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statistical techniques in business

Questions and Answers of Statistical Techniques in Business

Imagine a droplet of water, in equilibrium with its vapor, placed on a wire frame that can stretch the surface area of the droplet, keeping the temperature fixed. Assume that the whole system,
A boy blows a soap bubble of radius R which floats in the air a few moments before breaking. What is the difference in pressure between the air inside the bubble and the air outside the bubble when
The equation of state of a gas is given by the Berthelot equation (P+a/Tv)(-b)=RT. (a) Find values of the critical temperature Te, the critical molar volume vc, and the critical pressure Pc, in terms
A liquid crystal is composed of molecules which are elongated (and often have flat segments). It behaves like a liquid because the locations of the center of mass of the molecules have no long-range
Water has a latent heat of vaporization, Ah = 540 cal/gr. One mole of steam is kept at its condensation point under pressure at T = 373 K. The temperature is then lowered to T = 336 K, keeping the
A PVT system has a line of continuous phase transitions (a lambda line) separating two phases, I and II, of the system. The molar heat capacity cp and the thermal expansivity ap are different in the
Consider liquid mixture (!) of particles A and B coexisting in equilibrium with vapor mixture (g) of particles A and B. Show that the generalization of the Clausius-Clapeyron equation for the
Consider a binary mixture composed of two types of particles, A and B. For this system the fundamental equation for the Gibbs free energy is G=nH+B, the combined first and second laws are dG-S dT+ V
For a van der Waals gas, plot the isotherms in the P-V plane (P and V are the reduced pressure and volume) for reduced temperatures T = 0.5, T=1.0, and T=1.5. For T=0.5, is P = 0.1 the equilibrium
Deduce the Maxwell construction using stability properties of the Helmholtz free energy rather than the Gibbs free energy.
A system in its solid phase has a Helmholtz free energy per mole,a, B/Ty, and in its liquid phase it has a Helmholtz free energy per mole, a = A/Tv, where A and B are constants, v is the volume per
Consider a monatomic fluid along its liquid-gas coexistence curve. Compute the rate of change of chemical potential along the coexistence curve, (du/dT) coex where is the chemical potential and 7 is
Prove that the slope of the sublimation curve of a pure substance at the triple point must be greater than that of the vaporization curve at the triple point.
Find the coefficient of thermal expansion, coex = (1/v) (Ov/OT) coex for a gas maintained in equilibrium with its liquid phase. Find an approximate explicit expression for a coex, using the ideal gas
A condensible vapor has a molar entropy s = so+R In[C(v -b) (u+ (a/v))/2], where C and so are constants. (a) Compute the equation of state. (b) Compute the molar heat capacities,c, and cp. (c)
EXERCISE 3.6. Consider a binary mixture of particles of types 1 and 2, whose Gibbs free energy is given bywhere n = n + n, n and n are the mole numbers, and x and x2 are the mole fractions of
EXERCISE 3.5. Use purely mechanical arguments to derive Eq. (3.96) for a spherical liquid droplet, with radius R, floating in a gas. Assume that the liquid has pressure P, the gas has pressure Pg,
EXERCISE 3.4. Compute the critical exponents,a, 3, 6, and y for a gas whose equation of state is given by the van der Waals equation.
EXERCISE 3.3. In the neighborhood of the triple point of ammonia (NH3), the equation for the sublimation curve is In(P) = 27.79 - 3726/T and the equation for the vaporization curve is In(P) = 24.10 -
Exercise 3.2. Compute the molar heat capacity of a vapor along the vaporization curve.
Exercise 3.1. Prove that the latent heat must always be positive (heat is absorbed) when making a transition from a low-temperature phase to a high- temperature phase. G1 G11 G11 Gi Te T
Locate all period-3 points of the Baker map in the (p,q) plane.
Consider a harmonic oscillator with Hamiltonian = (1/2m)p+ mu. Assume that at time = 0 the oscillator is in an eigenstate of the momentum operator, (0) Po) (pol. (a) Write the Liouville equation in
Consider a system with one degree of freedom whose dynamics is governed by a Hamiltonian of the form H(p,q) = p+9=E, where E is the total energy. Assume that initially p(p,q, 0) = (1/)6(p)e. Solve
Consider a particle with mass m = 1 moving in an infinite square well potential, V(x) = 0 for -1 < x < 1 and V(x) = 0 otherwise. Assume that initially the particle lies at x=-1 with momentum, p = po
Consider a particle which bounces vertically in a gravitational field, as discussed in Exercise 6.1. Assume an initial probability distribution, p(p,z, 0) = 108(z)(1.0 p)(p -0.1)(e(x) is the
Consider a system of N uncoupled harmonic oscillators with Hamiltonian, H=(p/2m+kq/2). Assume that the system initially has a probability density P(p,q,0)=6(pi-Po)&(q-90). Compute the probability
EXERCISE 6.8. Compute the trace of the Baker map, ".
