Reaction Rates
Half-lives of Some Radioactive Isotopes. Radionuclide Half-life (Days) Radionuclide Half-life (Days) 3H 4.50 × 10 3 90Sr 1.00 × 10 4 14C 2.09 × 10 6 99Mo 2.79 32P 14.3 99mTc 0.250 35S 87.1 99Tc 7.70 × 10 6 42K 0.52 109Pd 0.570 45Ca 16.4 111In 2.81 47Ca 4.90 129I 6.30 × 10 9 59Fe 45.1 131I 8.00 57Co 270 135I 0.280 72Ga 0.59 207T1 3.33 × 10 −3 58mCo 0.38 207Bi 1.53 × 10 −3 58Co 72.0 226Ra 5.84 × 10 5 60Co 1.9 × 10 3 235U 2.60 × 10 11 64Cu 0.538 236U 8.72 × 10 −5 67Cu 2.58
initially present in solution to react, is given for a first-order reaction by the expression
t 1/2 = ln 2/ k 1
where the first-order rate constant k 1 has the units of reciprocal time. This is a general expression for all first-order reactions.
Overall Reaction Order
When the sucrose inversion reaction was later run in nonaqueous solvents it was recognized that a better description of the rate of disappearance of sucrose S is given by the following equations:
S + H + ⇌ SH +
Kc = [ SH + ]/[ S ][ H + ]
− d[S] / dt = k[SH + ][H 2 O] = kK c [S][H + ][H 2 O]
Thus, the reaction rate is first-order in sucrose, first-order in the catalyst H + , and first-order in H 2 O. The reaction is said to be "third-order overall," third because of the sum of the powers on the three concentration factors.
Temperature Dependence of Rates
In 1889 Svante Arrhenius noted that an increase in Kelvin temperature T caused the rate constant k of many reactions to increase according to the relation
k = Ae −Ea/RT
or
ln / k = ln A − Ea / RT
where the activation energy E a is related to the minimum amount of energy that a reactant molecule must acquire from collisions or some other form of excitation to go on to form reaction products. R is the perfect gas (ideal gas) constant.
If the rate constant k for a reaction is determined at several different temperatures, all that one needs to do to obtain a numerical value of E a is to construct a plot of ln k on the vertical axis versus 1/ T on the horizontal axis. If the chemical reaction obeys the Arrhenius equation, a straight line plot of the experimental data having a negative slope is obtained. The slope of this line is equal to − E a /R . Readers living in temperate climates will recall that the rate at which crickets chirp gradually declines in autumn as out-side temperatures become cooler. If the natural logarithm of the frequency of the chirping of crickets is plotted versus the reciprocal of the Kelvin temperature, the observer deduces from the slope of the resulting straight line that the activation energy for chirping is about E a = 5 × 10 4 joules/mole.
If k 1 is the rate constant for a given reaction at a Kelvin temperature T 1 , we may estimate the magnitude k 2 of the rate constant of that reaction at some other temperature T 2 from the following alternative form of the Arrhenius equation:
A familiar rule in chemistry states that "the rate of a chemical reaction doubles for each increase in temperature of ten degrees." From this second form of the Arrhenius equation it becomes clear that the moderate success of this rule of thumb proceeds from the fact that for many chemical reactions the activation energy E a has a magnitude in the general ballpark area of 5 × 10 4 joules/mole.