Calculate the diffusion constant of argon at 25C and (a) 1.00 Pa, (b) 100 kPa, (c) 10.0

Use the kinetic theory to justify the following observations: (a) the rate of a reaction in the gas phase depends on the energy with which two molecules collide, which in turn depends on their speeds; (b) in the Earths atmosphere, light gases, such as H2 and He, are rare but heavier gases, such as O2, CO2, and N2, are abundant.

Provide a molecular interpretation for each of the following processes: diffusion, thermal conduction, electric conduction, and viscosity.

Provide a molecular interpretation for the observation that the viscosity of a gas increases with temperature, whereas the viscosity of a liquid decreases with increasing temperature.

Discuss the mechanism of proton conduction in liquid water.

Limit the generality of the following expressions: (a) \(J=-D(\mathrm{~d} c / \mathrm{d} x)\), (b) \(D=k T / f\), and (c) \(D=k T / 6 \pi \eta a\).

Provide a molecular interpretation for the observation that mediated transport across a biological membrane leads to a maximum flux Jmax when the concentration of the transported species becomes very large.

Discuss how nuclear magnetic resonance spectroscopy, inelastic neutron scattering, and dynamic light scattering may be used to measure the mobility of molecules in liquids.

Determine the ratios of (a) the mean speeds, (b) the mean kinetic energies of H2 molecules and Hg atoms at 20C.

Determine the ratios of (a) the mean speeds, (b) the mean kinetic energies of He atoms and Hg atoms at 25C.

A 1.0 dm3 glass bulb contains 1.0 1023 H2 molecules. If the pressure exerted by the gas is 100 kPa, what are (a) the temperature of the gas, (b) the root mean square speeds of the molecules? (c) Would the temperature be different if they were O2 molecules?

The best laboratory vacuum pump can generate a vacuum of about 1 nTorr. At 25C and assuming that air consists of N2 molecules with a collision diameter of 395 pm, calculate (a) the mean speed of the molecules, (b) the mean free path, (c) the collision frequency in the gas.

At what pressure does the mean free path of argon at \(25^{\circ} \mathrm{C}\) become comparable to the size of a \(1 \mathrm{dm}^3\) vessel that contains it? Take \(\sigma=0.36 \mathrm{~nm}^2\).

At what pressure does the mean free path of argon at 25C become comparable to the diameters of the atoms themselves?

At an altitude of 20 km the temperature is 217 K and the pressure 0.050 atm. What is the mean free path of N2 molecules? ( = 0.43 nm2.)

At an altitude of 15 km the temperature is 217 K and the pressure 12.1 kPa. What is the mean free path of N2 molecules? ( = 0.43 nm2.)

How many collisions does a single Ar atom make in 1.0 s when the temperature is 25C and the pressure is (a) 10 atm, (b) 1.0 atm, (c) 1.0 atm?

How many collisions per second does an N2 molecule make at an altitude of 15 km? (See Exercise 21.4b for data.)

Calculate the mean free path of molecules in air using = 0.43 nm2 at 25C and (a) 10 atm, (b) 1.0 atm, (c) 1.0 atm.

Calculate the mean free path of carbon dioxide molecules using = 0.52 nm2 at 25C and (a) 15 atm, (b) 1.0 bar, (c) 1.0 Torr.

Use the Maxwell distribution of speeds to estimate the fraction of N2 molecules at 500 K that have speeds in the range 290 to 300 m s1.

Use the Maxwell distribution of speeds to estimate the fraction of CO2 molecules at 300 K that have speeds in the range 200 to 250 m s1.

A solid surface with dimensions 2.5 mm 3.0 mm is exposed to argon gas at 90 Pa and 500 K. How many collisions do the Ar atoms make with this surface in 15 s?

A solid surface with dimensions 3.5 mm 4.0 cm is exposed to helium gas at 111 Pa and 1500 K. How many collisions do the He atoms make with this surface in 10 s?

An effusion cell has a circular hole of diameter 2.50 mm. If the molar mass of the solid in the cell is 260 g mol1 and its vapour pressure is 0.835 Pa at 400 K, by how much will the mass of the solid decrease in a period of 2.00 h?

