Chapter 3, Errors: classification and propagation Video Solutions, A Student's Guide to Data and Error Analysis

Problem 1

Perform the following operations and give the result with standard deviation. The standard deviations of quantities are indicated by $\pm$; they are independent of each other.
(a) $15.000 /(5.0 \pm 0.1)$
(b) $(30.0 \pm 0.9) /(5.0 \pm 0.2)$
(c) $\log _{10}(1000 \pm 2)$
(d) $(20.0 \pm 0.3) \exp [-(2.00 \pm 0.01)]$

Amit Srivastava

Numerade Educator

02:10

Problem 2

The half-life time $\tau_{1 / 2}$ of a first-order chemical reaction is determined at four different temperatures. The temperatures are accurate; the standard uncertainties in $\tau_{1 / 2}$ are indicated:
$$
\begin{array}{lc}
\hline \text { Temperature }\left({ }^{\circ} \mathrm{C}\right) & \text { half-life } \tau_{1 / 2}(\mathrm{~s}) \\
510 & 2000 \pm 100 \\
540 & 600 \pm 40 \\
570 & 240 \pm 20 \\
600 & 90 \pm 10 \\
\hline
\end{array}
$$
Determine the rate constant $k$ (what unit?) and its standard uncertainty, as well as $\ln k$ and its standard uncertainty, at every temperature. Now plot $\ln k$ with error bars versus the reciprocal absolute temperature. Also, plot $k$ with appropriate error bars on a logarithmic scale versus the reciprocal absolute temperature. Compare the two plots.

Adriano Chikande

Numerade Educator

14:50

Problem 3

Suppose you determine the acceleration of gravity $g$ by measuring the oscillation period $T$ of a pendulum with length $l$. The value of $g$ follows from
$$
g=4 \pi^2 l / T^2
$$
You measure $T=2.007 \pm 0.002 \mathrm{~s}$ and $l=1.000 \pm 0.002 \mathrm{~m}$. Determine $g$ and its standard uncertainty.

Donald Albin

Numerade Educator

Problem 4

The Gibbs activation function for a chemical reaction $\Delta G^{\dagger}$ follows from the rate constant $k$ according to Eyring's equation
$$
k=\left(k_B T / h\right) \exp \left(-\Delta G^{\ddagger} / R T\right)
$$
Here $k_B$ is Boltzmann's constant, $h$ is Planck's constant and $R$ the gas constant (see the data sheet PHYSICAL CONSTANTS on page 209, or use the Python module physcon.py).
(a) If the rate constant $k$ has an uncertainty of $10 \%$, what is the resulting uncertainty in $\Delta G^{\dagger}$ ?
(b) Discuss how an uncertainty in the temperature propagates into $\Delta G^{\ddagger}$.
(c) If $\Delta G^\psi=30 \mathrm{~kJ} / \mathrm{mol}$ and $T=300 \mathrm{~K}$, how large is the uncertainty in $\Delta G^t$ as a result of an uncertainty of $5^{\circ} \mathrm{C}$ in the temperature?

Check back soon!

04:03

Problem 5

(this exercise relates to Appendix A2 on page 138)
Generate an array with 1000 samples of the volume of spheres, of which the radii are samples of a normal distribution with mean 1.0 mm and standard deviation 0.1 mm . Compare the mean of the distribution with the volume of a sphere with radius 1.0 mm and discuss whether the latter is a biased result. Discuss the significance of the bias. Plot the cumulative volume distribution on a probability scale.

Sheryl Ezze

Numerade Educator