Consider the following initial rate data for the decomposition of compound AB to give A and B:

$$\lfloor \mathrm { AB } ] _ { 0 } ( \mathrm { mol } / \mathrm { L } ) \quad \quad \text { Initial Rate (mol } \mathrm { L } ^ { - 1 } \mathrm { s } ^ { - 1 } )$$

$$\begin{array} { l l } { 0.200 } &&&&&& { 3.20 \times 10 ^ { - 3 } } \\ { 0.400 } &&&&&& { 1.28 \times 10 ^ { - 2 } } \\ { 0.600 } &&&&&& { 2.88 \times 10 ^ { - 2 } } \end{array}$$

Determine the half-life for the decomposition reaction initially having 1.00 M AB present.

Dinitrogen pentoxide decomposes in the gas phase to form nitrogen dioxide and oxygen gas. The reaction is first order in dinitrogen pentoxide and has a half-life of 2.81 hat 25$^{\circ} \mathrm{C}$. If a 1.5-L reaction vessel initially contains 745 torr of $N_2O_5$ at 25$^{\circ} \mathrm{C}$, what partial pressure of $O_2$ is present in the vessel after 215 minutes?

Dinitrogen pentoxide decomposes into nitrogen dioxide and oxygen. If 5.00 L of

$$\mathrm { N } _ { 2 } \mathrm { O } _ { 5 }$$

reacts at STP, what volume of

$$\mathrm { NO } _ { 2 }$$

is produced when measured at

$$64.5 ^ { \circ } C$$

and 1.76 atm?

Fill in the blanks. The sum of a finite geometric sequence with common ratio r ≠ 1 is given by ________.

A first-order reaction is 75.0% complete in 320 s. What are the first and second half-lives for this reaction?

Find an explicit formula for

$$\begin{equation*}U_W\begin{pmatrix}\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}\end{pmatrix},\end{equation*}$$

the orthogonal projection of $R^3$ on the plane $W$ defined by $x+y-2z=0$.

Find the limits.

$$\lim _ { x \rightarrow + \infty } x e ^ { - x }$$

The reaction $\mathrm { NO } ( g ) + \mathrm { O } _ { 3 } ( g ) \longrightarrow \mathrm { NO } _ { 2 } ( g ) + \mathrm { O } _ { 2 } ( g )$ was studied by performing two experiments. In the first experiment (results shown in following table), the rate of disappearance of NO was followed in a large excess of $\mathrm { O } _ { 3 . }$ (The $\left[ \mathrm { O } _ { 3 } \right]$ remains effectively constant at $1.0 \times 10 ^ { 14 }$ molecules/cm $^ { 3 } . )$

$$\begin{array} { c c } { \text { Time } } & { [ \mathrm { NO } ] } \\ { \text { (ms) } } & { \text { (molecules/cm } ^ { 3 } ) } \\ \hline { 0 } & { 6.0 \times 10 ^ { 8 } } \\ { 100 \pm 1 } & { 5.0 \times 10 ^ { 8 } } \\ { 500 \pm 1 } & { 2.4 \times 10 ^ { 8 } } \\ { 700 \pm 1 } & { 1.7 \times 10 ^ { 8 } } \\ { 1000 \pm 1 } & { 9.9 \times 10 ^ { 7 } } \end{array}$$

In the second experiment, $[ \mathrm { NO } ]$ was held constant at $2.0 \times 10 ^ { 14 }$ molecules/cm $^ { 3 } .$ The data for the disappearance of $\mathrm { O } _ { 3 }$ were as follows:

$$\begin{array} { c c } { \text { Time } } & { \left[ \mathrm { O } _ { 3 } \right] } \\ { ( \mathrm { ms } ) } & { \text { (molecules/cm } ^ { 3 } ) } \\ \hline { 0 } & { 1.0 \times 10 ^ { 10 } } \\ { 50 \pm 1 } & { 8.4 \times 10 ^ { 9 } } \\ { 100 \pm 1 } & { 7.0 \times 10 ^ { 9 } } \\ { 200 \pm 1 } & { 4.9 \times 10 ^ { 9 } } \\ { 300 \pm 1 } & { 3.4 \times 10 ^ { 9 } } \end{array}$$

What is the order with respect to each reactant?

The reaction $\mathrm { NO } ( g ) + \mathrm { O } _ { 3 } ( g ) \longrightarrow \mathrm { NO } _ { 2 } ( g ) + \mathrm { O } _ { 2 } ( g )$ was studied by performing two experiments. In the first experiment (results shown in following table), the rate of disappearance of NO was followed in a large excess of $\mathrm { O } _ { 3 . }$ (The $\left[ \mathrm { O } _ { 3 } \right]$ remains effectively constant at $1.0 \times 10 ^ { 14 }$ molecules/cm $^ { 3 } . )$

$$\begin{array} { c c } { \text { Time } } & { [ \mathrm { NO } ] } \\ { \text { (ms) } } & { \text { (molecules/cm } ^ { 3 } ) } \\ \hline { 0 } & { 6.0 \times 10 ^ { 8 } } \\ { 100 \pm 1 } & { 5.0 \times 10 ^ { 8 } } \\ { 500 \pm 1 } & { 2.4 \times 10 ^ { 8 } } \\ { 700 \pm 1 } & { 1.7 \times 10 ^ { 8 } } \\ { 1000 \pm 1 } & { 9.9 \times 10 ^ { 7 } } \end{array}$$

