Explain why the neutralization reaction of a strong acid and a weak base gives a weakly acidic solution.

What is wrong with the claim that a cyclical heat engine does 4.00 kJ of work on an input of 24.0 kJ of heat transfer while 16.0 kJ of heat transfers to the environment?

Write the chemical formula and Lewis structure of the following, each of which contains five carbon atoms: (a) an alkane (b) an alkene (c) an alkyne

Assume that the change in concentration of $\mathrm{N}_{2} \mathrm{O}_{4}$ is small enough to be neglected in the following problem. (a) Calculate the equilibrium concentration of both species in 1.00 L of a solution prepared from 0.129 mol of $\mathrm{N}_{2} \mathrm{O}_{4}$ with chloroform as the solvent. $\mathrm{N}_{2} \mathrm{O}_{4}(g) \rightleftharpoons 2 \mathrm{NO}_{2}(g)$ $K_{c}=1.07 \times 10^{-5}$ in chloroform (b) Show that the change is small enough to be neglected.

The equilibrium constant ($K_{C}$) for this reaction is 1.60 at 990 $^{\circ} \mathrm{C}$: $\mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(g)+\mathrm{CO}(g)$ Calculate the number of moles of each component in the final equilibrium mixture obtained from adding 1.00 mol of $\mathrm{H}_{2}$, 2.00 mol of $\mathrm{CO}_{2}$, 0.750 mol of $\mathrm{H}_{2} \mathrm{O}$, and 1.00 mol of CO to a 5.00-L container at 990 $^{\circ} \mathrm{C}$.

The $(2 \times 2)$ matrices listed in Eq. are used in several of the exercises that follow.

$$A=\left[\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right], \ B=\left[\begin{array}{cc} 0 & -1 \\ 1 & 3 \end{array}\right]$$

$$C=\left[\begin{array}{rr} -2 & 3 \\ 1 & 1 \end{array}\right], \ Z=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]$$

. Find a matrix D such that A + 2B + 2D = 3B.

Liquids and solids expand with increasing temperature, because the kinetic energy of a body’s atoms and molecules increases. Explain why some materials shrink with increasing temperature.

Are the concentrations of hydronium ion and hydroxide ion in a solution of an acid or a base in water directly proportional or inversely proportional? Explain your answer.

Calculate the concentration of sulfate ion when $\mathrm{BaSO}_{4}$ just begins to precipitate from a solution that is 0.0758 M in $\mathrm{Ba}^{2+}$.

(a) If a 500-mL glass beaker is filled to the brim with ethyl alcohol at a temperature of $5.00^{\circ} \mathrm{C},$ how much will overflow when its temperature reaches $22.0^{\circ} \mathrm{C} ?$ (b) How much less water would overflow under the same conditions?

Convert 40 kilometers per hour to miles per minute.

Which of the equations are linear? $x_{1}+2 x_{3}=3$

Consider the system

$$\mathbf{y}^{\prime}=\left(\begin{array}{ccc} 1 & -1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{array}\right) \mathbf{y}$$

. (a) Find the eigenvalues and eigenvectors. Use your numerical solver to sketch the half-lines generated by the exponential solutions $\mathbf{y}(t)=C_{i} e^{\lambda_{i} t} v_{i}$ for i=1, 2, and 3. (b) The six half-line solutions found in part (a) come in pairs that form straight lines. These three lines, taken two at a time, generate three planes. In turn these planes divide phase space into eight octants. Use your numerical solver to add solution trajectories with initial conditions in each of the eight octants. (c) Based on your phase portrait, what would be an appropriate name for the equilibrium point in this case?

Solve the system by transforming the augmented matrix to reduced echelon form.

$$\begin{array}{r} x_{1}+2 x_{2}=1 \\ 2 x_{1}+4 x_{2}=2 \\ -x_{1}-2 x_{2}=-1 \end{array}$$