EXERCISE 6.7. Compute the Wigner function for a system with a density operator, p=hab(e-are-b + e-bp e-ai).
EXERCISE 6.6. An ensemble of silver atoms (each with spin) is prepared so that 60% of the atoms are in the S =+ eigenstate of S, and 40% of the atoms are in the S, = -eigenstate of S, (S, and , are
EXERCISE 6.5. Consider a harmonic oscillator with Hamiltonian, = (1/2m)(p2+mw22). Assume that at time = 0 the oscillator is a state described by the density operator, (0) = hab(e-a e-bp + e-bp e-ax),
EXERCISE 6.4. Consider a dynamical flow on the two-dimensional unit square, 0< p< 1 and 0
EXERCISE 6.3. Compute the structure function for a gas of N noninteracting particles in a box of volume V. Assume that the system has a total energy E.
EXERCISE 6.2. A system of N particles has a Hamiltonian HP/2m+1)/2 V(q q). The phase function which gives the particle density at position R in configuration space is n(q, R) (q, - R). Write the
EXERCISE 6.1. Consider a particle which bounces elastically and vertically off the floor under the influence of gravity (assume no friction acts). At time t=0, the particle is located at z=0 and has
Problem 2.14. Compute the Helmholtz free energy for a van der Waals gas. The equation of state is (P+ (an/V2)) (V - nb) = nRT, where a and b are constants which depend on the type of gas and n is the
Problem 2.15. Prove that (a) T (Cp - Cv) = TV and (b) (Cp/Cy) = (KT/KS).
Problem 2.13. Compute the entropy, enthalpy, Helmholtz free energy, and Gibbs free energy of a paramagnetic substance and write them explicitly in terms of their natural variables when possible.
Problem 2.16. Show that Tds = c,(OT/OY),dY+cy (OT/ax)ydx, where x = x/n is the amount of extensive variable, X, per mole,c, is the heat capacity per mole at constant x, and cy is the heat capacity
Problem 2.12. Prove that CYN and Y,N T,N =T (*)(*)* -X. Y,N
Problem 2.17. Compute the molar heat capacity cp, the compressibilities KT and Ks, and the thermal expansivity ap for a monatomic van der Waals gas. Start from the fact that the mechanical equation
Problem 2.18. Compute the heat capacity at constant magnetic field CH,n, the susceptibilities XT, and XS,, and the thermal expansivity Q for a magnetic system, given that the mechanical equation of
Problem 2.19. A material is found to have a thermal expansivity ap = (R/Pv)+ (a/RT2v) and an isothermal compressibility T = (1/v)(Tf(P)+(b/P)), where v (V/n) is the molar volume. (a) Find f(P). (b)
Problem 2.20. Compute the efficiency of the reversible two heat engines in Fig. 2.20. Which engine is the most efficient? (Note that these are not Carnot cycles. The efficiency of a heat engine is n
Problem 2.21. It is found for a gas that T = Tvf (P) and ap = (Rv/P) + (Av/T), where T'is the temperature, v is the molar volume, P is the pressure, A is a constant, and f(P) is an unknown function
EXERCISE 2.7. Consider a fluid with electric potential, o, containing v different kinds of particles. Changes in the internal energy can be written, dU=4Q-PdV+ode + jdn. Find the amount of Gibbs free
Problem S2.8. Consider the reaction 2NH3 = N2 + 3H2, which occurs in the gas phase. Start initially with 2 mol of NH3 and 0 mol each of H and N2. Assume that the reaction occurs at temperature 7 and
Problem S2.7. Consider the reaction = 2HI=2H2 +12, which occurs in the gas phase. Start initially with 2 mol of HI and 0 mol each of H and 12. Assume that the reaction occurs at temperature 7 and
Problem S2.6. A solution of particles A and B has a Gibbs free energyInitially, the solution has n moles of A and ng moles of B. (a) If an amount, Ang, of B is added keeping the pressure and
Problem S2.4. An insulated box with fixed total volume V is partitioned into m insulated compartments, each containing an ideal gas of a different molecular species. Assume that each compartment has
Problem S2.3. Two containers, each of volume V, contain ideal gas held at temperature T and pressure P. The gas in chamber 1 consists of Na molecules of type a and N,b molecules of typeb. The gas in
Problem S2.2. An insulated box is partitioned into two compartments, each containing an ideal gas of different molecular species. Assume that each compartment has the same temperature but different
Problem S2.1. Consider a gas obeying the Dieterici equation of state,where a and b are constants. (a) Compute the Joule coefficient. (b) Compute the Joule- Kelvin coefficient. (c) For the throttling
Problem 2.22. A monomolecular liquid at volume V, and pressure PL is separated from a gas of the same substance by a rigid wall which is permeable to the molecules, but does not allow liquid to pass.
