"Mathematics is a language in which one cannot express unprecise or nebulous thoughts." - Henri Poincaire

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1 CHM 3400 Problem Set 1 Due date: Tuesday, September 2 nd Do all of the following problems. Show your work. (NOTE: Conversion factors between different pressure units are given in Table 0.1 of Atkins. Values for R in various units are given in Table 1.1 of Atkins. Note that 1 dm 3 = 1 L = cm 3 ; L = 1 m 3.) "Mathematics is a language in which one cannot express unprecise or nebulous thoughts." - Henri Poincaire 1) The density of oxygen gas (O 2, M = g/mol) in a new oxygen gas cylinder, measured at T = 20.0 C, is g/ml. Find the following: a) The number density of oxygen molecules in the gas cylinder (in units of molecules/cm 3 ). b) The calculated pressure of gas in the cylinder, using the ideal gas law. Give your answer in units of atm, to four significant figures. c) The calculated pressure of gas in the cylinder, using the van der Waals equation. Give your answer in units of atm, to four significant figures. The van der Waals coefficients for O 2 are a = L 2 atm/mol 2, b = L/mol (Atkins, Table 1.5). d) Which calculated value for pressure (ideal or van der Waals) do you expect to be closer to the true value for gas pressure for the conditions in the problem? Why? 2) At C and 16.0 kpa the mass density of phosphorus vapor is kg/m 3. What is the molecular formula of phosphorus under these conditions? Assume the ideal gas law applies. 3) Consider a mixture of argon (Ar, M = g/mol) and helium (He, M = 4.00 g/mol). The density of the mixture, measured at T = 22.8 C and p = 791. torr, is = g/l. What is X Ar, the mol fraction of argon in the gas mixture? You may assume that argon and helium behave ideally for the conditions in the problem. 4) The atmosphere of Mars is composed almost entirely of carbon dioxide (CO 2, M = g/mol). In this problem we will assume the atmosphere is pure CO 2. At the surface of Mars typical values for pressure and temperature are p = 4.5 torr, T = 220. K. Find the following for these conditions of pressure and temperature: a) c rms, the rms average speed of a CO 2 molcule in the atmosphere at surface level. Give your answer in units of m/s. b), the mean free path of a CO 2 molecule in the atmosphere at surface level. Give your answer in units of nm. c) z, the collision frequency for a CO 2 molecule in the atmosphere at surface level. Give your answer in units of collisions/s. The value for for CO 2 is given in Table 1.3 of Atkins. Note that if all of the quantities appearing in the equations for c rms,, or z are given in MKS units the values calculated using those equations will also have MKS units. Also, c,, and z are related to one another by the relationship = c rms/z. 5) A generic phase diagram for a typical pure chemical substance is given in Fig 5.12 of Atkins. a) What is the physical significance of the critical point in the phase diagram? b) The van der Waals equation is the simplest equation of state that can be used to describe critical behavior for pure substances. Use the values for the van der Waals a and b coefficients for carbon dioxide (Table 1.5 of Atkins) to estimate the values for the critical constants for CO 2 (p C, V C, and T C). Compare your values to the experimental values (p C = atm, V C = 94.0 cm 3 /mol, and T C = K), and briefly discuss the agreement between the experimental values and those predicted using the van der Waals coefficients.

2 6) The Maxwell distribution of molecular speeds in an ideal gas is given by the equation F(s) ds = 4 (M/2 RT) 3/2 s 2 exp(-ms 2 /2RT) ds (6.1) where s is the speed of a molecule (0 s < ) and F(s) ds is the fraction of molecules with speeds between s and s + ds. Starting with the above equation find an expression for c mp, the most probable speed of a molecule in a gas. HINT: The most probable speed corresponds to the peak in a plot of number of molecules vs speed (see Atkins, Fig 1.8). Using the condition for an extreme point in a function you can find an expression for c mp. 7) The van der Waals equation of state can be written p = nrt - an 2 (7.1) (V nb) V 2 a) Show that for a fixed number of moles of gas at a constant temperature, the behavior of a van der Waals gas approaches ideal gas behavior in the limit V. b) Find the value for V for a fixed number of moles of a van der Waals gas at a constant temperature in the limit p (high pressure limit). c) Find the value(s) of p (other than p 0) where the pressure of a van der Waals gas is equal to the pressure predicted by the ideal gas law.

