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

Discuss the physical origin of quantization of energy for a particle confined to motion around a ring.
The rotation of a molecule can be represented by the motion of a point mass moving over the surface of a sphere. Calculate the magnitude of its angular momentum when l=1 and the possible components of the angular momentum on an arbitrary axis. Express your results as multiples of h.
Determine the linear momentum and kinetic energy of a free electron described by the wavefunction eikx with k=3nm−1.
Here we explore further the idea introduced in Impact I8.1 that quantum mechanical effects need to be invoked in the description of the electronic properties of metallic nanocrystals, here modelled as three-dimensional boxes. (a) Set up the Schrödinger equation for a particle of mass m in a
Describe the features of the solution of the particle on a ring that appear in the solution of the particle on a sphere. What concept applies to the latter but not to the former?
Explain why the particle in a box and the harmonic oscillator are useful models for quantum mechanical systems: what chemically significant systems can they be used to represent?
Describe the features of the solution of the particle in a one-dimensional box that appear in the solutions of the particle in two- and three-dimensional boxes. What concept applies to the latter but not to a one-dimensional box?
A particle is confined to move in a one-dimensional box of length L.(a) If the particle is classical, show that the average value of x is 1/2 L and that the root-mean square value is L/31/2.(b) Show that for large values of n, a quantum particle approaches the classical values. This result is an
Write the wavefunction for a particle of mass 2.0 g travelling to the left with a kinetic energy of 20 J.
What is the physical reason for the existence of a zero-point vibrational energy?
Of the four assumptions made in Problem 8B.2, the last two are questionable. Suppose that the first two assumptions are still reasonable and that you have at your disposal a supply of myoglobin, a suitable buffer in which to suspend the protein, 12C16O, 13C16O, 12C18O, 13C18O, and an infrared
Describe the vector model of angular momentum in quantum mechanics. What features does it capture? What is its status as a model?
Calculate the mean kinetic energy of a harmonic oscillator by using the relations in Table 8B.1. Data in Table 8B.1. V 0 1 2 3 34 56 H₂(y) 1 2y 4y²-2 8y³-12y 16y¹-48y² +12 32y5-160y³+120y 64y6-480y¹+720y² - 120
Calculate the values of 〈x3〉 and 〈x4〉 for a harmonic oscillator by using the relations in Table 8B.1.Data in Table 8B.1. V 0 1 2 3 4 5 6 H₂(y) 1 2y 4y²-2 8y³-12y 16y¹-48y²+12 32y5-160y³+120y 64y6-480y¹+720y² - 120
The vibrational frequency of H2 is 131.9THz. What is the vibrational frequency of D2 (D=2H)?
The moment of inertia of a CH4 molecule is 5.27×10−47 kgm2. What is the minimum energy needed to start it rotating?
Determine the values of Δx=(〈x2〉−〈x〉2)1/2 and Δp=(〈p2〉−〈p〉2)1/2 for the ground state of(a) A particle in a box of length L(b) An harmonic oscillator. Discuss these quantities with reference to the uncertainty principle.
Use mathematical software to construct a harmonic oscillator wavepacket of the formwhere the wavefunctions and energies are those of a harmonic oscillator and with coefficients of your choice (for example, all equal). Explore how the wavepacket oscillates to and fro with time.
What is the magnitude of the angular momentum of a CH4 molecule when it is rotating with its minimum energy?
Calculate the expectation values of xˆ and xˆ2 for a particle in the state n=1 in a one-dimensional square-well potential.
What are the most probable displacements of a harmonic oscillator with v=1?
The number of states corresponding to a given energy plays a crucial role in atomic structure and thermodynamic properties. Determine the degeneracy of a body rotating with l=3.
What are the most likely locations of a particle in a box of length L in the state n=5?
Calculate the probability that an O-H bond treated as an harmonic oscillator will be found at a classically forbidden extension when v=1.
What is the value of n of a particle in a one-dimensional box such that the separation between neighbouring levels is equal to the mean energy of thermal motion (1/2 kT).
For a particle in a rectangular box with sides of length L1=L and L2=2L, find a state that is degenerate with the state n1=n2=2. Degeneracy is normally associated with symmetry; why, then, are these two states degenerate?
Suppose that the junction between two semiconductors can be represented by a barrier of height 2.0 eV and length 100pm. Calculate the transmission probability of an electron with energy 1.5 eV.
Write the ground-state electron configurations of the d-metals from scandium to zinc.
Identify the shortest and longest wavelength lines in the Lyman series.
