Questions and Answers of Mathematics Physical Chemistry

Use Mathematica to verify the partial fractions in the above example.
Find the values of the Cartesian coordinates that correspond to r = 5.00, θ = 45◦, ϕ = 135◦.
Draw vector diagrams and convince yourself that the two schemes presented for the construction of D = A − B give the same result.
Draw rough graphs of several functions from Table 6.1. Below each graph, on the same sheet of paper, make a rough graph of the derivative of the same function.Table 6.1Function, y =
Draw rough graphs of the third and fourth derivatives of the function whose graph is given in Figure 6.6. xpp -
Determine whether the following improper integrals converge. Evaluate the convergent integrals ʃπ0 tan (x)dx.a. ʃπ/20 tan (x) dx.b. ʃ10 (1/x) dx.
Show that no Maclaurin seriesf(x) = a0 + a1x + a2x2 +· · ·can be formed to represent the function f(x) = √x. Why is this?
Find the first few coefficients for the Maclaurin series for the functionf(x) = √1 + x.
Write an equation similar to Eq. (13.27) for the ??v operator whose symmetry element is the y?z plane. буха) (х1, У ,21) %3 (х1, — У1,21). (13.27)
Fit the data of the previous example to a quadratic function (polynomial of degree 2) and repeat thecalculation. Example 16.12. Fit the data in Table 16.1 to a fourth- degree polynomial. Obtain a
Carry out a linear least-squares fit on the following data, once with the intercept fixed at 2.00 and once without specifying the intercept:Compare your slopes and your correlation coefficients for
The Bouguer–Beer law (sometimes called the Lambert–Beer law or Beer’s law) states that A = alc, where A is the of a solution, defined as log10 (I0/I ), where I0 is the incident intensity of
Change the data set of Table 16.1 by adding a value of the vapor pressure at 70—¦C of 421 torr ± 40 torr. Find the least-squares line using both the unweighted and weighted
If a capacitor of capacitance C is discharged through a resistor of resistance R the voltage on the capacitor follows the formulaV(t) = V(0)e−t/RC.The following are data on the voltage as a
Take the data from the previous exercise and test for first order by carrying out an exponential fit using Excel. Find the value of the rate constant.Previous ExerciseThe following is a set of data
The following are (contrived) data for a chemical reaction of one substances.Time (min) Concentration (mol
The following is a set of data for the following reaction at 25 ◦C.(CH3)3CBr + H2O → (CH3)3COH + HBr.Using linear least squares, determine whether the reaction obeys first-order, second-order, or
Make a graph of the partial pressure of butadiene as a function of time, using the data in the previous problem. Find the slope of the tangent line at 33.00 min and deduce the rate constant from it.
Assuming that the reaction in Example 16.5 is first order, find the expected error in the rate constant, using the residuals as estimates of the errors.
Vaughan obtained the following data for the dimerization of butadiene at 326 ◦C.Time (min) Partial pressure of butadiene (atm) 0 to be
Sum the residuals in Example 16.5 and show that this sum vanishes in each of the three least-square fits.
Water rises in a clean glass capillary tube to a height h given bywhere r is the radius of the tube, Ï is the density of water, equal to 998.2 kg mˆ’3 at 20 —¦C, g is
Assume that the expected error in the logarithm of each concentration in Example 16.5 is equal to 0.010. Find the expected error in the rate constant, assuming the reaction to be first order.
The vibrational contribution to the molar heat capacity of a gas of nonlinear molecules is given in statistical mechanics by the formula i=1 " style="" class="fr-fic fr-dib">where ui =
Calculate the covariance for the following ordered pairs:y
The following is a set of student data on the vapor pressure of liquid ammonia, obtained in a physical chemistry laboratory course.(a) Find the indicated enthalpy change of
The following data give the vapor pressure of water at various temperatures. Transform the data, using ln (P) for the dependent variable and 1/T for the independent variable. Carry out the linear
In the cryoscopic determination of molar mass, the molar mass in kg molˆ’1is given bywhere W is the mass of the solvent, w is the mass of the unknown solute, Î”Tf is the amount
The van der Waals equation of state isFor carbon dioxide, a = 0.3640 Pa m6 molˆ’1 and b = 4.267 Ã— 10ˆ’5 m3 molˆ’1. Find the pressure of 0.7500 mol of carbon
Assume that you estimate the total systematic error in a melting temperature measurement as 0.20 ◦C at the 95% confidence level and that the random error has been determined to be 0.06 ◦C at the
Assuming that the ideal gas law holds, find the amount of nitrogen gas in a container ifP = 0.836 atm ± 0.003 atm,V = 0.01985 m3 ± 0.00008 m3,T = 298.3 K± 0.2 K.Find the expected error in the
Two time intervals have been clocked as t1 = 6.57 s ± 0.13 s and t2 = 75.12 s ± 0.17 s. Find the probable value of their sum and its probable error.