3 Solutions. 1) Before we begin, we will convert the density of the gas (in g/ml) to the molar volume. If we let = m/v be the density of the gas, and M be the molecular mass of the gas, then V m = V = m = M = (32.00 g/mol) = ml/mol = L/mol n n ( g/ml) a) Using the result above, the number density of molecules in the gas is N = 1 mol x molecule = x molecule/cm 3 V cm 3 mole b) From the ideal gas law p = nrt = RT = ( L atm/mol K) ( K) = atm V V m ( L/mol) c) From the van der Waals equation ( a = L 2 atm/mol 2 ; b = L/mol ) p = RT - a (V m - b) V m 2 = ( L atm/mol K) ( K) - (1.364 L 2 atm/mol 2 ) [ ( L/mol) - ( L/mol) ] ( L/mol) 2 = atm atm = atm The percent difference between the value calculated using the ideal gas law and that found using the van der Waals equation of state is % difference = % = 10.5 % which is a significant difference. d) We would expect the value calculated using the van der Waals equation to be closer to the true value for pressure than that found using the ideal gas law, because the van der Waals gas contains corrections for the approximations made in the ideal gas law. 2) If we assume that the vapor obeys the ideal gas law, then n = p = 16.0 x 10 3 N/m 2 = mol/m 3 V RT ( J/mol.K) (373.2 K) The molecular mass is M = (mass density) = x 10 3 g/m 3 = g/mol (molar density) mol/m 3 If we assume the vapor molecules have the formula P j, then (since M(P) = g/mol) j = g/mol = The formula for the vapor molecules is P g/mol

4 3) Assume you have L of gas. Based on the density, the mass of gas you have is m = g. The moles of gas may be found using the ideal gas law n = pv = (791. torr) (1. atm/760. torr) (1.000 L) = mol gas RT ( L atm/mol K) ( K) Let n total = n He + n Ar = mol be the total number of moles of gas. Then n He = n total n Ar m = n HeM He + n Ar M Ar = (n total n Ar)M He + n ArM Ar = n totalm He + n Ar(M Ar M He) or, solving for n Ar n Ar = m n totalm He = [ (0.662 g ( mol) (4.00 g/mol) ] = mol Ar (M Ar M He) [ (39.95 g/mol) (4.00 g/mol) ] The mol fraction of argon in the gas is then X Ar = n Ar = mol = n total mol 4) a) v rms = (3RT/M) 1/2 = [3( J/mol K)(220. K)/(44.01 x 10-3 kg/mol) ] 1/2 = 353. m/s b) N A = x molecule/mol = 0.52 nm 2 = 0.52 x m 2 p = 4.5 torr (1 atm/760 torr) ( x 10 5 Pa/atm) = Pa = RT = ( J/mol K) (220. K) N A p (6.022 x molecule/mol) (0.52 x m 2 ) (600.0Pa) = 9.74 x 10-6 m = nm c) z = c/ = (353. m/s) = 3.63 x 10 7 collisions/s (9.74 x 10-6 m) 5) a) The significance of the critical point is most easily seen by considering the critical temperature, T C. For a substance initially in the gas phase, the substance can be converted from gas to condensed phase (solid or liquid) by an isothermal reversible compression if T < T C. If T > T C an isothermal reversible compression will never result in a phase transition (that is, the properties of the substance will change continuously from those of a gas at low pressure to those of a liquid at high pressure, but without a phase transition ever taking place). p C and V C represent the values for pressure and temperature at the critical temperature on the liquid-gas phase boundary. b) For CO 2, a = L 2. atm/mol 2, and b = L/mol. So (eq 1.24) V C = 3 ( L/mol) = L/mol = cm 3 /mol (experimental value is 94.0 cm 3 /mol) T C = 8a = 8 (3.610 L 2. atm/mol 2 ) 27Rb 27 ( L. atm/mol. K)( L/mol) = K (experimental value is K)

