Problem Sheet 1 (covering Lectures 1-3)

1.   The following values and uncertainties were obtained in an experiment. The             scientists have left more significant figures than they need in the table. How should they report their measurements in the final report?

2. Identify the type of distribution that you would assume for these reported values.        Calculate the value of the standard uncertainty, u, that you would use for propagation of errors.

3. i) In your own words, explain why the Student t-distribution is used instead of the standard normal distribution. For this course, under what circumstances do we assume the normal     distribution and t-distribution are the same?

ii) What value of k would you use to convert the standard uncertainty, u, into the expanded uncertainty, U, in each of the cases below. Tables of standard normal distribution and           Student t-distribution can be found at the end of the Problem Sheet. Assume two-tailed       confidence intervals.

4. In the 1997 RACI titration competition, six participants who analysed sample ‘B’ (assigned value 0.1241 M) obtained the following results:

Concentration of acetic acid /M

0.1255

0.1252

0.1245

0.1243

0.1227

0.1256

i) Calculate the mean, median and standard deviation (at this point don’t round the           answers to significant figures), then compare the values of the mean and median and       discuss under which circumstances each gives a better estimate of the centre of the data.

ii) Calculate the expanded uncertainty (U) at 95% confidence and report the 95% confidence interval for the concentration of acetic acid.

5. The table below lists a series of measurements for four different properties, along with their error.

Combine the errors according to the following equations:

i) m=a+d

ii) n=b×c

iii) p=n/m

6. By inspection (no calculations!), propagate the error in the following expressions:

i)  c=a+b, where a=34± 15 m and b=5± 1 m

ii)  N = , whereNA =(6.023± 0.001)×1023mol– 1, m=2.00± 0.20g, M=16.000 ± 0.001 g mol– 1 (you can use a calculator to work out the numerical value of N, but no            calculation of errors!)

There is a nice take-home lesson here.... always look up fundamental constants (e.g. molecular weights, Avogadro constant) to more sig. figs. than your worst error. Then you won’t have to  propagate through any error to do with these quantities.