An effusion cell has a circular hole of diameter 3.00 mm. If the molar mass of the solid in the cell is 300 g mol1 and its vapour pressure is 0.224 Pa at 450 K, by how much will the mass of the solid decrease in a period of 24.00 h?

A manometer was connected to a bulb containing carbon dioxide under slight pressure. The gas was allowed to escape through a small pinhole, and the time for the manometer reading to drop from 75 cm to 50 cm was 52 s. When the experiment was repeated using nitrogen (for which M= 28.02 g mol1) the same fall took place in 42 s. Calculate the molar mass of carbon dioxide.

A manometer was connected to a bulb containing nitrogen under slight pressure. The gas was allowed to escape through a small pinhole, and the time for the manometer reading to drop from 65.1 cm to 42.1 cm was 18.5 s. When the experiment was repeated using a fluorocarbon gas, the same fall took place in 82.3 s. Calculate the molar mass of the fluorocarbon.

A space vehicle of internal volume 3.0 m3 is struck by a meteor and a hole of radius 0.10 mm is formed. If the oxygen pressure within the vehicle is initially 80 kPa and its temperature 298 K, how long will the pressure take to fall to 70 kPa?

A container of internal volume 22.0 m3 was punctured, and a hole of radius 0.050 mm was formed. If the nitrogen pressure within the vehicle is initially 122 kPa and its temperature 293 K, how long will the pressure take to fall to 105 kPa?

Calculate the flux of energy arising from a temperature gradient of \(2.5 \mathrm{~K} \mathrm{~m}^{-1}\) in a sample of argon in which the mean temperature is \(273 \mathrm{~K}\).

Calculate the flux of energy arising from a temperature gradient of 3.5 K m1 in a sample of hydrogen in which the mean temperature is 260 K.

Use the experimental value of the thermal conductivity of neon (Table 21.2) to estimate the collision cross-section of Ne atoms at 273 K.

Use the experimental value of the thermal conductivity of nitrogen (Table 21.2) to estimate the collision cross-section of N2 molecules at 298 K.

In a double-glazed window, the panes of glass are separated by 5.0 cm. What is the rate of transfer of heat by conduction from the warm room (25C) to the cold exterior (10C) through a window of area 1.0 m2? What power of heater is required to make good the loss of heat?

Two sheets of copper of area 1.50 m2 are separated by 10.0 cm. What is the rate of transfer of heat by conduction from the warm sheet (50C) to the cold sheet (10C). What is the rate of loss of heat?

Use the experimental value of the coefficient of viscosity for neon (Table 21.2) to estimate the collision cross-section of Ne atoms at 273 K.

Use the experimental value of the coefficient of viscosity for nitrogen (Table 21.2) to estimate the collision cross-section of the molecules at 273 K.

Calculate the inlet pressure required to maintain a flow rate of 9.5 105 dm3 h1 of nitrogen at 293 K flowing through a pipe of length 8.50 m and diameter 1.00 cm. The pressure of gas as it leaves the tube is 1.00 bar. The volume of the gas is measured at that pressure.

Calculate the inlet pressure required to maintain a flow rate of 8.70 cm3 s1 of nitrogen at 300 K flowing through a pipe of length 10.5 m and diameter 15 mm. The pressure of gas as it leaves the tube is 1.00 bar. The volume of the gas is measured at that pressure.

Calculate the viscosity of air at (a) 273 K, (b) 298 K, (c) 1000 K. Take 0.40 nm2. (The experimental values are 173 P at 273 K, 182 P at 20C, and 394 P at 600C.)

Calculate the viscosity of benzene vapour at (a) 273 K, (b) 298 K, (c) 1000 K. Take 0.88 nm2.

Calculate the thermal conductivities of (a) argon, (b) helium at 300 K and 1.0 mbar. Each gas is confined in a cubic vessel of side 10 cm, one wall being at 310 K and the one opposite at 295 K. What is the rate of flow of energy as heat from one wall to the other in each case?

