Consider a gas of molecules whose interaction energy u(r) is infinite for r < r0 and negative for r > r0, with a minimum value of –u0. Suppose further that κT ≫ u0, so you can approximate the Boltzmann factor for r > r0 using ex ≈ 1 + x. Show that under these conditions the second virial coefficient has the form B(T) = b − (a/κT), the same as what you found for a van der Waals gas in Problem. Write the van der Waals constants a and b in terms of r0 and u(r), and discuss the results briefly.

Problem:

Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial expansion,

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) |
B (cm3/mol) |

100 |
−160 |

200 |
− 35 |

300 |
− 4.2 |

400 |
9.0 |

500 |
16.9 |

600 |
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 expect B(T) to be negative at low temperatures but positive at high temperatures.

(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,

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 binomial expansion says that provided that |px| ≪ 1. Apply this approximation to the quantity [1 − (nb/V)]−1.)

(d) Plot a graph of the van der Waals prediction for B(T), choosing a and b so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)

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