## Find the relative uncertainty in the half-life.

1.  The flow rate of water through a cylindrical pipe is given by Q = πr 2v, where r is the radius of the pipe and v is the flow velocity.

a.  Assume that r = 3.00 ± 0.03 m and v = 4.0 ± 0.2 m/s. Estimate Q, and find the uncertainty in the estimate.

b.   Assume that r = 4.00 ± 0.04 m and v = 2.0 ± 0.1 m/s. Estimate Q, and find the uncertainty in the estimate.

c.    If r and v have not been measured, but it is known that the relative uncertainty in r is 1% and that the relative uncertainty in v is 5%, is it possible to compute the relative uncertainty in Q? If so, compute the relative uncertainty. If not, explain what additional information is needed.

2.    The conversion of cyclobutane (C4H8 ) to ethylene (C2H4 ) is a first-order reaction. This means that the concentration of cyclobutane at time t is given by ln C = ln C0 − kt, where C is the concentration at time t, C0 is the initial concentration, t is the time since the reaction started, and k is the rate constant. Assume that C0 = 0.2 mol/L with negligible uncertainty. After 300 seconds at a constant temperature, the concentration is measured to be C = 0.174 ± 0.005 mol/L. Assume that time can be measured with negligible uncertainty.

a.   Estimate the rate constant k, and find the uncertainty in the estimate.

b.  The units of k will be s−1 . Find the relative uncertainty in k. The half-life t1/2 of the reaction is the time it takes for the concentration to be reduced to one-half its initial value.

c.   The half-life is related to the rate constant by t1/2 = (ln 2)/k. Using the result found in part (a), find the uncertainty in the half-life.

d.  Find the relative uncertainty in the half-life.

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