Calculate the thermal conductivities of (a) neon, (b) nitrogen at 300 K and 15 mbar. Each gas is confined in a cubic vessel of side 15 cm, one wall being at 305 K and the one opposite at 295 K. What is the rate of flow of energy as heat from one wall to the other in each case?

The viscosity of carbon dioxide was measured by comparing its rate of flow through a long narrow tube (using Poiseuille's formula) with that of argon. For the same pressure differential, the same volume of carbon dioxide passed through the tube in \(55 \mathrm{~s}\) as argon in \(83 \mathrm{~s}\). The viscosity of argon at \(25^{\circ} \mathrm{C}\) is \(208 \mu \mathrm{P}\); what is the viscosity of carbon dioxide? Estimate the molecular diameter of carbon dioxide.

The viscosity of a chlorofluorocarbon (CFC) was measured by comparing its rate of flow through a long narrow tube (using Poiseuilles formula) with that of argon. For the same pressure differential, the same volume of the CFC passed through the tube in 72.0 s as argon in 18.0 s. The viscosity of argon at 25C is 208 P; what is the viscosity of the CFC? Estimate the molecular diameter of the CFC. Take M= 200 g mol1.

Calculate the thermal conductivity of argon (CV,m = 12.5 J K1 mol1, = 0.36 nm2) at room temperature (20C).

Calculate the thermal conductivity of nitrogen (CV,m = 20.8 J K1 mol1, = 0.43 nm2) at room temperature (20C).

Calculate the diffusion constant of argon at 25C and (a) 1.00 Pa, (b) 100 kPa, (c) 10.0 MPa. If a pressure gradient of 0.10 atm cm1 is established in a pipe, what is the flow of gas due to diffusion?

Calculate the diffusion constant of nitrogen at 25C and (a) 10.0 Pa, (b) 100 kPa, (c) 15.0 MPa. If a pressure gradient of 0.20 bar m1 is established in a pipe, what is the flow of gas due to diffusion?

The mobility of a chloride ion in aqueous solution at 25C is 7.91 108 m2 s1 V1. Calculate the molar ionic conductivity.

The mobility of an acetate ion in aqueous solution at 25C is 4.24 108 m2 s1 V1. Calculate the molar ionic conductivity.

The mobility of a Rb+ ion in aqueous solution is 7.92 108 m2 s1 V1 at 25C. The potential difference between two electrodes placed in the solution is 35.0 V. If the electrodes are 8.00 mm apart, what is the drift speed of the Rb+ ion?

The mobility of a Li+ ion in aqueous solution is 4.01 108 m2 s1 V1 at 25C. The potential difference between two electrodes placed in the solution is 12.0 V. If the electrodes are 1.00 cm apart, what is the drift speed of the ion?

What fraction of the total current is carried by Li+ when current flows through an aqueous solution of LiBr at 25C?

What fraction of the total current is carried by Cl when current flows through an aqueous solution of NaCl at 25C?

The limiting molar conductivities of KCl, KNO3, and AgNO3 are 14.99 mS m2 mol1, 14.50 mS m2 mol1, and 13.34 mS m2 mol1, respectively (all at 25C). What is the limiting molar conductivity of AgCl at this temperature?

The limiting molar conductivities of NaI, NaCH3CO2, and Mg(CH3CO2)2 are 12.69 mS m2 mol1, 9.10 mS m2 mol1, and 18.78 mS m2 mol1, respectively (all at 25C). What is the limiting molar conductivity of MgI2 at this temperature?

At 25C the molar ionic conductivities of Li+, Na+, and K+ are 3.87 mS m2 mol1, 5.01 mS m2 mol1, and 7.35 mS m2 mol1, respectively. What are their mobilities?

At 25C the molar ionic conductivities of F, Cl, and Br are 5.54 mS m2 mol1, 7.635 mS m2 mol1, and 7.81 mS m2 mol1, respectively. What are their mobilities?

The mobility of a NO3 ion in aqueous solution at 25C is 7.40 108 m2 s1 V1. Calculate its diffusion coefficient in water at 25C.

