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Physical Chemistry Thermodynamics And Kinetics 10th Edition Peter Atkins, Julio De Paula - Solutions

How is the method of combination differences used in rotation– vibration spectroscopy to determine rotational constants?
Suppose that three conformations are proposed for the nonlinear molecule H2O2 (1, 2, and 3). The infrared absorption spectrum of gaseous H2O2 has bands at 870, 1370, 2869, and 3417 cm−1. The Raman spectrum of the same sample has bands at 877, 1408, 1435, and 3407 cm−1. All bands correspond to
From the data in Exercise 12D.4(a), predict the fundamental vibrational wavenumbers of the deuterium halides.Data in Exercise 12D.4(a)The hydrogen halides have the following fundamental vibrational wavenumbers: 4141.3 cm−1 (HF); 2988.9 cm−1 (H35Cl); 2649.7 cm−1 (H81Br); 2309.5 cm−1 (H127I).
Use molecular modelling software and the computational method of your choice to construct molecular potential energy curves like the one shown in Fig. 12D.1. Consider the hydrogen halides (HF, HCl, HBr, and HI):(a) Plot the calculated energy of each molecule against the bond length, (b)
It is common to make measurements of absorbance at two wavelengths and use them to find the individual concentrations of two components A and B in a mixture. Show that the molar concentrations of A and B are Where A1 and A2 are absorbances of the mixture at wavelengths λ1 and λ2, and the
Account for the existence of a rotational zero-point energy in molecular hydrogen.
The following data were obtained for the absorption by Br2 in carbon tetrachloride using a 2.0mm cell. Calculate the molar absorption coefficient of bromine at the wavelength employed: [Br₂]/(mol dm-³) T/(per cent) 0.0010 81.4 0.0050 35.6 0.0100 12.7 0.0500 3.0x10-3
The spacing of lines in the microwave spectrum of 35Cl19F is 1.033 cm−1; calculate the moment of inertia and bond length of the molecule.
In what ways may the rotational and vibrational spectra of molecules change as a result of isotopic substitution?
When pyridine is added to a solution of iodine in carbon tetrachloride the 520nm band of absorption shifts toward 450nm. However, the absorbance of the solution at 490nm remains constant: this feature is called an isosbestic point. Show that an isosbestic point should occur when two absorbing
A 5.00mm cell was filled with a solution of a dye. The concentration of the dye was 18.5mmol dm−3. Calculate the molar absorption coefficient of the dye at this wavelength given that the transmission was 29 per cent. What will the transmittance be in a 2.50mm cell at the same wavelength?
Use appropriate electronic structure software to perform calculations on H2O and CO2 with basis sets of your or your instructor’s choosing.(a) Compute ground-state energies, equilibrium geometries and vibrational frequencies for each molecule.(b) Compute the magnitude of the dipole moment of H2O;
The wavenumber of the incident radiation in a Raman spectrometer is 20 623 cm−1. What is the wavenumber of the scattered Stokes radiation for the J=4←2 transition of 16O2?
Wachewsky et al. (J. Phys. Chem. 100, 11559 (1996)) examined the UV absorption spectrum of CH3I, a species of interest in connection with stratospheric ozone chemistry. They found the integrated absorption coefficient to be dependent on temperature and pressure to an extent inconsistent with
Which of the normal modes of CH2Cl2 (Exercise 12E.7(a)) are infrared active? Which are Raman active?Data in Exercise 12E.7(a)The molecule CH2Cl2 belongs to the point group C2v. The displacements of the atoms span 5A1+2A2+4B1+4B2. What are the symmetry species of the normal modes of vibration?
The rotational Raman spectrum of 35Cl2 shows a series of Stokes lines separated by 0.9752 cm−1 and a similar series of anti-Stokes lines. Calculate the bond length of the molecule.
The rotational terms of a symmetric top, allowing for centrifugal distortion, are commonly written(a) Develop an expression for the wavenumbers of the allowed rotational transitions.(b) The following transition frequencies (in gigahertz, GHz) were observed for CH3F:Determine the values of as many
What is the ratio of weights of populations due to the effects of nuclear statistics for 35Cl2?
The average spacing between the rotational lines of the P and R branches of 12C2 1H2 and 12C2 2H2 is 2.352 cm−1 and 1.696 cm−1, respectively. Estimate the CC and CH bond lengths.
Absorptions in the v=1←0 vibration–rotation spectrum of 1H35Cl were observed at the following wavenumbers (in cm−1):Assign the rotational quantum numbers and use the method of combination differences to determine the rotational constants of the two vibrational levels.