In order to determine the intrinsic viscosity [Î·] of a solution of polyvinyl alcohol, the viscosities of several solutions with different concentrations are measured. The intrinsic
Apply the Q test to the 39.75 ◦C data point appended to the data set of the previous example.
Assume that the H–O–H bond angles in various crystalline hydrates have been measured to be 108◦, 109◦, 110◦, 103◦, 111◦, and 107◦. Give your estimate of the correct bond angle and its
The following measurements of a given variable have been obtained: 68.25, 68.36, 68.12, 68.40, 69.70, 68.53, 68.18, 68.32. Apply the Q test to see if one of the data points can be disregarded.
Find the mean and the sample standard deviation for the following set of values: 2.876 m, 2.881 m, 2.864 m, 2.879 m, 2.872 m, 2.889 m, 2.869 m. Determine how many values lie below (x) − sx and how
The following measurements of a given variable have been obtained: 23.2, 24.5, 23.8, 23.2, 23.9, 23.5, 24.0. Apply the Q test to see if one of the data points can be disregarded. Calculate the mean
A sample of seven individuals has the following set of annual incomes: $40,000, $41,000, $41,000, $62,000, $65,000, $125,000, and $650,000. Find the mean income, the median income, and the mode of
A certain harmonic oscillator has a position given as a function of time byz = (0.150 m)[sin (Ï‰t)],whereThe value of the force constant k is 0.455 N mˆ’1 and the mass of the
Find the value of the z coordinate after 1.00 s and find the time-average value of the z coordinate of the particle in the previous example for the first 1.00 s of fall if the initial position is z =
Evaluate the most probable speed for nitrogen molecules at 298.15 K.
A certain harmonic oscillator has a position given byz = (0.150 m)[sin (Ï‰t)],whereThe value of the force constant k is 0.455 N mˆ’1 and the mass of the oscillator m is 0.544
Evaluate vrms for N2 gas at 298.15 K.
A sample of 10 sheets of paper has been selected randomly from a ream (500 sheets) of paper. The width and length of each sheet of the sample were measured, with the following results:(a) Calculate
Evaluate of (ν) for N2 gas at 298.15 K.
Find the third and fourth moments (defined in the previous problem) for the uniform probability distribution such that all values of x in the range ˆ’5.00 ‰º x ‰º 5.00
Find the expression for (v2x)1/2, the rootmean-square value of vx, and the expression for the standard deviation of vx.
The nth moment of a probability distribution is defined by
Find the expectation values for px and p2x for the lowest-energy state of our particle in a box of length L. Find the standard deviation.
Find the probability that x lies between μ − 1.500σ and μ + 1.500σ for a Gaussian distribution.
For the lowest-energy state of a particle in a box of length L, find the probability that the particle will be found between L/4 and 3L/4.
Assume that a random variable, x, is governed by the probability distribution (a version of the Lorentzian function)where x ranges from ˆ’10.000 to 10.000.(a) Find the mean value of x and
Show that the fraction of a population lying between μ−1.96σ and μ+1.96σ is equal to 0.950 for the Gaussian distribution.
Assume that a random variable, x, is governed by the probability distribution (a version of the Lorentzian function)where x ranges from ˆ’6.000 to 6.000.(a) Find the mean value of x and its
Calculate the mean and standard deviation of the Gaussian distribution, showing that μ is the mean and that σ is the standard deviation.
Assume that a random variable, x, is governed by the probability distributionwhere x ranges from 1.00 to 10.00.(a) Find the mean value of x and its variance and standard deviation.(b) Find the
If x ranges from 0.00 to 10.00 and if f (x) = cx2, find the value of c so that f (x) is normalized. Find the mean value of x, the root-mean-square value of x, and the standard deviation.
Consider the uniform probability distribution such that all values of x are equally probable in the range −5.00 ≺ x ≺ 5.00. Find the mean and the standard deviation. Compare these values with
Find the mean and the standard deviation for the distribution of “heads” coins in the case of 10 throws of an unbiased coin. Find the probability that a single toss will give a value within one
Calculate the mean and the standard deviation of all of the possible cases of ten throws for the biased coin in the previous problem.Previous ProblemAssume that a certain biased coin has a 51.0%
Calculate the probability that “heads” will come up 60 times if an unbiased coin is tossed 100 times.
Assume that a certain biased coin has a 51.0% probability of coming up “heads” when thrown.(a) Find the probability that in 10 throws five “heads” will occur.(b) Find the probability that in
Assume the following discrete probability distribution:Find the mean and the standard deviation. Find the probability that x lies between Œ©xŒª ˆ’Ïƒx and
Use Cramer’s rule to solve the simultaneous equations4x + y = 21,2x − 3y = −11.
Find the values of x2 and x3 for the previous example.
Find the third eigenvector for the previous example.
Show that the second eigenvector in the previous example is an eigenvector.