The mobility of a \(\mathrm{CH}_3 \mathrm{CO}_2^{-}\) ion in aqueous solution at \(25^{\circ} \mathrm{C}\) is \(4.24 \times 10^{-8} \mathrm{~m}^2 \mathrm{~s}^{-1} \mathrm{~V}^{-1}\). Calculate its diffusion coefficient in water at \(25^{\circ} \mathrm{C}\).

The diffusion coefficient of \(\mathrm{CCl}_4\) in heptane at \(25^{\circ} \mathrm{C}\) is \(3.17 \times 10^{-9} \mathrm{~m}^2 \mathrm{s}^{-1}\). Estimate the time required for a \(\mathrm{CCl}_4\) molecule to have a root mean square displacement of \(5.0 \mathrm{~mm}\).

The diffusion coefficient of I2 in hexane at 25C is 4.05 109 m2 s1. Estimate the time required for an iodine molecule to have a root mean square displacement of 1.0 cm.

Estimate the effective radius of a sucrose molecule in water 25C given that its diffusion coefficient is 5.2 1010 m2 s1 and that the viscosity of water is 1.00 cP.

Estimate the effective radius of a glycine molecule in water at \(25^{\circ} \mathrm{C}\) given that its diffusion coefficient is \(1.055 \times 10^{-9} \mathrm{~m}^2 \mathrm{~s}^{-1}\) and that the viscosity of water is \(1.00 \mathrm{cP}\).

The diffusion coefficient for molecular iodine in benzene is 2.13 109 m2 s1. How long does a molecule take to jump through about one molecular diameter (approximately the fundamental jump length for translational motion)?

The diffusion coefficient for CCl4 in heptane is 3.17 109 m2 s1. How long does a molecule take to jump through about one molecular diameter (approximately the fundamental jump length for translational motion)?

What are the root mean square distances travelled by an iodine molecule in benzene and by a sucrose molecule in water at 25C in 1.0 s?

About how long, on average, does it take for the molecules in Exercise 21.31a to drift to a point (a) 1.0 mm, (b) 1.0 cm from their starting points?

Conductivities are often measured by comparing the resistance of a cell filled with the sample to its resistance when filled with some standard solution, such as aqueous potassium chloride. The conductivity of water is 76 mS m1 at 25C and the conductivity of 0.100 mol dm3 KCl(aq) is 1.1639 S m1. A cell had a resistance of 33.21 when filled with 0.100 mol dm3 KCl(aq) and 300.0 when filled with 0.100 mol dm3 CH3COOH. What is the molar conductivity of acetic acid at that concentration and temperature?

The resistances of a series of aqueous NaCl solutions, formed by successive dilution of a sample, were measured in a cell with cell constant (the constant C in the relation = C/R) equal to 0.2063 cm1. The following values were found: c/(mol dm3) 0.00050 0.0010 0.0050 0.010 0.020 0.050 R/ 3314 1669 342.1 174.1 89.08 37.14 Verify that the molar conductivity follows the Kohlrausch law and find the limiting molar conductivity. Determine the coefficient K . Use the value of K (which should depend only on the nature, not the identity of the ions) and the information that (Na+) = 5.01 mS m2 mol1 and (I) = 7.68 mS m2 mol1 to predict (a) the molar conductivity, (b) the conductivity, (c) the resistance it would show in the cell, of 0.010 mol dm3 NaI(aq) at 25C.

After correction for the water conductivity, the conductivity of a saturated aqueous solution of AgCl at 25C was found to be 0.1887 mS m1. What is the solubility of silver chloride at this temperature?

What are the drift speeds of Li+, Na+, and K+ in water when a potential difference of 10 V is applied across a 1.00-cm conductivity cell? How long would it take an ion to move from one electrode to the other? In conductivity measurements it is normal to use alternating current: what are the displacements of the ions in (a) centimetres, (b) solvent diameters, about 300 pm, during a half cycle of 1.0 kHz applied potential?