What is the Doppler-shifted wavelength of a red (680nm) traffic light approached at 60 km h−1?
Estimate the lifetime of a state that gives rise to a line of width (i) 0.20 cm−1, (ii) 2.0 cm−1.
The Gaussian shape of a Doppler-broadened spectral line reflects the Maxwell distribution of speeds in the sample at the temperature of the experiment. In a spectrometer that makes use of phase-sensitive detection the output signal is proportional to the first derivative of the signal intensity,
Suppose that the internuclear distance may be written R=Re+x where Re is the equilibrium bond length. Also suppose that the potential well is symmetrical and confines the oscillator to small displacements. Deduce expressions for 1/〈R〉2, 1/〈R2〉, and 〈1/R2〉 to the lowest non-zero power of
A molecule in a liquid undergoes about 1.0×1013 collisions in each second. Suppose that (i) every collision is effective in deactivating the molecule vibrationally and (ii) that one collision in 100 is effective. Calculate the width (in cm−1) of vibrational transitions in the molecule.
Continue the development of Problem 12D.12 by using the virial expression to relate 〈x2〉 to the vibrational quantum number. Does your result imply that the rotational constant increases or decreases as the oscillator becomes excited to higher quantum states. What would be the effect of
Using mathematical software, elaborate on the results of Example 12A.2 by:(a) Exploring the effect of varying the wavenumbers and intensities of the three components of the radiation on the shape of the interferogram;(b) Calculating the Fourier transforms of the functions you generated in part
The analysis of combination differences summarized in the text considered the R and P branches. Extend the analysis to the O and S branches of a Raman spectrum.
Explain the origin of the term symbol 3Σg− for the ground state of dioxygen.
J.G. Dojahn et al. (J. Phys. Chem. 100, 9649 (1996)) characterized the potential energy curves of the ground and electronic states of homonuclear diatomic halogen anions. These anions have a 2Σu+ ground state and 2Πg, 2Πu, and 2Σg+ excited states. To which of the excited states are
Describe the mechanism of fluorescence. In what respects is a fluorescence spectrum not the exact mirror image of the corresponding absorption spectrum?
The line marked A in Fig. 13.3 is the fluorescence spectrum of benzophenone in solid solution in ethanol at low temperatures observed when the sample is illuminated with 360nm ultraviolet radiation. What can be said about the vibrational energy levels of the carbonyl group in (i) its ground
Describe the principles of (a) continuous-wave and (b) pulsed laser action.
Consider an evacuated laser cavity of length 1.0 cm. What are the allowed wavelengths and frequencies of the resonant modes?
Use a group theoretical argument to decide which of the following transitions are electric-dipole allowed:(a) The π*←π transition in ethene,(b) The π*←n transition in a carbonyl group in a C2v environment.
Explain the basis of the Franck–Condon principle and how it leads to the formation of a vibrational progression.
Which of the following transitions are electric-dipole allowed? (i) 2Π↔2Π, (ii) 1Σ↔1Σ, (iii) Σ↔Δ, (iv) Σ+↔Σ−, (v) Σ+↔Σ+.
The oxygen molecule absorbs ultraviolet radiation in a transition from its 3Σg− ground electronic state to an excited state that is energetically close to a dissociative 5Πu state. The absorption band has a relatively large experimental linewidth. Account for this observation.
How might you use a Q-switched or mode-locked laser in the study of a very fast chemical reaction that can be initiated by absorption of light?
Use mathematical software or an electronic spreadsheet to simulate the output of a mode-locked laser (that is, plots such as that shown in Fig. 13C.8) for L=30 cm and N=100 and 1000.Data in Fig. 13C.8 Intensity, I m 1 hud Time, ct/2L 2 3 m m 5
Use molecule (4) as a model of the trans conformation of the chromophore found in rhodopsin. In this model, the methyl group bound to the nitrogen atom of the protonated Schiff ’s base replaces the protein.(a) Using molecular modelling software and the computational method of your instructor’s
A certain laser can generate radiation in pulses, each of which delivers an energy of 0.10mJ, with peak power of 5.0MW and average power of 7.0 kW. What are the pulse duration and repetition frequency?
The highest kinetic energy electrons in the photoelectron spectrum of H2O using 21.22 eV radiation are at about 9 eV and show a large vibrational spacing of 0.41 eV. The symmetric stretching mode of the neutral H2O molecule lies at 3652 cm−1.(a) What conclusions can be drawn from the nature of
The R-branch of the 1Πu ←1Σg+ transition of H2 shows a band head at the very low value of J=1. The rotational constant of the ground state is 60.80 cm−1. What is the rotational constant of the upper state? Has the bond length increased or decreased in the transition?
