The thermal decomposition of phosphine $\left(\mathrm{PH}_3\right)$ into phosphorus and molecular hydrogen is a first-order reaction:

$$4 \mathrm{PH}_3(\mathrm{~g}) \longrightarrow \mathrm{P}_4(\mathrm{~g})+6 \mathrm{H}_2(\mathrm{~g})$$

the time required for 95 percent of the phosphine to decompose..

In an experimental study of a bimolecular reaction in aqueous solution, the second-order rate constant was measured at $25^{\circ} \mathrm{C}$ and at a variety of ionic strengths, the results are tabulated below. It is known that a singly charged ion is involved in the rate-determining step. What is the charge on the other ion involved?

$$\begin{matrix} \text{$I$} & \text{0.0025} & \text{0.0037} & \text{0.0045} & \text{0.0065} & \text{0.0085}\\ \text{$k_{r} /\left(\mathrm{dm}^{3} \mathrm{mol}^{-1} \mathrm{s}^{-1}\right)$} & \text{1.05} & \text{1.12} & \text{1.16} & \text{1.18} & \text{1.26}\\ \end{matrix}$$

Let $\left(1-x^{2}\right) y^{\prime \prime}+2 x y^{\prime}-2 y=0 \quad(-1<x<1) ; y_{1}(x)=x$. So, a differential equation and one solution $y_1$ are given. Use the method of reduction of order to find a second linearly independent solution $y_2$.

You are designing the fuel requirements for a small fusion electric-generating plant. Assume 33 percent conversion to electric energy. For the deuterium-tritium (D-T) fusion reaction in Example 7-18, calculate the number of reactions per second that are necessary to generate $1.00 \mathrm{~kW}$ of electric power.

The reaction

$$\mathrm{SO}_2 \mathrm{Cl}_2(\mathrm{~g}) \longrightarrow \mathrm{SO}_2(\mathrm{~g})+\mathrm{Cl}_2(\mathrm{~g})$$

is first order in $\mathrm{SO}_2 \mathrm{Cl}_2$. Using the following kinetic data, determine the magnitude of the first-order rate constant:

Egg protein albumin is precipitated when an egg is cooked in boiling $\left(100^{\circ} \mathrm{C}\right)$ water. $E_0$ for this first-order reaction is $52.0 \mathrm{~kJ} / \mathrm{mol}$. Estimate the time to prepare a 3-minute egg at an altitude at which water boils at $90^{\circ} \mathrm{C}$.

Rank the following compounds in decreasing order of their reactivity toward the $\mathrm{S}_{\mathrm{N}} 2$ reaction with sodium ethoxide $\left(\mathrm{Na}^{+-} \mathrm{OCH}_{2} \mathrm{CH}_{3}\right)$ in ethanol. methyl chloride. tert-butyl iodide. neopentyl bromide. isopropyl bromide. methyl iodide. ethyl chloride.

(a) Consider the second propagation step for peroxide-promoted HBr addition to an alkene. Calculate the $\Delta H^{\circ}$ for this reaction. (b) Calculate the $\Delta H^{\circ}$ for the same step using HCl instead of HBr. (c) Use your calculation to explain why no peroxide effect is observed for the addition of HCl to an alkene.

The following problem, a second-order equation of the form x" + f(x, x') = 0, corresponding to a certain mass-and-spring system, is given. Find and classify the critical points of the equivalent first-order system. $x^{\prime \prime}-8 x+2 x^{3}=0$: Here the linear part of the force is repulsive rather than attractive (as for an ordinary spring). Verify that the critical points resemble. Thus there are two stable equilibrium points and three types of periodic oscillations.

In a horse race, a quinela bet is won if you selected the two horses that finish first and second, and they can be selected in any order. The 140th running of the Kentucky Derby had a field of 19 horses. What is the probability of winning a quinela bet if random horse selections are made?

Use second order Taylor polynomials $P_{2}(x)$ for the given function about the point specified to approximate the indicated value. Estimate the error, and write the smallest interval you can be sure contains the value. $f(x)=\tan ^{-1} x$ about 1; approximate $\tan ^{-1}(0.97)$.

Hydrogen peroxide undergoes a first-order decomposition to water and $O_2$ in aqueous solution. The rate constant at $25^\circ C$ is $7.40 \times {10}^{-4}/s$. Calculate the volume of $O_2$ obtained from the decomposition reaction of $1.00 mol$ $H_2O_2$ at $25^\circ C$ and 740 mmHg after 20.0 min.

If a Butterworth filter has nine second-order stages, its roll-off rate is
a. $20 \mathrm{~dB}$ per decade
b. $40 \mathrm{~dB}$ per decade
c. $180 \mathrm{~dB}$ per decade
d. $360 \mathrm{~dB}$ per decade

Monochromatic x-rays $(\lambda=0.166 \mathrm{nm})$ from a nickel target are incident on a potassium chloride (KCl) crystal surface. The spacing between planes of atoms in KCl is 0.314 nm. At what angle (relative to the surface) should the beam be directed for a second-order maximum to be observed?