The mobilities of H+ and Cl at 25C in water are 3.623 107 m2 s1 V1 and 7.91 108 m2 s1 V1, respectively. What proportion of the current is carried by the protons in 103 m HCl(aq)? What fraction do they carry when the NaCl is added to the acid so that the solution is 1.0 mol dm3 in the salt? Note how concentration as well as mobility governs the transport of current.

In a moving boundary experiment on KCl the apparatus consisted of a tube of internal diameter 4.146 mm, and it contained aqueous KCl at a concentration of 0.021 mol dm3. A steady current of 18.2 mA was passed, and the boundary advanced as follows: t/s 200 400 600 800 1000 x/mm 64 128 192 254 318 Find the transport number of K+, its mobility, and its ionic conductivity.

The proton possesses abnormal mobility in water, but does it behave normally in liquid ammonia? To investigate this question, a movingboundary technique was used to determine the transport number of NH4 + in liquid ammonia (the analogue of H3O+ in liquid water) at 40C (J. Baldwin, J. Evans, and J.B. Gill, J. Chem. Soc. A, 3389 (1971)). A steady current of 5.000 mA was passed for 2500 s, during which time the boundary formed between mercury(II) iodide and ammonium iodide solutions in ammonia moved 286.9 mm in a 0.013 65 mol kg1 solution and 92.03 mm in a 0.042 55 mol kg1 solution. Calculate the transport number of NH4 + at these concentrations, and comment on the mobility of the proton in liquid ammonia. The bore of the tube is 4.146 mm and the density of liquid ammonia is 0.682 g cm3.

A dilute solution of potassium permanganate in water at 25C was prepared. The solution was in a horizontal tube of length 10 cm, and at first there was a linear gradation of intensity of the purple solution from the left (where the concentration was 0.100 mol dm3) to the right (where the concentration was 0.050 mol dm3). What is the magnitude and sign of the thermodynamic force acting on the solute (a) close to the left face of the container, (b) in the middle, (c) close to the right face? Give the force per mole and force per molecule in each case.

Estimate the diffusion coefficients and the effective hydrodynamic radii of the alkali metal cations in water from their mobilities at 25C. Estimate the approximate number of water molecules that are dragged along by the cations. Ionic radii are given Table 20.3.

Nuclear magnetic resonance can be used to determine the mobility of molecules in liquids. A set of measurements on methane in carbon tetrachloride showed that its diffusion coefficient is 2.05 109 m2 s1 at 0C and 2.89 109 m2 s1 at 25C. Deduce what information you can about the mobility of methane in carbon tetrachloride.

A concentrated sucrose solution is poured into a cylinder of diameter 5.0 cm. The solution consisted of 10 g of sugar in 5.0 cm3 of water. A further 1.0 dm3 of water is then poured very carefully on top of the layer, without disturbing the layer. Ignore gravitational effects, and pay attention only to diffusional processes. Find the concentration at 5.0 cm above the lower layer after a lapse of (a) 10 s, (b) 1.0 y.

In a series of observations on the displacement of rubber latex spheres of radius 0.212 m, the mean square displacements after selected time intervals were on average as follows: t/s 30 60 90 120 1012_x2 _/m2 88.2 113.5 128 144 These results were originally used to find the value of Avogadros constant, but there are now better ways of determining NA, so the data can be used to find another quantity. Find the effective viscosity of water at the temperature of this experiment (25C).

A.K. Srivastava, R.A. Samant, and S.D. Patankar (J. Chem. Eng. Data 41, 431 (1996)) measured the conductance of several salts in a binary solvent mixture of water and a dipolar aprotic solvent 1,3-dioxolan-2-one (ethylene carbonate). They report the following conductances at 25C in a solvent 80 per cent 1,3-dioxolan-2-one by mass: NaI c/(mmol dm3) 32.02 20.28 12.06 8.64 2.85 1.24 0.83 m/(S cm2 mol1) 50.26 51.99 54.01 55.75 57.99 58.44 58.67 KI c/(mmol dm3) 17.68 10.8 87.19 2.67 1.28 0.83 0.19 m /(S cm2 mol1) 42.45 45.91 47.53 51.81 54.09 55.78 57.42 Calculatem for NaI and KI in this solvent and (Na) (K). Compare your results to the analogous quantities in aqueous solution using Table 21.5 in the Data section.