Discuss the origin of the series of lines in the emission spectra of hydrogen. What region of the electromagnetic spectrum is associated with each of the series shown in Fig. 9C.1?Data in Fig. 9C.1 S Balmer Lyman Paschen р 15 328 (H) 20 571 (H) 23 039 (H) 24 380 (H) 82 259 97 492 102 824 d
An electron in the ground-state He+ ion undergoes a transition to a state described by the wavefunction R4,1(r)Y1,1(θ,φ).(a) Describe the transition using term symbols.(b) Compute the wavelength, frequency, and wavenumber of the transition.(c) By how much does the mean radius of the electron
Describe and account for the variation of first ionization energies along Period 2 of the periodic table. Would you expect the same variation in Period 3?
Compute the wavelength, frequency, and wavenumber of the n = 2 → n= 1 transition in He+.
Highly excited atoms have electrons with large principal quantum numbers. Such Rydberg atoms have unique properties and are of interest to astrophysicists.(a) For hydrogen atoms with large n, derive a relation for the separation of energy levels.(b) Calculate this separation for n = 100; also
Which of the following transitions are allowed in the normal electronic emission spectrum of an atom: (i) 2s → 1s, (ii) 2p → 1s, (iii) 3d → 2p?
A series of lines in the spectrum of neutral Li atoms rise from combinations of 1s22p1 2P with 1s2nd1 2D and occur at 610.36nm, 460.29nm, and 413.23nm. The d orbitals are hydrogenic. It is known that the 2P term lies at 670.78nm above the ground state, which is 1s22s1 2S. Calculate the ionization
An emission line from K atoms is found to have two closely spaced components, one at 766.70nm and the other at 770.11nm. Account for this observation, and deduce what information you can.
Which of the following transitions between terms are allowed in the normal electronic emission spectrum of a many-electron atom: (i) 3D2 → 3P1, (ii) 3P2 → 1S0, (iii) 3F4 → 3D3?
Locate the radial nodes and nodal planes of each of the 3p orbitals of a hydrogenic atom of atomic number Z. To locate the angular nodes, give the angle that the plane makes with the z-axis.
The Bohr model of the atom is specified in Problem 9A.8.(a) What features of it are untenable according to quantum mechanics?(b) How does the Bohr ground state differ from the actual ground state?(c) Is there an experimental distinction between the Bohr and quantum mechanical models of the ground
Write down the expression for the radial distribution function of a 2s electron in a hydrogenic atom of atomic number Z and determine the radius at which the electron is most likely to be found.
(a) Express (∂CV/∂V)T as a second derivative of U and find its relation to (∂U/∂V)T and (∂Cp/∂p)T as a second derivative of H and find its relation to (∂H/∂p)T.(b) From these relations show that (∂CV/∂V)T =0 and (∂Cp/∂p)T =0 for a perfect gas.
A sample consisting of 1.00mol Ar is expanded isothermally at 20 °C from 10.0dm3 to 30.0dm3 (i) Reversibly, (ii) Against a constant external pressure equal to the final pressure of the gas, (iii) Freely (against zero external pressure). For the three processes calculate q, w, and ΔU.
The following data show how the standard molar constant-pressure heat capacity of sulfur dioxide varies with temperature. By how much does the standard molar enthalpy of SO2(g) increase when the temperature is raised from 298.15K to 1500K? T/K 300 500 700 900 1100 1300 1500 Cm/(JK-¹mol-¹) 39.909
Describe two calorimetric methods for the determination of enthalpy changes that accompany chemical processes.
In 2006, the Intergovernmental Panel on Climate Change (IPCC) considered a global average temperature rise of 1.0–3.5 °C likely by the year 2100, with 2.0 °C its best estimate. Predict the average rise in sea level due to thermal expansion of sea water based on temperature rises of 1.0 °C, 2.0
Why are adiabats steeper than isotherms?
Use the equipartition principle to estimate the values of γ =Cp/CV for gaseous ammonia and methane. Do this calculation with and without the vibrational contribution to the energy. Which is closer to the expected experimental value at 25 °C?
Give examples of state functions and discuss why they play a critical role in thermodynamics.
Describe and distinguish the various uses of the words ‘system’ and ‘state’ in physical chemistry.
Estimate the internal pressure, πT, of water vapour at 1.00 bar and 400K, treating it as a van der Waals gas.
From the enthalpy of combustion data in Table 2C.1 for the alkanes methane through octane, test the extent to which the relation ΔcH⦵ = k{(M/(g mol−1)}n holds and find the numerical values for k and n. Predict ΔcH⦵ for decane and compare to the known value.Data in table 2C.1
Why is the heat capacity at constant pressure of a substance normally greater than its heat capacity at constant volume?
Distinguish between ‘standard state’ and ‘reference state’, and indicate their applications.
