roblem 1.17. Even at low density, real gases don\'t quite obey the ideal gas aw.

ID: 1636372 • Letter: R

Question

roblem 1.17. Even at low density, real gases don't quite obey the ideal gas aw. A systematic way to account for deviations from ideal behavior is the virial expansion, C(T) PV = nRT(1 + B(T) where the functions B(T), C(T), and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations it's sufficient to omit the third term and concentrate on the second, whose coefficient B(T) is called the second virial coefficient (the first coefficient being 1). Here are some measured values of the second virial coefficient for nitrogen (N2): T (K) 100 200 300 400 500 600 B (cm3/mol) 160 35 -4.2 9.0 16.9 21.3 (a) For each temperature in the table, compute the second term in the virial equation, B(T)/(V/n), for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions. (b) Think about the forces between molecules, and explain why we might ex- pect B(T) to be negative at low temperatures but positive at high temper- atures. (c) Any proposed relation between P, V, and T, like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation, (P4 )(V-nb) = nRT where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients (B and C) for a gas obeying the van der Waals equation, in terms of a and b. (Hint: The binom that (1 +)p(p-1)x2, provided that lpzl

Explanation / Answer

P V = n R T (1 + B/(V/n)) = n R T (1 + n B/V) = n R T + n^2 R T B/V

==> V = n R T/P + n^2 R T (B/P)/V

==> V^2 = (n R T/P) V + n^2 R T B/P

==> V^2 - (n R T/P) V - n^2 R T B/P = 0

==> V = ((n R T/P) + sqrt((n R T/P)^2 + 4*(n^2 R T B/P)))/2

V = (n/2P) (sqrt(RT) sqrt(4 B P + RT) + RT)

n = 1

R = 8.314

P = 1 atm = 1.01e5 Pa

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T = 100 K...... B = -160e-6

V = (1/(2*1.01e5)) (sqrt(8.314*(100)) sqrt(4*(-160*10^(-6))*(1.01e5) + 8.314*(100)) + 8.314*(100))

V = 0.00806845