A. Fenghour, W.A. Wakeham, V. Vesovic, J.T.R. Watson, J. Millat, and E. Vogel (J. Phys. Chem. Ref. Data 24, 1649 (1995)) have compiled an extensive table of viscosity coefficients for ammonia in the liquid and vapour phases. Deduce the effective molecular diameter of NH3 based on each of the following vapour-phase viscosity coefficients: (a) = 9.08 106 kg m1 s1 at 270 K and 1.00 bar; (b) = 1.749 105 kg m1 s1 at 490 K and 10.0 bar.

G. Bakale, K. Lacmann, and W.F. Schmidt ( J. Phys. Chem. 100, 12477 (1996)) measured the mobility of singly charged C 60 ions in a variety of nonpolar solvents. In cyclohexane at 22C, the mobility is 1.1 cm2 V1 s1. Estimate the effective radius of the C 60 ion. The viscosity of the solvent is 0.93 103 kg m1 s1. Comment. The researchers interpreted the substantial difference between this number and the van der Waals radius of neutral C60 in terms of a solvation layer around the ion.

Start from the MaxwellBoltzmann distribution and derive an expression for the most probable speed of a gas of molecules at a temperature T. Go on to demonstrate the validity of the equipartition conclusion that the average translational kinetic energy of molecules free to move in three dimensions is 32kT.

Consider molecules that are confined to move in a plane (a twodimensional gas). Calculate the distribution of speeds and determine the mean speed of the molecules at a temperature T.

A specially constructed velocity-selector accepts a beam of molecules from an oven at a temperature T but blocks the passage of molecules with a speed greater than the mean. What is the mean speed of the emerging beam, relative to the initial value, treated as a one-dimensional problem?

What is the proportion of gas molecules having (a) more than, (b) less than the root mean square speed? (c) What are the proportions having speeds greater and smaller than the mean speed?

Calculate the fractions of molecules in a gas that have a speed in a range v at the speed nc* relative to those in the same range at c* itself? This calculation can be used to estimate the fraction of very energetic molecules (which is important for reactions). Evaluate the ratio for n = 3 and n = 4.

Derive an expression that shows how the pressure of a gas inside an effusion oven (a heated chamber with a small hole in one wall) varies with time if the oven is not replenished as the gas escapes. Then show that t1/2, the time required for the pressure to decrease to half its initial value, is independent of the initial pressure. Hint. Begin by setting up a differential equation relating dp/dt to p = NkT/V, and then integrating it.

Show how the ratio of two transport numbers t and t for two cations in a mixture depends on their concentrations c and c and their mobilities u and u.

Confirm that eqn 21.72 is a solution of the diffusion equation with the correct initial value.

The diffusion equation is valid when many elementary steps are taken in the time interval of interest, but the random walk calculation lets us discuss distributions for short times as well as for long. Use eqn 21.84 to calculate the probability of being six paces from the origin (that is, at x = 6) after (a) four, (b) six, (c) twelve steps.

Use mathematical software to calculate P in a one-dimensional random walk, and evaluate the probability of being at x = n for n = 6, 10, 14, . . . , 60. Compare the numerical value with the analytical value in the limit of a large number of steps. At what value of n is the discrepancy no more than 0.1 per cent?

Supply the intermediate mathematical steps in Justification 21.7.