Why do heat capacities play a role in the expressions for adiabatic expansion?
Describe the distinction between heat and work in thermodynamic and molecular terms, the latter in terms of populations and energy levels.
Which of (i) Pressure (ii) Temperature (iii) Work (iv) Enthalpy are state functions?
Explain why a perfect gas does not have an inversion temperature.
When 3.0mol O2 is heated at a constant pressure of 3.25 atm, its temperature increases from 260K to 285K. Given that the molar heat capacity of O2 at constant pressure is 29.4 J K−1 mol−1, calculate q, ΔH, and ΔU.
A sample consisting of 2.5mol of perfect gas molecules with Cp,m =20.8 J K−1 mol−1 is initially at 240 kPa and 325K. It undergoes reversible adiabatic expansion until its pressure reaches 150 kPa. Calculate the final volume and temperature and the work done.
Identify varieties of additional work.
A better model of a DNA molecule is the ‘one-dimensional freely jointed chain’, in which a rigid unit of length l can only make an angle of 0° or 180 ° with an adjacent unit. In this case, the restoring force of a chain extended by x=nl is given byWhere k is Boltzmann’s constant.(a) What is
A chemical reaction takes place in a container of cross-sectional area 75.0 cm2. As a result of the reaction, a piston is pushed out through 25.0 cm against an external pressure of 150 kPa. Calculate the work done by the system.
A thermodynamic study of DyCl3 (E.H.P. Cordfunke et al., J. Chem. Thermodynamics 28, 1387 (1996)) determined its standard enthalpy of formation from the following informationDetermine ΔfH⦵(DyCl3, s) from these data. (1) DyCl3(s)→DyCl3(aq, in 4.0 M HCl) A,H=-180.06 kJ mol-¹ (2) Dy(s)+3 HCl(aq,
Use mathematical software, a spreadsheet, or the Living graphs on the web site for this book to:(a) Calculate the work of isothermal reversible expansion of 1.0mol CO2(g) at 298K from 1.0m3 to 3.0m3 on the basis that it obeys the van der Waals equation of state.(b) Explore how the parameter γ
The isothermal compressibility of water at 293K is 4.96×10−5 atm−1. Calculate the pressure that must be applied in order to increase its density by 0.10 per cent.
Rearrange the van der Waals equation of state, p=nRT/(V – nb) – n2a/V2, to give an expression for T as a function of p and V (with n constant). Calculate (∂T/∂p)V and confirm that (∂T/∂p)V =1/(∂p/∂T)V . Go on to confirm Euler’s chain relation (Mathematical background 2).
Calculate the final pressure of a sample of carbon dioxide that expands reversibly and adiabatically from 67.4 kPa and 0.50 dm3 to a final volume of 2.00 dm3. Take γ =1.4.
Suppose that a DNA molecule resists being extended from an equilibrium, more compact conformation with a restoring force F=−kfx, where x is the difference in the end-to-end distance of the chain from an equilibrium value and kf is the force constant. Use this model to write an expression for the
In biological cells that have a plentiful supply of oxygen, glucose is oxidized completely to CO2 and H2O by a process called aerobic oxidation. Muscle cells may be deprived of O2 during vigorous exercise and, in that case, one molecule of glucose is converted to two molecules of lactic acid
During a hypothetical process, the entropy of a system increases by 125 JK−1 while the entropy of the surroundings decreases by 125 JK−1. Is the process spontaneous?
Consider a perfect gas contained in a cylinder and separated by a frictionless adiabatic piston into two sections A and B. All changes in B are isothermal; that is, a thermostat surrounds B to keep its temperature constant. There is 2.00mol of the gas molecules in each section. Initially TA =TB
Suppose that 2.5mmol N2(g) occupies 42 cm3 at 300K and expands isothermally to 600 cm3. Calculate ΔG for the process.
Calculate ΔrG⦵ (375K) for the reaction 2 CO(g)+O2(g)→2 CO2(g) from the value of ΔrG⦵ (298K), ΔrH⦵ (298K), and the Gibbs–Helmholtz equation.
A gaseous sample consisting of 1.00mol molecules is described by the equation of state pVm =RT(1+Bp). Initially at 373K, it undergoes Joule–Thomson expansion from 100 atm to 1.00 atm. Given that Cp,m = 5/2 R, μ =0.21K atm−1, B=−0.525(K/T) atm−1 and that these are constant over the
Discuss why the standard entropies of ions in solution may be positive, negative, or zero.
Suppose that the hexagonal molecule C6HnF6–n has a residual entropy on account of the similarity of the H and F atoms. Calculate the residual for each value of n.
