Each practice problem below requires that you correctly apply propagation of errors to report the correct value and its precision (including the error). You should attempt all seven problems prior to our schedule lab meeting on this topic (and submit prior to the meeting on Canvas). This pre-lab assignment will prepare you to participate in the lab meeting by presenting problems on the board, answering questions from your peers, and asking your own questions.
You will submit your work and final answers after the in-lab problem session. Your answers to these problems may be hand-written or typed. However, any submissions that are deemed illegible, unclear, or poorly organized will not be accepted.
Note
(1) We will give you a MatLab template to get you help you in using MatLab for this problem set. However, you may use whatever technology (calculators, software, etc.) you like to aid in working these problems.
(2) It is important to show how your calculations are set-up and, where appropriate, give a sample calculation. Simply giving the final answer is not acceptable.
(3) In reporting all results, you should always round your reported answer to be consistent with your estimated uncertainty. For example: a calculated result of 56.2189 ± 0.0386 would be reported as 56.20 ± 0.04 where 0.04 is the standard deviation, 95% confidence level, or propagated uncertainty.
The seven problems are below. Note that there are Hints that you can click on for each problem.
PROBLEM 1: Statistics
A student makes five independent measurements of each of the linear dimensions of a box. The results are tabulated as follows:
| Length (cm) |
Width (cm) |
Height (cm) |
| 1.010 |
5.376 |
2.001 |
| 1.015 |
5.339 |
2.105 |
| 1.013 |
5.385 |
1.985 |
| 0.999 |
5.375 |
1.989 |
| 1.009 |
5.369 |
2.008 |
A. Calculate the average values of length, width, and height, and calculate the standard deviation in each of these three quantities. Report the values appropriately, considering the calcuated precision.
B. Calculate the variance of the length, width, and height.
C. Calculate the 95% confidence limits to the measurements of length, width, and height.
D. Calculate the 95% confidence limits to the average volume obtained from the independent measurements of length, width, and height.
- Hint
-
This problems is one where there are five replicate measurements of length, width, and height. That's enough to use some basic statistics to determine our confidence of each measurement. Normally the standard deviation is an appropriate indicator of the data's precision in a case like this where there are multiple measurements. The problem also asks for the variance and the 95% confidence limit. Be sure to report all values in the appropriate number of digits, considering the calculated precision. The equation for variance and for 95% confidence can be found in Section 3.2.
PROBLEM 2: A Simple Case for Propogation of Error
A student weighs a sample using an analytical balance that has an estimated precision of ±0.5 mg. From the data below, calculate the weight of the sample and the range of uncertainty in the net weight.
| |
Mass (g) |
| Sample + flask |
5.1647 ± 0.0005 |
| Tared flask |
2.3712 ± 0.0005 |
- Hint
-
This is a simple propagation of error problem – any time we add, multiply, divide, or subtract multiple measurements, the final value's error is a propagation (or a combo of) the error associated with each measurement. So, you just need to recognize this fact, and then find the appropriate equation to propagate error in the case when values are subtracted. The appropriate equation is given in Section 3.2 (also on pg 57 of Shoemaker, equation 55...but in the Shoemaker text, you'd need to take the square root of both sides of the equation).
In this case, the function (F) that determines the sample weight is \(F = \text{Mass of Sample in Flask }-\text{ Mass of empty flask} = x - y\)
Apply Equation 3.2.7 where \(a, b= 1\) (and thus can be ignored), and the uncertainty in the measurements are \(\Delta x = \Delta y = 0.0005 \; g\). Specifically for this case, the uncertainty is:
\[\Delta F = \sqrt{(0.0005)^2 + (0.0005)^2}\]
PROBLEM 3: Statistics and Rejection of Discordant Data
An investigator measures the photon emission in the reaction
\( Na^{+} + O^{-} \rightarrow Na^{*} + O \overset{h\nu }{\rightarrow} Na + O \)
in which the emitted photon (at 589 nm) is counted by means of a photomultiplier. Here are some typical "run" data
|
Run #
|
Signal + Background
(counts/sec)
|
Background
(counts/sec)
|
| 1 |
1500 |
1000 |
| 2 |
1200 |
1250 |
| 3 |
1375 |
1100 |
| 4 |
1425 |
900 |
| 5 |
1280 |
1200 |
| 6 |
1490 |
1363 |
Evaluate the data using statistical methods only and calculate:
A. The average signal value.
B. The standard deviation and variance in the average signal value.
C. The 95% confidence limits to the average signal value.
D. Using the Q test, determine whether the results from any one run can be rejected as discordant data at the 90% confidence level
- Hint
-
This is a noisy data set where the signal is just barely detectable. The problem explicitly asks for calculation of variance (variance is just the square of standard deviation), 95% confidence limit (see problem 1), and application of the Q-test to determine whether any of the data points should be rejected. The Q-test is described in Section 3.3.1.
PROBLEM 4: Least Squares Analysis and Rejection of Discordant Data
A. Fit a straight line:
\( y = a + bx \)
to the following set of data points by the linear least-squares method
x 26 30 44 50 62 68 74
y 92 85 78 81 54 51 40
and calculate the coefficients \(a\) and \(b\) using the Equations (3.4.5) and (3.4.6) from Section 3.4.
B. Plot the x,y data and the least-squares line, and submit this with your report.
C. Calculate the standard deviation of the coefficients, \(a\) and \(b\) using Equations (3.4.11), (3.4.12) and (3.4.13) from from Section 3.4 (note that the lsq script does this automatically for you, and you are welcome to use it!).
D. Calculate the correlation coefficient, using equation (3.4.14) from Section 3.4. What is the significance of the sign of r?
E. Apply the Q-test to the y-residuals (see Equation (3.4.2) from Section 3.4), to determine whether there is a statistical basis for rejection of any of the data points as an outlier, (I) at the 96% confidence level (II) at the 90% confidence level.
F. If a data point is rejected, repeat the regression analysis (steps A-D) using the remaining data.
G. Comment quantitatively on the effect of data removal on the regression analysis.
- Hint
-
You will need to plot the x,y data and then perform a linear fit. You are explicitly instructed to determine the standard deviation of the slope and y-intercept of the least squares fit linear function. There are three ways I know to determine these values: (1) Excel, using the "LINEST" array function. (2) MatLab, using the lsq custom script or the Statistics and Machine Learning Toolkit (3) Logger pro does it automatically with linear fitting. The LSQ script is recomended.
Hints for individual parts:
- A,B) - just plot the data and best fit line and get the equation.
- C, D) use lsq (Matlab)
- E) Following the linear regression analysis, perform a Q-test at two different confidence intervals - 96% and 90%. For this, you will need the critical Q-values for these two confidence limits (see Section 3.4.1).
PROBLEM 5: Error Propagation
In a chemistry lab, students obtain the following data and are asked to find the molecular weight (M) of methanol using PV = nRT. Determine the uncertainty in the result using a propagation of error treatment.
| |
Pressure (methanol) |
Temperature |
mass (methanol) |
| Trial 1 |
72.5 ± 0.5 mmHg |
295.10 ± 0.05 K |
0.1331 ± 0.0002 g |
| Trial 2 |
72.0 ± 0.5 mmHg |
295.05 ± 0.05 K |
0.1329 ± 0.0002 g |
The volume (V) in both trials was 1.05678 liter, with no uncertainty given. You must decide on the uncertainty in V. If necessary, perform appropriate conversions of the units for each quantity, including R, so that the final value of M is reported in the correct units.
- Hint
-
This is a propagation of error problem, where we are multiplying and dividing individual measurements to find the value of interest. In this case, it is the determination of the molecular weight (which I'm going to represent as MW) of methanol from the mass of methanol, pressure, temperature, and volume measurements.