Problem S2.5. A tiny sack made of membrane permeable to water but not NaCl (sodium chloride) is filled with a 1% solution (by weight) of NaCl and water and is immersed in an open beaker of pure water
Problem 2.11. For a low-density gas the virial expansion can be terminated at first order in the density and the equation of state iswhere B(T) is the second virial coefficient. The heat capacity
Problem 2.10. Two vessels, insulated from the outside world, one of volume V and the other of volume V2, contain equal numbers N of the same ideal gas. The gas in each vessel is orginally at
EXERCISE 2.10. An experiment is performed in which the osmotic pressure of a solution, containing suc moles of sucrose (C12H22O11) and 1 kg of water (H2O), is found to have the following values [2]:
EXERCISE 2.9. A mixture of particles, A and B, has a Gibbs free energy of the formwhere n = n + NB, XA = n, and xB = nB (n indicates mole number), and are functions only of P and T. Plot the region
EXERCISE 2.8. Compute the molar heat capacities,c, and cp, the compressibilities, KT and K, and the thermal expansivity, ap, for a monatomic ideal gas. Start from the fact that the molar entropy of
EXERCISE 2.6. Consider a system which has the capacity to do work, AW-YaX+dW'. Assume that processes take place spontaneously so that ds = (1/T)4Q+d;S, where d;S is a differential element of entropy
EXERCISE 2.1. Consider the differential do = (x + y)dx + xdy. (a) Show that it is an exact differential. (b) Integrate do between the points A and B in the figure below, along the two different
EXERCISE 2.5. Compute the Helmholtz free energy for n moles of a monatomic ideal gas and express it in terms of its natural variables. The mechanical equation of state is PV = nRT and the entropy is
EXERCISE 2.4. Compute the enthalpy for n moles of a monatomic ideal gas and express it in terms of its natural variables. The mechanical equation of state is PV = nRT and the entropy is S = nR + nR
EXERCISE 2.3. The entropy of n moles of a monatomic ideal gas is S=(5/2)nRnR in[(V/Vo) (no/n) (T/To)/2], where Vo, no, and To are constants (this is called the Sackur-Tetrode equation). The
EXERCISE 2.11. Consider the reaction N2O4 = 2NO2, which occurs in the gas phase. Start initially with 1 mol of N2O4 and no NO2. Assume that the reaction occurs at temperature T and pressure P. Use
EXERCISE 2.12. Consider a vessel held at constant temperature T and pressure P, separated into two disjoint compartments, I and II, by a membrane. In each compartment there is a well-stirred, dilute
Problem 2.1. Test the following differentials for exactness. For those cases in which the differential is exact, find the function u(x, y). (a) dua = + (b) dub (c) duc == = xdy (y-x)dx + (x + y)dy.
Problem 2.9. Blackbody radiation in a box of volume V and at temperature T has internal energy U = aVT4 and pressure P = (1/3)aT4, where a is the Stefan-Boltzmann constant. (a) What is the
Problem 2.8. Experimentally one finds that for a rubber bandwhere J is the tension, a = 1.0 x 10 dyne/K, and Lo = 0.5 m is the length of the band when no tension is applied. The mass of the rubber
Problem 2.7. Compute the efficiency of the heat engine shown in Fig. 2.19. The engine uses a rubber band whose equation of state is J = aLT, where a is a constant, J is the tension, L is the length
Problem 2.6. One kilogram of water is compressed isothermally at 20C from 1 atm to 20 atm. (a) How much work is required? (b) How much heat is ejected? Assume that the average isothermal
Problem 2.5. Find the efficiency of the engine shown in Fig. 2.18. Assume that the operating substance is an ideal monatomic gas. Express your answer in terms of V and V2. (The processes 12 and 3 4
Problem 2.3. Electromagnetic radiation in an evacuated vessel of volume V at equilibrium with the walls at temperature 7 (black body radiation) behaves like a gas of photons having internal energy U
Problem 2.4. A Carnot engine uses a paramagnetic substance as its working substance. The equation of state is M = (nDH/T), where M is the magnetization, H is the magnetic field, n is the number of
Problem 2.2. Consider the two differentials (1) du = (2xy + x)dx + xdy and (2) duzy(x-2y)dx-xdy. For both differentials, find the change in u(x, y) between two points, (a,b) and (x, y). Compute the
Exercise 2.2. Compute the efficiency of a Carnot cycle (shown in the figure below) which uses a monatomic ideal gas as an operating substance. P Th P2 P P3 Te 3 V V4 V2 V3
1.1 Show that for a quasistatic adiabatic process in a perfect gas, with constant specific heats,(γ ≡ Cp / Cy)
1.2 The molar energy of a monatomic gas which obeys van der Waals’ equation is given by where V is the molar volume at temperature T, and a is a constant. Initially one mole of the gas is at the
