjustable parameters can the fit experimental data. Perhaps the most well-used form is parameters for ue (12-3) where σ is the distance at which u(r) 0, and ε is the depth of the well. The exponent n is usually taken to be an integer between 9 and 15, but for historical reasons. 12 still the most popular value. The r-6 is included in Eq. (12-37) so that ur) has a correct asymptotic form. For n = 12, u(r) is called the Lennard-Jones 6-12 potential unut for 6-12 p0 so sd auso inoioflisoo lsiniv (12-38)

Answer 1

ennavdt- Jn pohal: n 1 (3.94. 5 ylo. ma1i b - cQ γ3.lux 3 12 Cd 3r 3 433245x10 a = 3.85x10 a= 3.25 xio 23