So, first…how would we do this?
\[M W=\frac{\text { mass of methanol in grams }}{\text { moles methanol }}=\frac{m}{n}\]
The mass of methanol was measured in the experiment, and the number of moles can be determined from \(PV=nRT\). The calculation involves multiplication and division to determine \(n=\dfrac{PV}{RT}\), and then involves division again to determine \(MW=\dfrac{m}/{n}.
We can substitute into the equation for \(MW\) and get everything into one combined equation:
\[MW=\frac{m}{n}={m}{\frac{PV}{RT}}={mRT}{PV}\]
Once you recognize that it is simply a propagation of errors problem in which the factors are multiplied and divided, you can determine the error in the molecular weight value using Equation 3.2.8.
The errors in P, T, and m are explicitly given. The error in V is not explicitly given, but we can assume that the error is indicated in the number of digits given in the volume measurement. The volume given is 1.05678 liters or 1,056.78 mL - so we assume we are uncertain in the last digit (the 8) by roughly half a digit…. to +/- 0.00005 L or +/- 0.05 mL.
Note: Pay attention to units. Depending on which units you choose for R, unit conversion may be necessary.
PROBLEM 6: Error Propagation
The rate constant for the reaction
\( 2DI \rightarrow D_{2} + I_{2} \)
is measured at two temperatures with the given estimated uncertainties:
\( k_{660}=(1.2 \pm 0.2)\times 10^{-3}\frac{liter}{mol\; sec} \) at \( 660\pm 2 K \)
\( k_{660}=(1.8 \pm 0.2)\times 10^{-3}\frac{liter}{mol\; sec} \) at \( 720\pm 2 K \)
A. Calculate the activation energy Eact for the reaction using
\( \large {ln\left ( \frac{k_{2}}{k_{1}} \right )=-\frac{E_{act}}{R}\left (\frac{1}{T_{2}} -\frac{1}{T_{1}} \right )} \)
B. Assuming the above equation is a correct expression for Eact, what is the uncertainty in Eact?
- Hint
-
The calculation of \(E_{act}\) using the given data is relatively straightforward. The error in the value of \(E_{act}\) is a propagation of the uncertainty of the two \(k\) values and two \(T\) values. The function is complicated enough that it should be addressed using partial derivatives (Equation 3.2.6). There is no one simple shortcut in this case, but if the function is broken down into simpler functions, then error can be determined in a step-wise fashion using the shortcuts.
PROBLEM 7: Error Propagation
In the method of Clement and Desormes, the heat capacity ratio of an ideal gas is determined from the manometrically determined values of P1, the gas pressure at temperature T1; P2, the gas pressure immediately after a reversible adiabatic expansion when the temperature is T2; and, P3, the gas pressure after the gas has warmed back up to temperature T1. The equation used to calculate the heat capacity ratio is
\( \large {\gamma=\frac{C_{P}}{C_{V}}=\frac{\frac{C_{V}}{R}+1}{\frac{C_{V}}{R}}} \)
where
\( \large {\frac{C_{V}}{R}=\frac{P_{2}\left [ 1-\left ( \frac{P_{3}}{P_{1}} \right ) \right ]}{P_{3}-P_{2}}} \)
Calculate the uncertainty in the heat capacity ratio γ, given the following data:
P1 = 763.1 ± 0.2 mmHg
P2 = 757.0 ± 0.1 mmHg
P3 = 758.7 ± 0.2 mmHg
- Hint
-
This is propogation of error problem where the error in the value of \(\gamma\) is a propagation of the uncertainty of the three \(P\) measurements. The calcaution of \(\gamma\) involves a complex function for which no short cuts are available. I recomend using partial derivatives to arrive at a proper estimation of error in the heat capacity ratio. Some students have broken the function into separate parts and used shortcuts for each, and have arrived at a reasonable estimation of error…though I find this to be more work than doing it using partial derivatives).
