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physical chemistry
Physical Chemistry 3rd edition Thomas Engel, Philip Reid - Solutions
Using the distribution of particle translational kinetic energy provided in Problem P33.19, derive expressions for the average and most-probable translational kinetic energies for a collection of gaseous particles.
Starting with the Maxwell speed distribution, demonstrate that the probability distribution for translational kinetic energy for εTr>> kT is given by: 3/2 -е, кт /RT 1/2 dɛr SE,Jder, - 2л ет deр лkT
Derive the Maxwell–Boltzmann distribution using the Boltzmann distribution introduced in statistical mechanics. Begin by developing the expression for the distribution in translational kinetic energy in one dimension and then extend it to three dimensions.
Demonstrate that the Maxwell–Boltzmann speed distribution is normalized.
A molecular beam apparatus employs supersonic jets that allow gas molecules to expand from a gas reservoir held at a specific temperature and pressure into a vacuum through a small orifice. Expansion of the gas allows for achieving internal temperatures of roughly 10 K. The expansion can be treated
For N2 at 298 K, what fraction of molecules has a speed between 200. and 300.m s–1? What is this fraction if the gas temperature is 500. K?
The escape velocity from the Earth’s surface is given by vE = (2gR)1/2, where g is gravitational acceleration (9.807 m s–2) and R is the radius of the Earth (6.37 × 106 m).a. At what temperature will νmp for N2 be equal to the escape velocity?b. How does the answer for part (a) change if the
For O2 at 1 atm and 298 K, what fraction of molecules has a speed that is greater than vrms?
The speed of sound is given by V sound = √ykT / m = √yRT / M, where y = Cp / Cv.a. What is the speed of sound in Ne, Kr, and Ar at 1000. K?b. At what temperature will the speed of sound in Kr equal the speed of sound in Ar at 1000. K?x. MNe = 0.0202 kg mol–1 MKr = 0.0840 kg mol–1 MAr
The probability that a particle will have a velocity in the x direction in the range of ˆ’vx0and vx0is given by m? /2kT dv% 1/2 f(-v, s V, s V, ) = 2rkT 1/2 v mv /2kT dv x rkT
Determine the temperature at which ν ave for Kr is equal to that of Ne at 298 K.
At what temperature is the νrms of Ar equal to that of SF6 at 298 K? Perform the same calculation for νmp.
As mentioned in Section 33.3, the only differences between the quantities νmp, νave, and νrms involve constants.a. Derive the expressions for νave and vrms relative to νmp provided in the text.b. Your result from part (a) will involve quantities that are independent of gas-specific quantities
a. What is the average time required for H2 to travel 1.00 meter at 298 K and 1 atm?b. How much longer does it take N2 to travel 1 m on average relative to H2 under these same conditions?c. (Challenging) What fraction of N2 particles will require more than this average time to travel 1
How far on average does O2 travel in 1 second at 298 K and 1 atm? How does this distance compare to that of Kr under identical conditions?
Compare the average speed and average kinetic energy of O2 with that of CCl4 at 298 K.Mo2 = 0.0320 kg mol-1 MccI4 = 0.154 kg mol-1
Compute νave for H2O, HOD, and D2O. Do you need to perform the same calculation each time, or can you derive an expression that relates the ratio of average speeds for two gases to their respective masses?
Compute νmp, νave, and νrms for O2 at 300. and 500. K. How would your answers change for H2?
Determine νmp ,νave, and νrms for the following species at 298 K:a. Neb. Krc. CH4d. C2H6e. C60Note: J kg−1 ≡ m2 s−2.
For carbon monoxide, the calculated molar entropy was more negative than the experimental value. Why?
Consider the following energy levels and associated degeneracies for atomic Fe:a. Determine the electronic contribution to CV for atomic Fe at 150. K assuming that only the first two levels contribute to CV .b. How does your answer to part (a) change if the n = 2 level is included in the
The speed of sound is given by the relationshipwhere CP is the constant pressure heat capacity (equal to CV + R), R is the ideal gas constant, T is temperature, and M is molar mass.a. What is the expression for the speed of sound for an ideal monatomic gas?b. What is the expression for the speed of
The measured molar heat capacities for crystalline KCl are as follows at the indicated temperatures:a. Explain why the high-temperature limit for CV is apparently twofold greater than that predicted by the Dulong€“Petit law.b. Determine if the Einstein model is applicable to ionic solids.