A dilute solution of a weak (1,1)-electrolyte contains both neutral ion pairs and ions in equilibrium \(\left(\mathrm{AB} \rightleftharpoons \mathrm{A}^{+}+\mathrm{B}^{-}\right)\). Prove that molar conductivities are related to the degree of ionization by the equations: \(\frac{1}{\Lambda_{\mathrm{m}}}=\frac{1}{\Lambda_{\mathrm{m}}(\alpha)}+\frac{(1-\alpha) \Lambda_{\mathrm{m}}^{\circ}}{\alpha^2 \Lambda_{\mathrm{m}}(\alpha)^2}\) \(\Lambda_{\mathrm{m}}(\alpha)=\lambda_{+}+\lambda_{-}=\Lambda_{\mathrm{m}}^{\circ}-\mathcal{K}(\alpha c)^{1 / 2}\) where \(\Lambda_{\mathrm{m}}^{\circ}\) is the molar conductivity at infinite dilution and \(\mathscr{K}\) is the constant in Kohlrausch's law (eqn 21.29).

Calculate the escape velocity (the minimum initial velocity that will take an object to infinity) from the surface of a planet of radius R. What is the value for (a) the Earth, R = 6.37 106 m, g = 9.81 m s2, (b) Mars, R = 3.38 106 m, mMars /mEarth = 0.108. At what temperatures do H2, He, and O2 molecules have mean speeds equal to their escape speeds? What proportion of the molecules have enough speed to escape when the temperature is (a) 240 K, (b) 1500 K? Calculations of this kind are very important in considering the composition of planetary atmospheres.

Interstellar space is a medium quite different from the gaseous environments we commonly encounter on Earth. For instance, a typical density of the medium is about 1 atom cm3 and that atom is typically H; the effective temperature due to stellar background radiation is about 10 000 K. Estimate the diffusion coefficient and thermal conductivity of H under these conditions. Comment. Energy is in fact transferred much more effectively by radiation.

The principal components of the atmosphere of the Earth are diatomic molecules, which can rotate as well as translate. Given that the translational kinetic energy density of the atmosphere is 0.15 J cm3, what is the total kinetic energy density, including rotation?

In the standard model of stellar structure (I. Nicholson, The sun. Rand McNally, New York (1982)), the interior of the Sun is thought to consist of 36 per cent H and 64 per cent He by mass, at a density of 158 g cm3. Both atoms are completely ionized. The approximate dimensions of the nuclei can be calculated from the formula rnucleus = 1.4 1015 A1/3 m, where A is the mass number. The size of the free electron, re 1018 m, is negligible compared to the size of the nuclei. (a) Calculate the excluded volume in 1.0 cm3 of the stellar interior and on that basis decide upon the applicability of the perfect gas law to this system. (b) The standard model suggests that the pressure in the stellar interior is 2.5 1011 atm. Calculate the temperature of the Suns interior based on the perfect gas model. The generally accepted standard model value is 16 MK. (c) Would a van der Waals type of equation (with a = 0) give a better value for T?

Enrico Fermi, the great Italian scientist, was a master at making good approximate calculations based on little or no actual data. Hence, such calculations are often called Fermi calculations. Do a Fermi calculation on how long it would take for a gaseous air-borne cold virus of molar mass 100 kg mol1 to travel the distance between two conversing people 1.0 m a part by diffusion in still air.

The diffusion coefficient of a particular kind of t-RNA molecule is D = 1.0 1011 m2 s1 in the medium of a cell interior. How long does it take molecules produced in the cell nucleus to reach the walls of the cell at a distance 1.0 m, corresponding to the radius of the cell?

In this problem, we examine a model for the transport of oxygen from air in the lungs to blood. First, show that, for the initial and boundary conditions c(x,t) = c(x,0) = c0, (0 < x < ) and c(0,t) = cs, (0 t) where c0 and cs are constants, the concentration, c(x,t), of a species is given by c(x,t) = c0 + (cs c0){1 erf} (x,t) = where erf is the error function (Justification 9.4) and the concentration c(x,t) evolves by diffusion from the yz-plane of constant concentration, such as might occur if a condensed phase is absorbing a species from a gas phase. Now draw graphs of concentration profiles at several different times of your choice for the diffusion of oxygen into water at 298 K (when D = 2.10 109m2 s1) on a spatial scale comparable to passage of oxygen from lungs through alveoli into the blood. Use c0 = 0 and set cs equal to the solubility of oxygen in water. Hint. Use mathematical software.