5 p C = a = (3.610 L 2. atm/mol 2 ) = atm (experimental value is atm) 27b 2 27 ( L/mol) 2 The values for the critical temperature and pressure calculated using the van der Waals coefficients are within a few tenths of a percent of the experimental value. The value for V C calculated from the van der Waals coefficients is 37% higher than the experimental value. The van der Waals equation typically does a good job in predicting T C and p C, but usually overestimates the value for V C. 6) The Maxwell distribution for speeds in an ideal gas is given by the expression F(s) ds = 4 (M/2 RT) 3/2 s 2 exp(-ms 2 /2RT) ds = A s 2 exp(-bs 2 ) ds where A = 4 (M/2 RT) 3/2 B = M/2RT At an extreme point in F(s) the first derivative will be equal to zero. So df(s)/ds = 0 = d/ds { A s 2 exp(-bs 2 ) } = A {2s exp(-bs 2 ) + s 2 exp(-bs 2 ) (- 2Bs) } = 2As exp(-bs 2 ) { 1 Bs 2 } The above will be equal to zero when the term in brackets is equal to zero. Therefore 1 Bs 2 = 0 1 = Bs 2 s 2 = 1/B, or s = c mp = (1/B) 1/2 = (2RT/M) 1/2 is the most probable speed for a molecule in an ideal gas. Note that we have, strictly speaking, only found an extreme point for the function. To show that it is a maximum we could use the second derivative test (maximum if d 2 F(s)/ds < 0). However, based on our discussion of the Maxwell distribution in class we know the extreme point we have found is a maximum. 7) a) If we multiply both sides of eq 7.1 by V, we get pv = (nrt) V - an 2 (V nb) V If we look at the right side of the above equation in the limit V V/(V nb) 1 an 2 /V 0 Therefore, in the limit V (nrt) V - an 2 (nrt) (1) nrt (V nb) V and so in this limit, pv nrt, the ideal gas law.

6 b) The van der Waals equation says p = nrt - an 2 (V nb) V 2 If we examine the above equation we can see that as V nb (from above) the first term on the right blows up (becomes infinitely large) while the second term approaches a constant value of a/b 2. Therefore V nb in the limit of high pressure. then c) If the pressure calculated from the ideal gas law and the van der Waals equation of state are the same, p = nrt = nrt - an 2 V (V nb) V 2 If we multiply both sides of this equation by (V/nRT), we get or 1 = V - an (V nb) VRT V = an (V nb) VRT If we cross multiply, we get RTV 2 = an (V nb) = anv abn 2 V 2 (an/rt)v + (abn 2 /RT) = 0 The values of V corresponding to places where the ideal gas law and van der Waals equation give the same result can be found using the quadratic equation V = (an/rt) [(an/rt) 2 (4abn 2 /RT)] 1/2 2 = an { 1 [ 1 (4bRT/a) ] 1/2 } 2RT There are two possibilities 1) If (4bRT/a) > 1 (that is, if T > a/4br) then the square root term above will be the square root of a negative number, and there will be no place (other than p 0) where the ideal gas law and the van der Waals equation predict the same value for pressure. 2) If (4bRT/a) < 1 (that is, if T < a/4br) then the square root term above will be the square root of a positive number, and there will be two values of V where the ideal gas law and van der Waals equation predict the same value for pressure. This will correspond to values of pressure given by or, finally p = nrt = (nrt) 2RT 1 V an { 1 [ 1 (4bRT/a) ] 1/2 } p = 2R 2 T 2 a { 1 [ 1 (4bRT/a) ] 1/2 }