The molar constant volume heat capacity for I2 (g) is 28.6 J mol−1 K−1. What is the vibrational contribution to the heat capacity? You can assume that the contribution from the electronic degrees of freedom is negligible.
Inspection of the thermodynamic tables in the back of the text reveals that many molecules have quite similar constant volume heat capacities.a. The value of CV, m for Ar (g) at standard temperature and pressure is 12.48 J mol–1 K–1, identical to gaseous He (g). Using statistical mechanics,
Consider rotation about the C~C bond in ethane. A crude model for torsion about this bond is the €œfree rotor€ model where rotation is considered unhindered. In this model the energy levels along the torsional degree of freedom are given by:In this expression I is the moment of inertia.
Determine the molar entropy for 1 mol of gaseous Ar at 200, 300, and 500. K and V = 1000 cm3 assuming that Ar can be treated as an ideal gas. How does the result of this calculation change if the gas is Kr instead of Ar?
The standard molar entropy of O2 is 205.14J mol–1 K–1 at P = 1.00 atm. Using this information, determine the bond length of O2. For this molecule, v̅ = 1580.cm−1, and the ground electronic state degeneracy is three.
Determine the standard molar entropy of N2O, a linear triatomic molecule at 298 K and P = 1.00 atm. For this molecule, B = 0.419 cm–1 and v̅1 = 1285 cm-1, v̅2 = 589 cm-1 (doubly degenerate), and v̅3 = 2224 cm-1.
Determine the standard molar entropy of OClO at 298.15 K and P = 1.00 atm, a nonlinear triatomic molecule where BA = 1.06 cm–1, BB = 0.31cm–1, BC = 0.29 cm–1 and v̅1 = 938 cm-1, ν̅2 = 450 cm-1, and ν̅3 = 1100 cm-1.
Determine the standard molar entropy for the hydroxyl radical, OH•, for which v̅ = 3735 cm−1 and B = 18.9 cm−1, and the ground electronic state is doubly degenerate. P = 1.00 atm.
Determine the standard molar entropy of N2 (ν̅ = 2359 cm-1 and B = 2.00 cm-1, g0 = 1) and the entropy when P = 1.00 atm, but T = 2500.K.
Define the mean free path. How does this quantity vary with number density, particle diameter, and average particle speed?
Determine the standard molar entropy of H35Cl at 298 K, where B = 10.58 cm–1, ν̅ = 2886 cm−1, the ground-state electronic level degeneracy is 1, and P = 1.00 atm.
Derive the expression for the standard molar entropy of a monatomic gas restricted to two-dimensional translational motion. (You are deriving the two-dimensional version of the Sackur–Tetrode equation.)
The standard molar entropy of CO is 197.7 J mol−1 K−1. How much of this value is due to rotational and vibrational motion of CO?
The standard molar entropy of the tropospheric pollutant NO2 is 240.1J mol–1 K–1. How much of this value is due to rotational motion? The vibrational frequencies of NO2 are 1318, 750, and 1618 cm−1 and the ground electronic state is doubly degenerate.
Determine the standard molar entropy of CO2, where B = 0.39 cm−1 and P = 1.00 atm. You can ignore the vibrational and electronic contributions to the standard molar entropy in this calculation.
Consider the molecule NNO, which has a rotational constant nearly identical to CO2. Would you expect the standard molar entropy for NNO to be greater or less than CO2? If greater, can you provide a rough estimation of how much greater?
What is the difference between z11 and z12?
Entropy, heat, and temperature are related through the following expression:This expression can be rearranged to provide an expression for T in terms of heat and weight (W, the number of microstates for a given configuration of energy):a. Substitute the expression for W for a two-level system into
The molecule NO has a ground electronic level that is doubly degenerate, and a first excited level at 121.1cm−1 that is also two-fold degenerate. Determine the contribution of electronic degrees of freedom to the standard molar entropy of NO. Compare your result to R ln (4). What is the
Determine the residual molar entropies for molecular crystals of the following:a. 35Cl37Clb. CFCl3c. CF2Cl2d. CO2
Using the Helmholtz energy, demonstrate that the pressure for an ideal polyatomic gas is identical to that derived for an ideal monatomic gas in the text.
Derive an expression for the standard molar enthalpy of an ideal monatomic gas by evaluation of the statistical mechanical expression for enthalpy as opposed to the thermodynamic argument provided in Example Problem 32.7.
Demonstrate that the molar enthalpy is equal to the molar energy for a collection of one-dimensional harmonic oscillators.