The complex ion [Fe(OH2)6]3+ has an electronic absorption spectrum with a maximum at 700nm. Estimate a value of ΔO for the complex.
Estimate the magnitude of the transition dipole moment of a chargetransfer transition modelled as the migration of an electron from a H1s orbital on one atom to another H1s orbital on an atom a distance R away. Approximate the transition moment by −eRS where S is the overlap integral of the two
Figure 13.2 shows the UV-visible absorption spectra of a selection of amino acids. Suggest reasons for their different appearances in terms of the structures of the molecules. Data in Figure 13.2 Absorbance Tyr Gly Cys 220 Trp Wavelength, Wnm 280
3-Buten-2-one (3) has a strong absorption at 213nm and a weaker absorption at 320nm. Justify these features and assign the ultraviolet absorption transitions. 3 3-Buten-2-one
Suppose that we can model a charge-transfer transition in a onedimensional system as a process in which a Gaussian wavefunction centred on x=0 and width a makes a transition to another Gaussian wavefunction of the same width centred on x = 1/2 a. Evaluate the transition moment ∫ψf xψidx.
What is the maximum number of phases that can be in equilibrium in a ternary system?
The addition of a small amount of a salt, such as (NH4)2SO4, to a solution containing a charged protein increases the solubility of the protein in water. This observation is called the salting-in effect. However, the addition of large amounts of salt can decrease the solubility of the protein to
Derive an expression for the temperature coefficient of the solubility, dxB/dT, and plot it as a function of temperature for several values of the enthalpy of fusion.
Explain why reactions that are not redox reactions may be used to generate an electric current.
Can you identify other contributions to the cell potential when a current is being drawn from the cell?
The work function for metallic rubidium is 2.09 eV. Calculate the kinetic energy and the speed of the electrons ejected by light of wavelength (i) 650nm, (ii) 195nm.
Calculate the de Broglie wavelength of (i) a mass of 1.0 g travelling at 1.0 cms−1, (ii) the same, travelling at 100 kms−1, (iii) an He atom travelling at 1000ms−1 (a typical speed at room temperature).
Describe the variation of the separation of the vibrational energy levels with the mass and force constant of the harmonic oscillator.
In what ways does the quantum mechanical description of a harmonic oscillator merge with its classical description at high quantum numbers?
Calculate the minimum excitation energy of a proton constrained to rotate in a circle of radius 100pm around a fixed point.
Evaluate the z-component of the angular momentum and the kinetic energy of a particle on a ring that is described by the (unnormalized) wavefunctions (a) eiφ, (b) e−2iφ, (c) cos φ, and (d) (cos χ)eiφ+(sin χ)e−iφ.
Use the data in Exercise 8C.4(a) to calculate the energy needed to excite a CH4 molecule from a state with l=1 to a state with l=2.Data in Exercise 8C.4(a)The moment of inertia of a CH4 molecule is 5.27×10−47 kgm2. What is the minimum energy needed to start it rotating?
Draw scale vector diagrams to represent the states (i) l=1, ml=+1, (ii) l=2, ml=0.
What feature of molecular orbital theory is responsible for bond formation?
Calculate the (molar) energy of electrostatic repulsion between two hydrogen nuclei at the separation in H2 (74.1pm). The result is the energy that must be overcome by the attraction from the electrons that form the bond. Does the gravitational attraction between them play any significant role?
Draw diagrams to show the various orientations in which a p orbital and a d orbital on adjacent atoms may form bonding and antibonding molecular orbitals.
Write the valence-bond wavefunction for the single bond in HF.
Why is spin-pairing so common a features of bond formation (in the context of molecular orbital theory)?
Suppose that a molecular orbital has the (unnormalized) form 0.145A+0.844B. Find a linear combination of the orbitals A and B that is orthogonal to this combination and determine the normalization constants of both combinations using S=0.250.
Outline the rules of the building-up principle for homonuclear diatomic molecules.
Why are promotion and hybridization invoked in valence-bond theory?
Show that two equivalent hybrid orbitals of the form spλ make an angle θ to each other, then λ = −1/cosθ . Plot a graph of λ against θ and confirm that θ = 180° when no s orbital is included and θ = 120° when λ = 2.
What is the role of the Born–Oppenheimer approximation in molecular orbital theory?
Describe the various types of hybrid orbitals and how they are used to describe the bonding in alkanes, alkenes, and alkynes. How does hybridization explain that in allene, CH2=C=CH2, the two CH2 groups lie in perpendicular planes?