The following expressions have been used to establish criteria for spontaneous change: dAT,V <0 and dGT,p<0. Discuss the origin, significance, and applicability of each criterion.
A certain ideal heat engine uses water at the triple point as the hot source and an organic liquid as the cold sink. It withdraws 10.00 kJ of heat from the hot source and generates 3.00 kJ of work. What is the temperature of the organic liquid?
Calculate the molar internal energy, molar entropy, and molar Helmholtz energy of a collection of harmonic oscillators and plot your expressions as a function of T/θV, where θV =hV/k.
Discuss the relationship between the thermodynamic and statistical definitions of entropy.
Under what circumstances, and why, can the spontaneity of a process be discussed in terms of the properties of the system alone? These problems were provided by Charles Trapp and Carmen Giunta.
Calculate ΔS (for the system) when the state of 3.00mol of perfect gas atoms, for which Cp,m = 5/2 R, is changed from 25 °C and 1.00 atm to 125 °C and 5.00 atm. How do you rationalize the sign of ΔS?
Discuss the significance of the terms ‘dispersal’ and ‘disorder’ in the context of the Second Law.
Calculate the change in entropy when 100 kJ of energy is transferred reversibly and isothermally as heat to a large block of copper at (i) 0 °C, (ii) 50 °C.
Estimate the change in the Gibbs energy and molar Gibbs energy of 1.0dm3 of octane when the pressure acting on it is increased from 1.0 atm to 100 atm. The mass density of octane is 0.703 g cm−3.
Use mathematical software, a spreadsheet, or the Living graphs on the web site for this book to: (a) Evaluate the change in entropy of 1.00mol CO2(g) on expansion from 0.001m3 to 0.010m3 at 298K, treated as a van der Waals gas.(b) Allow for the temperature dependence of the heat capacity by
Use standard Gibbs energies of formation to calculate the standard reaction Gibbs energies at 298K of the reactions (i) 2 CH3CHO(g) + O₂(g) →2CH3COOH(1) (ii) 2 AgCl(s) + Br₂ (1)→2 AgBr(s)+Cl₂(g) (iii) Hg(1)+Cl₂(g) →HgCl₂ (s)
Discuss the relationships between the various formulations of the Second Law of thermodynamics.
Two empirical equations of state of a real gas are as follows:Evaluate (∂S/∂V)T for each gas. For an isothermal expansion, for which kind of gas (also consider a perfect gas) will ΔS be greatest? Explain your conclusion. RT a van der Waals: p=V-b V m m RTe-a/RTVm Dieterici: p= V-b
Which of F2(g) and I2(g) is likely to have the higher standard molar entropy at 298K?
The molar heat capacity of anhydrous potassium hexacyanoferrate(II) varies with temperature as follows: Calculate the molar enthalpy relative to its value at T=0 and the Third-Law entropy at each of these temperatures. T/K Cp.m/(JK-¹ mol-¹) T/K Cp.m/(JK-¹ mol-¹) T/K Cp.m/(JK-¹
Account for deviations from Trouton’s rule for liquids such as water and ethanol. Is their entropy of vaporization larger or smaller than 85 JK−1mol−1? Why?
Calculate the change in entropy when 15 g of carbon dioxide gas is allowed to expand from 1.0dm3 to 3.0dm3 at 300K.
The expressions that apply to the treatment of refrigerators (Problem 3A.4) also describe the behaviour of heat pumps, where warmth is obtained from the back of a refrigerator while its front is being used to cool the outside world. Heat pumps are popular home heating devices because they are very
The compound 1,3,5-trichloro-2,4,6-trifluorobenzene is an intermediate in the conversion of hexachlorobenzene to hexafluorobenzene, and its thermodynamic properties have been examined by measuring its heat capacity over a wide temperature range (R.L. Andon and J.F. Martin, J. Chem. Soc. Faraday
Predict the enthalpy of vaporization of benzene from its normal boiling point, 80.1 °C.
(a) Use the Maxwell relations to express the derivatives (∂S/∂V)T , (∂V/∂S)p, (∂p/∂S)V , and (∂V/∂S)p in terms of the heat capacities, the expansion coefficient α=(1/V)(∂V/∂T)p, and the isothermal compressibility, κT =−(1/V) (∂V/∂p)T .(b) The Joule coefficient, μJ
The molar heat capacity of chloroform (trichloromethane, CHCl3) in the range 240K to 330K is given by Cp,m/(JK−1mol−1)=91.47+7.5×10−2 (T/K).
Derive an expression for the molar entropy of a monatomic solid on the basis of the Einstein and Debye models and plot the molar entropy against the temperature (use T/θ in each case, with θ the Einstein or Debye temperature). Use the following expressions for the temperature-dependence of the

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