Calculate the standard Helmholtz energy for molar ensembles of Ne and Kr at 298 K.
What is the vibrational contribution to the Helmholtz and Gibbs energies from a molar ensemble of one-dimensional harmonic oscillators?
Determine the standard molar Gibbs energy for 35Cl35Cl, where v̅ = 560. cm−1, B = 0.244 cm−1, and the ground electronic state is non-degenerate.
Determine the rotational and vibrational contributions to the standard molar Gibbs energy for N2O (NNO), a linear triatomic molecule, where B = 0.419 cm−1 and v̅1 = 1285 cm-1, v̅2 = 589 cm-1 (doubly degenerate), and v̅3 = 2224 cm-1.
Determine the equilibrium constant for the dissociation of sodium at 298 K:Na2 (g) ⇄ 2Na (g)For Na2, B = 0.155 cm−1, v̅ = 159 cm−1, the dissociation energy is 70.4 kJ/mol, and the ground-state electronic degeneracy for Na is 2.
The isotope exchange reaction for Cl2 is as follows:35Cl35Cl + 37Cl37Cl ⇄ 237Cl35ClThe equilibrium constant for this reaction is ∼4. Furthermore, the equilibrium constant for similar isotope-exchange reactions is also close to this value. Demonstrate why this would be so.
Consider the following isotope-exchange reaction:DCl (g) + HBr (g) ‡„ DBr (g) + HCl (g)The amount of each species at equilibrium can be measured using proton and deuterium NMR (see Journal of Chemical Education 73 [1996]: 99). Using the spectroscopic information below, determine
The equilibrium between hydrogen cyanide (HCN) and its isomer hydrogen isocyanide (HNC) is important in interstellar chemistry:HCN (g) ‡„ HNC (g)A long-standing €œpuzzle€ regarding this reaction is that in space (T = 2.75 K) surprisingly large amounts of HNC
In “Direct Measurement of the Size of the Helium Dimer” by F. Luo, C. F. Geise, and W. R. Gentry [J. Chemical Physics 104 (1996): 1151], evidence for the helium dimer is presented. As one can imagine, the chemical bond in the dimer is extremely weak, with an estimated value of only 8.3mJ/mol.a.
Why is probability used to describe the velocity and speed of gas molecules?
What is the most-probable velocity for a one-dimensional gas velocity distribution? Why?
Provide a physical explanation as to why the Maxwell speed distribution approaches zero at high speeds. Why is f (ν) = 0 at ν = 0?
How would the Maxwell speed distributions for versus Kr compare if the gases were at the same temperature?
Imagine that you are performing an experiment using a molecular beam using Ar at a given temperature. If you switch the gas to Kr, will you have to increase or decrease the temperature of the gas to achieve a distribution of speeds identical to Ar?
How does the average speed of a collection of gas particles vary with particle mass and temperature?
Does the average kinetic energy of a particle depend on particle mass?
Arrange vmp, vave, and vrms in order from largest to smallest speed.
Why does the mean free path depend on the collisional cross section? Would an increase in number density, N̂ increase or decrease the mean free path?
In effusion, how does the frequency of collisions with the opening scale with molecular mass?
What is the typical length scale for a molecular diameter?
What is the contribution to the internal energy from translations for an ideal monatomic gas confined to move on a surface? What is the expected contribution from the equipartition theorem?
For a system of energy levels, εm = m2α, where α is a constant with units of energy and m = 0, 1, 2, ⋅ ⋅ ⋅, ∞. What is the internal energy and heat capacity of this system in the high-temperature limit?
(Challenging) Building on the concept of equipartition, demonstrate that for any energy term of the form ax2, where α is a constant, the contribution to the internal energy is equal to 1/2 kT by evaluating the following expression: ax²e dx 00 xx /kT dx
Consider the following table of diatomic molecules and associated rotational constants:a. Calculate the rotational temperature for each molecule.b. Assuming that these species remain gaseous at 100 K, for which species is the equipartition theorem prediction for the rotational contribution to the
The lowest four energy levels for atomic vanadium (V) have the following energies and degeneracies:What is the contribution to the average energy from electronic degrees of freedom for V when T = 298 K? Level (n) Energy (cm-1) Degeneracy 4 137.38 6 323.46 552.96 3 10
The three lowest energy levels for atomic carbon (C) have the following energies and degeneracies:What is the contribution to the average molar energy from the electronic degrees of freedom for C when T = 100.K? Energy (cm-1) Level (n) Degeneracy 16.4 43.5 3 1 2.