Describe the structure of a P2 molecule in valence-bond terms. Why is P4 a more stable form of molecular phosphorus than P2?
Identify the g or u character of bonding and antibonding π orbitals formed by side-by-side overlap of p atomic orbitals.
Examine whether occupation of the bonding orbital with one electron (as calculated in the preceding problem) has a greater or lesser bonding effect than occupation of the antibonding orbital with one electron. Is that true at all internuclear separations?
What is the justification for treating s and p atomic orbital contributions to molecular orbitals separately?
Why is spin-pairing so common a features of bond formation (in the context of valence-bond theory)?
Describe the bonding in 1,3-butadiene using hybrid orbitals.
To what extent can orbital overlap be related to bond strength?
What are the consequences of resonance?
Show that the linear combinations h1=s+px+py+pz and h2=s−px−py+pz are mutually orthogonal.
Normalize the sp2 hybrid orbital h=s+21/2p given that the s and p orbitals are each normalized to 1.
A sample consisting of 1.00mol of perfect gas atoms, for which CV,m = 3/2 , initially at p1=1.00 atm and T1=300K, is heated reversibly to 400K at constant volume. Calculate the final pressure, ΔU, q, and w.
The standard enthalpy of combustion of ethyl acetate (CH3COOC2H5) is –2231 kJmol−1 at 298K and its standard molar entropy is 259.4 JK−1mol−1. Calculate the standard Gibbs energy of formation of the compound at 298K.
The following temperature–composition data were obtained for a mixture of octane (O) and methylbenzene (M) at 1.00 atm, where x is the mole fraction in the liquid and y the mole fraction in the vapour at equilibrium.The boiling points are 110.6 °C and 125.6 °C for M and O, respectively. Plot
The mean activity coefficients for aqueous solutions of NaCl at 25 °C are given opposite. Confirm that they support the Debye–Hückel limiting law and that an improved fit is obtained with the Davies equation. b/(mmol kg ¹) V+ 1.0 2.0 0.9649 0.9519 0.9275 5.0 10.0 0.9024 20.0 0.8712
Why do the activity coefficients of ions in solution differ from 1? Why are they less than 1 in dilute solutions?
Explain the concept of partial molar quantity, and justify the remark that the partial molar property of a solute depends on the properties of the solvent too.
Francesconi, Lunelli, and Comelli studied the liquid–vapour equilibria of trichloromethane and 1,2-epoxybutane at several temperatures (Francesconi et al., J. Chem. Eng. Data 41, 310 (1996)). Among their data are the following measurements of the mole fractions of trichloromethane in the liquid
Mark the following features on triangular coordinates: (i) the point (0.2, 0.2, 0.6), (ii) the point (0, 0.2, 0.8), (iii) the point at which all three mole fractions are the same.
The experimental values of the partial molar volume of a salt in water are found to fit the expression vB =5.117+19.121x1/2, where vB =VB/(cm3mol−1) and x is the numerical value of the molality of B (x=b/b⦵). Use the Gibbs– Duhem equation to derive an equation for the molar volume of water in
Explain what is meant by a regular solution; what additional features distinguish a real solution from a regular solution?
What are the contributions that account for the difference between activity and concentration?
Substances A and B are both volatile liquids with pA∗ =300 Torr, pB∗ =250 Torr, and KB =200 Torr (concentration expressed in mole fraction). When xA =0.9, bB =2.22mol kg−1, pA =250 Torr, and pB =25 Torr. Calculate the activities and activity coefficients of A and B. Use the mole fraction,
What molecular features determine whether a mixture of two liquids will show high- and low-boiling azeotropic behaviour?
Sketch the phase diagram of the system NH3/N2H4 given that the two substances do not form a compound with each other, that NH3 freezes at –78 °C and N2H4 freezes at +2 °C, and that a eutectic is formed when the mole fraction of N2H4 is 0.07 and that the eutectic melts at –80 °C.
Consider the plot of log γ± against I1/2 with B=1.50 and C=0 in the Davies equation as a representation of experimental data for a certain MX electrolyte. Over what range of ionic strengths does the application of the limiting law lead to an error in the value of the activity coefficient of less
An, Zhao, Jiang, and Shen investigated the liquid–liquid coexistence curve of N,N-dimethylacetamide and heptane (X. An et al., J. Chem. Thermodynamics 28, 1221 (1996)). Mole fractions of N,N-dimethylacetamide in the upper (x1) and lower (x2) phases of a two-phase region are given opposite as a
Explain how thermodynamics relates non-expansion work to a change in composition of a system.

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