Use the rules of matrix multiplication to show that Eq. (14.3) is identical with Eqs. (14.1) and (14.2).a11x + a12 y = c1, (14.1)a21x + a22 y = c2. (14.2)AX = C, (14.3)
Solve the set of equations, using Cramer’s rule:3x1 + x2 + x3 = 19,x1 − 2x2 + 3x3 = 13,x1 + 2x2 + 2x3 = 23.Verify your result using Mathematica.
Solve the set of simultaneous equations:3x + 4y + 5z = 1,4x − 3y + 6z = 3,7x + 2y − 6z = 2.
Solve the set of simultaneous equations:y + z = 1,x + z = 2,x + y = 3.
Find the value of x1that satisfies the set of equations 10 X1 1-1 1 1 1 -1 1 [1 1 6. 1 х2 4 Хз -1] [ x4 ||
Solve the set of equations, using Gauss–Jordan elimination:x1 + x2 = 6,2x2 − x3 = 1,x1 + 2x2 = 5.Use Mathematica to confirm your solution.
Determine whether the set of four equations in three unknowns can be solved:x1 + x2 + x3 = 12,4x1 + 2x2 + 8x3 = 52,3x1 + 3x2 + x3 = 25,2x1 + x2 + 4x3 = 26.
Solve the equation:3x1 + 4x2 + 5x3 = 25,4x1 + 3x2 − 6x3 = −7,x1 + x2 + x3 = 6.
Solve the simultaneous equations by matrix inversion2x1 + x2 = 4,x1 + 2x2 + x3 = 7,x2 + 2x3 = 8.
Solve the equation: 1 1 3 6. 1 X1 2111 5 X2 12 3 4 10 хз [7 [201 4] Lx4. ||
Use Gauss–Jordan elimination to solve the set of simultaneous equations in the previous example. The same row operations will be required that were used in Example 13.16.2x1 + x2 = 1x1 + 2x2 + x3 =
Decide whether the following set of equations has a solution. Solve the equations if it does:3x + 4y + z = 13,4x + 3y + 2z = 10,7x + 7y + 3z = 23.
Find expressions for x and y in terms of z for the set of equations2x + 3y − 12z = 0,x + y − z = 0,2x − 3y = 0.
Solve the set of equations by matrix inversion. If available, use Mathematica to invert the matrix:2x1 + 4x2 + x3 = 40,x1 + 6x2 + 2x3 = 55,3x1 + x2 + x3 = 23.
Find the eigenvalues and eigenvectors of the matrix 111 1 1
Find the eigenvalues and eigenvectors of the matrix 10 1 1 1 0
The HÃ¼ckel secular equation for the hydrogen molecule isDetermine the two orbital energies in terms of Î± and Î². = 0. α
List as many sources of error as you can for some of the following measurements. Classify each one as systematic or random and estimate the magnitude of each source of error.(a) The measurement of
Find the eigenvalues and eigenvectors of the matrixDoes this matrix have an inverse? 10 10 1 10 1
In the HÃ¼ckel treatment of the cyclopropenyl radical, the basis functions are the three 2pz atomic orbitals, which we denote by f1, f2, and f3.Ïˆ = c1 f1 + c2 f2 + c3 f3.The
Pick a few pairs of 2 by 2 submatrices from Eq. (13.72) and show that they multiply in the same way as the 3 by 3 matrices. [100 0 10 E, 001 |-1/2 -V3/2 0 V3/2 -1/2 0= A, -1/2 V3/2 0 -/3/2 –1/2 0 =
Show by matrix multiplication that two matrices with a 2 by 2 block and two 1 by 1 blocks produce another matrix with a 2 by 2 block and two 1 by 1 blocks when multiplied together.
Verify several of the entries in the multiplication table by matrix multiplication of the matrices in Eq. (13.72). 100 0 10 = E, 00 1 -1/2 -/3/2 0 Ĉ3 + V3/2 -1/2 0 = A, -1/2 3/2 0 C + -3/2 -1/2 0 =
By transcribing Table 13.1 with appropriate changes in symbols, generate the multiplication table for the matrices in Eq. (13.65).Table 13.1A(BC) = (AB)C. (13.65) ốc дь ôa ốc ĉa дь ĉa ớc
Obtain the inverse of the following matrix by hand. Then use Mathematica to verify your answer. 130 304 120
Expand the following determinant by minors: 3 2 0 7 -1 5 2 3 4
Find the matrix P that results from the similarity transformationP = Xˆ’1QX,where 23 х 2 1 4 3
Use Mathematica or another software package to verify the inverse found in the preceding example.
Show that A(B + C) = AB + AC for the example matrices in the previous problem.
Show that (AB)C = A(BC) for the matrices: 01 2 3 14 C=| A=! -2 0 1 3 21 3 1 B= -4 -4 2 3 1 23 3 1 -2
Show that the properties of Eqs. (13.42) and (13.43) are obeyed by the particular matrices [10 1 123 -3 1 1 -2 -3 A=| 4 5 6 7 8 9 C=| 0 3 –2 B= 27 -7
Find the product 10 2 0 -1 1 3

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