Consider an ensemble of units in which the first excited electronic state at energy ε1is m1-fold degenerate, and the energy of the ground state is m0 -fold degenerate with energy ε0.a. Demonstrate that if ε0 = 0, the expression for the electronic partition
Calculate the molar internal energy of He, Ne, and Ar under standard thermodynamic conditions. Do you need to redo the entire calculation for each species?
Determine the internal energy of HCl (B = 10.59 cm–1 and ν̅ = 2886 cm–1) under standard thermodynamic conditions.
How would you expect the internal energy of79BrF to compare to that of79Br35Cl at 298 K? Check your answer by using the following data: V (cm-1) B (cm-1) 19BIF 671 0.356 79 Br3SC1 0.153 420
Determine the vibrational contribution to CV for a mole of HCl (ν̅ = 2886 cm–1) over a temperature range from 500. to 5000. K in 500.-K intervals and plot your result. At what temperature do you expect to reach the high-temperature limit for the vibrational contribution to CV?
Determine the vibrational contribution to CV for HCN, where ν̅1 = 2041 cm-1, ν̅2 = 712 cm-1 (doubly degenerate), and ν̅3 = 3369 cm-1 at T = 298, 500, and 1000.K.
Carbon dioxide has attracted much recent interest as a greenhouse gas. Determine the vibrational contribution to CV for CO2, where ν̅1 = 2349 cm-1, ν̅2 = 667 cm-1 (doubly degenerate), and ν̅3 = 1333 cm-1 at T = 260.K.
The three lowest energy levels for atomic carbon (C) have the following energies and degeneracies:Determine the electronic contribution to CV for atomic C at 100. K. Level (n) Energy (cm) Degeneracy 1 1 16.4 3 43.5 3. 2.
For an ensemble consisting of a mole of particles having two energy levels separated by 1000. cm−1, at what temperature will the internal energy equal 3.00 kJ?
Consider two separate molar ensembles of particles characterized by the energy-level diagram provided in the text. Derive expressions for the internal energy for each ensemble. At 298 K, which ensemble is expected to have the greatest internal energy? 600 600- 300 300- Energy (cm-1) Energy
For a two-level system where v = 1.50 × 1013 s−1, determine the temperature at which the internal energy is equal to 0.25 Nhν, or 1/2 the limiting value of 0.50 Nhν.
In exploring the variation of internal energy with temperature for a two-level system with the ground and excited state separated by an energy of hv, the total energy was found to plateau at a value of 0.5 Nhv. Repeat this analysis for an ensemble consisting of N particles having three energy
Assume you have an equilibrium expression that involves monatomic species only. What difference in energy between reactants and products would you use in the expression for KP?
The statistical mechanical expression for KP consisted of two general parts. What are these parts, and what energetic degrees of freedom do they refer to?
For the equilibrium involving the dissociation of a diatomic, what energetic degrees of freedom were considered for the diatomic and for the atomic constituents?
Why should the equilibrium constant be dependent on the difference in Gibbs energy? How is this relationships described using statistical mechanics?
What is the definition of “zero” energy employed in constructing the statistical mechanical expression for the equilibrium constant? Why was this definition necessary?
Which thermodynamic quantity is used to derive the ideal gas law for a monatomic gas? What molecular partition function is employed in this derivation? Why?
What thermodynamic property of what particular system does the Sackur–Tetrode equation describe?
How does the Boltzmann formula provide an understanding of the third law of thermodynamics?
What is the Boltzmann formula, and how can it be used to predict residual entropy?
Describe the model used to determine the heat capacity of atomic crystals.
Why do electronic degrees of freedom generally not contribute to the constant volume heat capacity?
The molar constant volume heat capacity of N2 is 20.8 J mol−1 K−1. What is this value in terms of R? Can you make sense of this value?
When are rotational degrees of freedom expected to contribute R or 3/2 R (linear and nonlinear, respectively) to the molar constant volume heat capacity? When will a vibrational degree of freedom contribute R to the molar heat capacity?
The constant volume heat capacity for all monatomic gases is 12.48 J mol–1 K–1. Why?
Write down the contribution to the constant volume heat capacity from translations and rotations for an ideal monatomic, diatomic, and nonlinear polyatomic gas, assuming that the high-temperature limit is appropriate for the rotational degrees of freedom.
List the energetic degrees of freedom for which the contribution to the internal energy determined by statistical mechanics is equal to the prediction of the equipartition theorem at 298 K.
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