36 1 The Behavior of Gases and LiquidsWhen two phases of a single substance are at equilibrium, the pressure is a function
only of the temperature. A phase diagram for a pure substance contains three curves
representing this dependence for the solid–liquid, solid–gas, and liquid–gas equilibria.
These three curves meet at a point called the triple point. The liquid–vapor coexistence
curve terminates at the critical point. Above the critical temperature, no gas–liquid
phase transition occurs and there is only one fluid phase. The law of corresponding
states was introduced, according to which all substances obey the same equation of
state in terms of reduced variablesADDITIONAL PROBLEMS
1.45 Assume that when Julius Caesar exhaled for the last time
he exhaled 1.0 L of air.
a.Estimate the number of nitrogen molecules that were
exhaled in Julius Caesar’s last breath. The mole fraction
of nitrogen in air is approximately 0.78. (The mole
fraction of a substance is its amount in moles divided
by the total amount of all substances.)
b.Estimate the number of nitrogen molecules out of those
in part a that are now present in the physical chemistry
classroom. Assume that sufficient time has elapsed for
the nitrogen molecules that Julius Caesar exhaled to mix
randomly in the entire atmosphere of the earth. Take the
average barometric pressure as 1.00 bar and estimate
the total mass of the atmosphere from its pressure and
the area of the earth. The radius of the earth is roughly
4000 miles. State any additional assumptions, including
your estimates of the room dimensions.1.46 a.Manipulate the van der Waals equation of state into the
virial form of Eq. (1.3-3). Use the identity
1
1 −x 1 +x+x^2 +x^3 + ···b.For each of the temperatures in Table A.4 at which a
value of the second virial coefficient is given for argon,
calculate the value of the second virial coefficient from
the values of the van der Waals parameters. Calculate
the percent error for each value, assuming that the
values in Table A.4 are correct.
c.In terms of intermolecular forces, what does it mean
when a second virial coefficient is positive? What does
it mean when a second virial coefficient is negative?
Draw a graph of the second virial coefficient of argon
as a function of temperature, and comment on the
temperature dependence.d.Calculate the value of the third virial coefficient of
argon at 0◦C and at 50◦C, assuming that the van der
Waals equation is a correct description.
e.Calculate the value of the compression factor of argon
at 0◦C and a molar volume of 2.271 L mol−^1 .Doit
once using the ideal gas equation, once using the van
der Waals equation, once using the virial equation of
state truncated at the second virial coefficient and using
the correct value of the second virial coefficient, once
using the virial equation of state truncated at the second
virial coefficient and using the value of the second
virial coefficient from the van der Waals parameters,
and once using the virial equation of state truncated at
the third virial coefficient and using the values of the
virial coefficients from the van der Waals parameters.
1.47 The volume of a sample of liquid water at constant
pressure can be represented by the formulaV(tC)V(0◦C)(1+α′tC+β′t^2 C+γ′tC^3 +δ′tC^4 )whereα′,β′,γ′, andδ′are constants andtCis the Celsius
temperature.
a.Find an expression for the coefficient of thermal
expansion as a function oftC.
b.Two different sets of values are used: The first set is
said to be valid from 0◦Cto33◦C:α′− 6. 4268 × 10 −^5 (◦C)−^1 ,
β′ 8. 505266 × 10 −^6 (◦C)−^2 ,
γ′− 6. 78977 × 10 −^8 (◦C)−^3 ,
δ′ 4. 01209 × 10 −^10 (◦C)−^4.The second set is said to be valid from 0◦Cto80◦C: