Chemistry 660 - EXTRA PROBLEMS
Prof. James Ingle
CHAPTER 2
2-17. A plasma is used to excite sodium atoms. The signals with a standard and blank solution are 0.56 V and 0.09 V, respectively. When the photodetector shutter is closed, the signal is 0.02 V. Specify or calculate the dark signal, the background emission signal, and the analyte emission signal.
2.18. A photoluminescence experiment is conducted on copper atoms in a flame. The total luminescence signal and blank luminescence signals are 1.34 V and 0.15 V, respectively. With the light source shutter blocked, the signal is 0.10 V with the blank or sample solution. Calculate the analyte luminescence signal, the analyte emission signal, and the background scattering signal. Assume the background luminescence is negligible.
2.19. With the light source shutter open, the total reference and sample signals are 1.34 and 0.63 V, respectively. The readout signal is 0.34 V with the light source shutter closed. In a separate experiment the analyte emission signal is determined to be 0.03 V. Calculate the true transmittance and absorbance. Assume all luminescence signals are negligible.
2-20. In a flame emission experiment for potassium, the analyte emission signal was 0.40 V and the blank signal was 0.10 V. What was the total emission signal for the sample?
2.21. In a fluorescence procedure for vitamin B2, the total luminescence signal and blank luminescence signals are 1.50 V and 0.20 V, respectively. With the light source shutter blocked, the signal is 0.05 V. Calculate the analyte luminescence signal and the background luminescence signal. Assume that scattering is negligible. Thermal emission from molecules in solution is normally considered to be insignificant.
2.22. Consider an absorption experiment. With the light source shutter open, the total reference and sample signals are 2.10 and 1.05 V, respectively. The readout signal is 0.10 V with the blank and the light source shutter closed. In a separate experiment the analyte luminescence signal is determined to be 0.05 V. Calculate the true transmittance and absorbance. Assume that background luminescence and all emission signals are negligible.
2.23 Potassium is one of the easier-to-excite elements with a transition to the first excited state at 766.5 nm. What temperature is needed to excite 0.1% and 0.2% of the potassium atoms. At the temperature needed to excite 0.1% to the first excited state, what fraction of the atoms are in the second excited state (8 = 404.7). Use reasonable assumptions. For this calculation, assume that the statistical weights of all states are the same. The energy levels in the question are really doublets but we will ignore that fact in this problem.
2.24. A spherical source with a radius of 1.0 cm emits a total radiant power of 1.0 W. The source is 1.0 m from a square receptor with an area of 2.0 cm2 that directly faces the source. Calculate the following: a) the source radiant intensity, b) the source radiance (consider what the projected area of a sphere is), c) the solid angle of the source viewed by the receptor, d) the radiant power striking the receptor, e) the irradiance at the surface of the receptor .
CHAPTER 3
3.31. Calculate the number density of a 1.0 micromolar solution of glucose.
3.32. For sapphire (data in appendix B), calculate the angle of refraction for a 589 nm light beam entering from air at a 10 degree angle with respect to the normal to the surface of the crystal. Calculate the critical angle for internal reflection for going from sapphire into air.
3.X Calculate the groove spacing of the CD that we looked at in class with the laser (from the distance to the wall and the distance between the zero and first order images. The laser has a wavelength of 532 nm.
CHAPTER 4
4-19. Calculate the spectral radiance of a blackbody at 4000 K at 500 nm (use figure 4-1 to double check your that your number is near the right value). By what factor does the spectral radiance at 500 nm increase if the above blackbody temperature is increased 200 K?
4-20. Explain for a laser based on a 4-level system, why it is not necessary to empty more than half the ground state to get lasing action.
4-21. a) Calculate the spectral radiance of a blackbody at 3000 K at 600 nm (use figure 4-1 to double check your that your number is near the right value). b). What must the blackbody temperature be to achieve the same spectral radiance at 300 nm that was found in part a at 600nm?
4-22. Explain what characteristics of a given type of laser (e.g., wavelength, output radiant power) you would expect to change and how if just the length of the laser cavity was doubled.
CHAPTER 5
5-21. A photon rate of 1.0 x 106 s-1 is incident on a PD detector. The quantum efficiency is 0.10. The integration time is 10.0 s. Calculate the following:
a. The output signal photoelectron rate rs and the output photocurrent is.
b. The number of electrons ns counted in 10 s.
c. The shot noise in nS and the S/N.
d. The bandwidth constant (K), the shot noise in is and the S/N. (should be the same S/N as calculated in c).
e. What would the answers be for a-d if the incident photon rate was 100 times greater? Calculate based on multiplying the original values by the appropriate factor rather than by calculating from scratch.
f. If the detector were a PMT with a current gain (m) of 1.0 x 106 an it was connected to a signal processor with a transfer function (G) of 1.0 x 106 V/A, what would be the signal voltage (ES), the bandwidth constant K, the shot noise in ES, and the S/N? Assume the secondary emission factor is 0.3.
5-22. This problem follows up on problem 5-21, parts a-d (PD detector with 10-s integration time). Hence, the shot noise calculated in problem 5-21 will be present it all parts of this problem.
a. First consider the addition of some signal flicker noise and assume that the signal flicker factor is 0.0020. In counts and current calculate the signal flicker noise, the total noise (st), and the S/N. By what factor is the S/N degraded by signal flicker noise?
b. Now consider adding some blank noise (assume the signal flicker noise above is absent and the integration time remains at 10 s). Assume that the dark electron rate is 1000 electrons/s. There is not background signal. In counts and current calculate the blank (dark) noise and the total noise (st). Calculate the S/N. Is the S/N significantly degraded by dark noise?
c. Dark (detector) noise often limits low level light measurements. Calculate the output signal photoelectron rate rs and photocurrent where the S/N = 1 (integration time remains at 10 s). What photon rate does this correspond to (note that this is the NEP)?
5-23. The purpose of this problem is to consider the effect of amplifier noise on the S/N. Assume that the PD in problem 1 is connected to a 1.0 x 106 W resistor to get a voltage signal (remember W has units of V/A). The fundamental noise from a resistor is called Johnson noise (p. 148 in text book, refer to the list of corrections for an on this page) and the formula at 25 C is sJ = (1.6 x 10-20RDf)1/2 and sJ has units of V (the constant has units of
W -1 Hz-1). Ignore signal flicker noise or dark current noise. Calculate the following.
a. The analytical signal ES and the Johnson noise in V. What does this Johnson noise correspond to in terms of number of electrons? With a 1-s integration time, Df = 1/(2t) = 0.05 Hz.
b. The signal shot in terms of V (from 5-21d, you have it in A)
c. The total noise (signal shot + Johnson noise) in V.
d. The S/N. How much does Johnson noise affect the S/N?
e. The total noise and S/N for the PMT (prob 5-21f). You already have Es and the signal shot noise as G was previously specified to be 106 which it would be with the above resistor. Which detector (PMT or PD) would you use?
5-24. A signal photon rate of 2.0 x 107 s-1 is incident on a pixel PDA detector. The quantum efficiency is 0.50. The integration time is 10.0 s. Calculate the following:
a. The rate of generation of electron-hole pairs rS .
b. The number of electron-hole pairs nS counted in 10 s.
c. The shot noise in nS and the S/N (assume only signal shot noise is present).
d. What would the answers be for a-d if the incident photon rate was 100 times greater? Calculate based on multiplying the original values by the appropriate factor rather than by calculating from scratch.
e. Now consider the addition of some signal flicker noise and assume that the signal flicker factor is 0.0010 (same photon rate as in parts a-c). In counts calculate the signal flicker noise, the total noise (st), and the S/N. By what factor is the S/N degraded by signal flicker noise?
5-25. This problem follows up on problem 5-24, parts a-c (PDA detector with 10-s integration time). Hence, the signal and signal shot noise calculated in problem 5-24 will be present it all parts of this problem. For this problem, assume the signal flicker noise in 5-24e is absent and the integration time remains at 10 s.
a. Now consider adding some blank noise. Assume that the dark electron rate is 1000 electrons/s. There is no background signal. In counts, calculate the blank (dark) noise and the total noise (st). Calculate the S/N. Is the S/N significantly degraded by dark noise?
b. Dark (detector) noise often limits low level light measurements. Calculate the signal electron-hole pair rate rS where the S/N = 1 (integration time remains at 10 s) (think about approximations to make it easier) (note that this is the NEP)?
c. Finally, some background signal and noise will be added. A background photon rate of 2.0 x 107 s-1 is incident on a pixel PDA detector. Calculate the S/B and the S/N. Can obtain a good S/N with a low S/B? Under what conditions will one obtain a low S/N with a good S/B?
5-26 An intensifier (see figure 4-27 in the textbook) is placed between the source of light and the PDA in problem 5-24. At the wavelength monitored, it has the same quantum efficiency as the PDA and produces a light gain of 1000 (an average of 1000 photons output to the PDA pixel for every photon incident on the photocathode of the intensifier).
a. Calculate the S/N if the incident photon rate is the same as that which produces the signal electron-hole rate calculated for a S/N of 1 in problem 5-25a.
b. What would dark current generated at the intensifier photocathode do to the S/N.
5-27. A signal photon rate of 5.0 x 107 s-1 is incident on a PMT which is monitored with a photon counting system. The quantum efficiency is 0.20. The integration time is 10.0 s. Calculate the following:
a. The rate of generation of photoelectrons (or electron packets at the anode) rS .
b. The number of electron packets at the anode nS counted in 10 s.
c. The shot noise in nS and the S/N (assume only signal shot noise is present).
d. What would the answers be for a-d if the incident photon rate was 100 times greater? Calculate based on multiplying the original values by the appropriate factor rather than by calculating from scratch.
e. Now consider the addition of some signal flicker noise and assume that the signal flicker factor is 0.0020 (same photon rate as in parts a-c). In counts calculate the signal flicker noise and the total noise (st). Calculate the S/N. By what factor is the S/N degraded by signal flicker noise?
5-28. This problem follows up on problem 5-24, parts a-c (PMT detector with 10-s integration time). Hence, the signal and signal shot noise calculated in problem 5-27 will be present it all parts of this problem. For this problem, assume the signal flicker noise in 5-27e is absent and the integration time remains at 10 s.
a. Now consider adding some blank noise. Assume that the dark electron rate is 10,000 electrons/s. There is no background signal. In counts, calculate the blank (dark) noise and the total noise (st). Calculate the S/N. Is the S/N significantly degraded by dark noise?
b. Dark (detector) noise often limits low level light measurements. Calculate the signal photoelectron pulse rate rS and signal photon level (s-1) where the S/N = 1 (integration time remains at 10 s) (think about approximations to make it easier) (note that this is the NEP)?
c. Finally, some background signal and noise will be added. A background photon rate of 1.0 x 108 s-1 is incident on the PMT (in addition to the signal photons and dark thermal electrons). Calculate the background noise and blank noise in counts. Calculate the S/B and the S/N. Can one obtain a good S/N with a low S/B? Under what conditions will one obtain a low S/N with a good S/B?
5-29. In an absorption experiment, identify the possible dominant noise from the following information.
a. In a plot of the RSD in A versus A from 0 to 3 AU, no minimum is observed.
b. When the absorbance increases from 1.0 to 2.0, the noise in absorbance units increases by a factor of 101/2.
5-30. If the minimum in the plot of RSD in A versus A is between 1.5 and 2, what noise sources are limiting in that absorbance region.
5-31. If the total noise in an emission measurement is found to be proportional to the square root of the total emission signal as the analyte concentration is varied, what are are the possible limiting noise sources (be specific - just shot noise or just flicker noise is not adequate)?
CHAPTER 6
6-13. The following atomic emission measurements for Ca were made on an ICP spectrophotometer.
Solution
Signal (EtE or Ebk), V
100 ng/mL standard 1.05 100 ng/mL standard 1.03 100 ng/mL standard 1.04 100 ng/mL standard 1.08 blank 0.00 blank 0.01 blank -0.01 blank 0.00 blank 0.02 sample 0.53 sample 0.52 sample 0.55 Calculate the following:
a. The mean signal and standard deviation for all three solutions.
b. The calibration sensitivity (m) (based on one point here).
c. The detection limit based on a confidence factor of 3.
d. The concentration of Ca in the sample and the range of Ca concentration for which the true value is contained with a 98% level of confidence.
e. The concentration of Ca in the sample was determined by a reference method to be 52.0 ng/mL. Do the data indicate there might be a systematic error or not?
6-14. Solutions of an 1) analyte, 2) analyte plus a potential interferent, 3) blank, and 4) just interferent are prepared. Solutions 1) and 2) give the same total and analytical signal. However the signal for solution 4) is greater than that for solution 3). In terms of additive and multiplicative interferences, what appears inconsistent and what is a plausible explanation?
6-15. In an emission experiment, the detection limit for Na is determined to be 3 nM based on a confidence factor of 3, 10 blank measurements, and a calibration curve with a slope of 1.00 V/µM. Four measurements of a sample with a known Na concentration of 1.00 µM indicates a mean concentration of 1.05 µM and a standard deviation of 0.020 µM.
a. Calculate the standard deviation of the blank signal.
b. Do the data suggest that the difference between the true and measured mean in the sample is due to random or systematic error? Justify your answer.
6-16. Ten paired measurements of the sample and blank with absorption spectrophotometry yielded a mean analyte concentration of 5.00 nM with a standard deviation of 0.100 nM. Calculate the concentration range that contains the true mean concentration with a confidence level of 98% and of 95%.
6- 17. Consider a sample solution and a 1:2 dilution and a 1:4 dilution of the sample solution. All three solutions are analyzed with a given technique and a good calibration curve. The analyte concentration in the original sample is calculated using the correct dilution factor. An interference is present in the sample and the concentration determined with all dilutions of the sample is too high in all cases. Would you expect the percent error in the determined concentration to vary with the sample solution dilution or to be independent of the sample dilution. Discuss the answer first assuming that the interference is additive and second if it is multiplicative. There is not a precise answer to this problem, and I am looking for a thoughtful discussion.
6-18. If a concomitant in sample attenuates the photoluminescence from the analyte by absorbing some of the analyte luminescence before it exits the sample cell, is the concomitant an additive or multiplicative interference? Explain your answer.
CHAPTER 7
7-17. Continuing with problem 5, calculate the collisional half-width in nanometers for Ca. Assume that N2 is the primary perturber with a number density of 2.0 x 1018 cm-3 and the collisional cross section is 2.0 x 10-14 cm2.
7-18. Calculate and compare the free ground state atom + ion number density in a flame and in a plasma with pneumatic nebulization for an analyte concentration of 1.0 micromolar. Assume that the overall atomization efficiency is controlled by the nebulizer which has a nebulization efficiency of 3%. For the flame assume that the sample solution flow rate is 5 mL/min, the total gas flow is 10 L/min, and the gas expansion factor is 10. For the plasma, assume that the sample flow rate is 1 mL/min, the nebulizer gas flow is 1.0 L/min, and the gas expansion factor is 20 (higher because of the greater temperature). The gas flow rate that is used to calculate the dilution is less with an ICP plasma because the analyte remains in the central channel of the plasma and does not mix much with the outer plasma gas.
7-19. Calculate the Doppler half-width in picometers for Cu (324.7 nm transition) in an ICP at 6000 K.
7-20. Continuing with problem 7-19, calculate the collisional half-width in Hz and picometers for Cu. Assume that Ar is the primary perturber with a number density of 1.0 x 1018 cm-3 and the collisional cross section is 2.0 x 10-14 cm2.
7-21. If the temperature in a given atomizer could be increased, how would you expect the absorption coefficient at the maximum and the absorbance measured with an HCL to change if the overall atomization efficiency is not affected.?
7-22. Calculate ion number density of the analyte atom in a plasma with pneumatic nebulization for an analyte concentration of 1.0 nanomolar. Assume that the overall atomization efficiency is controlled by the nebulizer which has a nebulization efficiency of 3.0% and a free atom fraction for the ion corresponding to half of the total metal species being ionized. Assume that the sample flow rate is 1.0 mL/min, the nebulizer gas flow rate is 1.0 L/min, and the gas expansion factor is 20. The gas flow rate that is used to calculate the dilution is not the plasma gas flow rate because the analyte remains in the central channel of the plasma and does not mix much with the outer plasma gas.
7-23. If 0.010 mL of a 1.0 micromolar solution of As is atomized completely atomized (assume an overall atomization efficiency of 1) into 1.0 L of gas, what is the number density for As. What concentration of As would provide a number density of 1 atom per cm3.
CHAPTER 8
8-15. Compare the S/N characteristics of a simultaneous and sequential ICP (same ICP and nebulizer). Assume that 9 elements are to be determined in the same total time of 9 s (i.e., the integration time per element would be 1 s for the sequential instrument). Assume the following: 1) a PMT is used for the sequential instrument in a photon counting mode, 2) a CTD is used for the simultaneous instrument, 3) the absolute signal level (electron-hole pairs per unit time) at a given wavelength for a given pixel in the simultaneous instrument is 1/10 of the photoelectron pulse rate with the sequential instrumental (the combination of a detector element area that is 1/50 the size but a quantum efficiency which is 5 times higher), 4) the dark signal level per pixel (dark counts per unit time) and the readout noise are the same for both photodetectors. For a given element, indicate S/N (simultaneous)/S/N (sequential) for the following situations (exact answers may not be possible in all cases):
a. readout noise limited (noise independent of integration time)
b. detector noise limited (dark current shot noise)
c. background emission shot noise limited
d. background emission flicker noise limited
8-16. Consider the example for ICP/MS on page 8-13 of the lecture notes. What would be the detection limit of the element with a 1-s integration time for a more complex sample where the background count rate was 2000 s-1?
8-17. Compare the S/N characteristics of a simultaneous and sequential ICP (same ICP and nebulizer). Assume that 1) 4 elements are to be determined in the same total time of 4 s (i.e., the integration time per element would be 1 s for the sequential instrument), 2) measurements are limited by a) background shot noise or b) signal shot noise. By what factor will the S/N be enhanced for a given element with the simultaneous instrument for case a) and b)?
8-18. See page 8-12 of lecture notes. For the best detection limit, what isotope of Fe would you determine with ICP/MS? Which isotope would be the worse if Ni was present?
8-19. see page 8-12 of lecture notes. For the best detection limit, what isotope of Cu would you determine with ICP/MS? Which isotope would be the worse choice if Zn was present at comparable levels?
revised by JDI 11/20/03