1.3 Calculate the work done on 1 mole of a perfect gas in an adiabatic quasistatic compression from volume Vl to V2.
1.4 The enthalpy H is defined by H=E + PV. Express the heat capacity at constant pressure in terms of H.
1.5 One mole of a perfect gas performs a quasistatic cycle which consists of the following four successive stages: (i) from the state(P1, V2) at constant pressure to the state (P1, V2), (ii) at
1.6 The same as problem 1.5 with the four stages of the cycle: (i) at constant volume from (T1, V1) to (T2, V1), (ii) isothermally to (T2,V2), (iii) at constant volume to (T1, V2), (iv) isothermally
1.7 Calculate the change in internal energy when 1 mole of liquid water at 1 atm and 100 °C is evaporated to water vapour at the same pressure and temperature, given that the molar volumes of the
2.1 In a monatomic crystalline solid each atom can occupy either a regular lattice site or an interstitial site. The energy of an atom at an interstitial site exceeds the energy of an atom at a
2.2 A system consists of N weakly interacting subsystems. Each subsystem possesses only two energy levels E1 and E2 each of them non-degenerate.(i) Draw rough sketches (i.e. from common sense, not
2.3 A system possesses three energy levels E1 = ε, E2 = 2ε and E3= 3ε, with degeneracies g(E1) = g(E3) = 1, g(E2) = 2. Find the heat capacity of the system.
2.4 Consider the problem of the Schottky defects studied in section .For a system consisting of N atoms and possessing n such defects write down the Helmholtz free energy F(n). For a system in
2.5 One knows from spectroscopy that the nitrogen molecule possesses a sequence of vibrationally excited states with energies Er= ħω(r + ), r = 0, 1, 2, . . . . If the level spacing ħω is 0.3 eV,
2.6 The first excited state of the helium atom lies at an energy 19.82 eV above the ground state. If this excited state is three-fold degenerate while the ground state is non-degenerate, find the
2.7 The following may be considered a very simple one-dimensional model of rubber. A chain consists of n( 1) links each of lengtha. The links all lie along the x-axis but they may double back on
2.8 It follows from Eq. (2.32) that the partition function of a macroscopic system is given by
3.1 A crystal contains N atoms which possess spin 1 and magnetic moment μ. Placed in a uniform magnetic field the atoms can orient themselves in three directions: parallel, perpendicular and
3.2 A magnetic specimen consists of N atoms, each having a magnetic moment μ. In an applied homogeneous magnetic field each atom may orient itself parallel or antiparallel to the field, the energy
3.3 A paramagnetic crystal contains N magnetic ions which possess spin and a magnetic dipole moment μ. The crystal is placed in a heat bath at temperature Tand a magnetic field is applied to the
3.4 The magnetic moment of nuclei is of the order of 10–26A m2.Estimate the magnetic field required at 0.01 K to produce appreciable alignment of such nuclei.
4.1 1,000 g of water at 20 °C are placed in thermal contact with a heat bath at 80 °C. What is the change in entropy of the total system(water plus heat bath) when equilibrium has been
4.2 Two vessels A and B each contain N molecules of the same perfect monatomic gas. Initially the two vessels are thermally isolated from each other, the two gases being at the same pressure P, and
4.3. Two vessels contain the same number N of molecules of the same perfect gas. Initially the two vessels are isolated from each other, the gases being at the same temperature T but at different
4.4 When 1 mole of nitrous oxide decomposes into nitrogen and oxygen, the system being at 25 °C and 1 atm both initially and finally, the entropy S of the system increases by 76 J/K and its enthalpy

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