In the second experiment, $[ \mathrm { NO } ]$ was held constant at $2.0 \times 10 ^ { 14 }$ molecules/cm $^ { 3 } .$ The data for the disappearance of $\mathrm { O } _ { 3 }$ were as follows:

$$\begin{array} { c c } { \text { Time } } & { \left[ \mathrm { O } _ { 3 } \right] } \\ { ( \mathrm { ms } ) } & { \text { (molecules/cm } ^ { 3 } ) } \\ \hline { 0 } & { 1.0 \times 10 ^ { 10 } } \\ { 50 \pm 1 } & { 8.4 \times 10 ^ { 9 } } \\ { 100 \pm 1 } & { 7.0 \times 10 ^ { 9 } } \\ { 200 \pm 1 } & { 4.9 \times 10 ^ { 9 } } \\ { 300 \pm 1 } & { 3.4 \times 10 ^ { 9 } } \end{array}$$

What is the value of the rate constant for the overall rate law? $\text {Rate} = k [ \mathrm { NO } ] ^ { x } [ \mathrm { O } _ { 3 } ] ^ { y }.$

The reaction $\mathrm { NO } ( g ) + \mathrm { O } _ { 3 } ( g ) \longrightarrow \mathrm { NO } _ { 2 } ( g ) + \mathrm { O } _ { 2 } ( g )$ was studied by performing two experiments. In the first experiment (results shown in following table), the rate of disappearance of NO was followed in a large excess of $\mathrm { O } _ { 3 . }$ (The $\left[ \mathrm { O } _ { 3 } \right]$ remains effectively constant at $1.0 \times 10 ^ { 14 }$ molecules/cm $^ { 3 } . )$

$$\begin{array} { c c } { \text { Time } } & { [ \mathrm { NO } ] } \\ { \text { (ms) } } & { \text { (molecules/cm } ^ { 3 } ) } \\ \hline { 0 } & { 6.0 \times 10 ^ { 8 } } \\ { 100 \pm 1 } & { 5.0 \times 10 ^ { 8 } } \\ { 500 \pm 1 } & { 2.4 \times 10 ^ { 8 } } \\ { 700 \pm 1 } & { 1.7 \times 10 ^ { 8 } } \\ { 1000 \pm 1 } & { 9.9 \times 10 ^ { 7 } } \end{array}$$

In the second experiment, $[ \mathrm { NO } ]$ was held constant at $2.0 \times 10 ^ { 14 }$ molecules/cm $^ { 3 } .$ The data for the disappearance of $\mathrm { O } _ { 3 }$ were as follows:

$$\begin{array} { c c } { \text { Time } } & { \left[ \mathrm { O } _ { 3 } \right] } \\ { ( \mathrm { ms } ) } & { \text { (molecules/cm } ^ { 3 } ) } \\ \hline { 0 } & { 1.0 \times 10 ^ { 10 } } \\ { 50 \pm 1 } & { 8.4 \times 10 ^ { 9 } } \\ { 100 \pm 1 } & { 7.0 \times 10 ^ { 9 } } \\ { 200 \pm 1 } & { 4.9 \times 10 ^ { 9 } } \\ { 300 \pm 1 } & { 3.4 \times 10 ^ { 9 } } \end{array}$$

What is the value of the rate constant obtained from each set of experiments? $\text {Rate} = k ^ { \prime } [ \mathrm { NO } ] ^ { x } \quad \text {Rate} = k ^ { \prime \prime } \left[ \mathrm { O } _ { 3 } \right] ^ { y }.$

DDT (molar mass = 354.49 g/mol) was a widely used insecticide that was banned from use in the United States in 1973. This ban was brought about due to the persistence of DDT in many different ecosystems, leading to high accumulations of the substance in many birds of prey. The insecticide was shown to cause a thinning of egg shells, pushing many birds toward extinction. If a 20-L drum of DDT was spilled into a pond, resulting in a DDT concentration of $8.75 \times 10 ^ { - 5 }\ \mathrm { M } ,$ how long would it take for the levels of DDT to reach a concentration of $1.41 \times 10 ^ { - 7 }\ \mathrm { M }$ (a level that is generally assumed safe in mammals)? Assume the decomposition of DDT is a first-order process with a half-life of 56.0 days.

Tell whether the two ratios form a proportion. Explain.

$$\frac { 3 } { 4 } \text { and } \frac { 12 } { 15 }$$

Evaluate each function for the given input values. For g(x)=4x-9, find g(x) when x=3 and when x=5.

Estimate each sum or difference. Explain how you determined your estimate. 3.77 + 1.39