Estimate the pressure, in atmospheres, at the following locations: Ogden Utah (4

Question

Estimate the pressure, in atmospheres, at the following locations: Ogden Utah (4700 ft or 1430 m above sea level): Leadville, Colorado (10, 150 ft, 3090 m): Mt. Whitney, California (14, 5000 ft, 4420 m): Mt, Everest, Nepal/ Tibet (29, 000 ft. 8850 m). (Assume that the pressure at sea level is 1 atm.) 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. PV = nRT(1 + B(T)/(V/n) + C(T)/(V/n)^2 + ...). 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 l). Here are some measured values of the second virial coefficient for nitrogen (N_2): (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. (P + an^2/V^2)(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. (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.)

Explanation / Answer

17. b

The virial equation

PV = nRT( 1 + B(T)/(V/n) + C(T)/(V/n)^2 .. )

at low temperatures, gas molecues are closer to each other due to less potential energy , so forces of interaction are high

so, PV/nRT < 1 for low temperatures, as high intermolecular forces mean the gas occupies less volume than its ideal counterpart

for high temperatures, the gas molecules move away from each other, less intermolecular forces are there, and gases tensd to occupy more volume than their ideal gas counterpart

so, PV/nRT > 1

hence B(T) is +ve

c. Wander wall's equation of state

(P + an^2/V^2) (V - nb) = nRT

PV(1 + an^2/V^2*P)(1 - nb/V) = nRT

PV = nRT(1 + an^2/V^2*P)^-1 * (1 - nb/V)^-1

so from binomial expansion

(1 + an^2/V^2*P)^-1 = 1 - an^2/V^2*P + (an^2/V^2*P ) ^2 ...

(1 - nb/V)^-1 = 1 + nb/V - (nb/V)^2 ..

PV = nRT(1 - an^2/V^2*P + (an^2/V^2*P ) ^2 ...)( 1 + nb/V - (nb/V)^2 .. )

PV = nRT(1 - an^2/V^2*P)( 1 + nb/V )

PV = nRT(1 + nb/V - an^2/V^2P - an^3*b/V^3P )

comparing with virial equaiton

B(T) = b

C(T) = -a/P

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