Identify whether each of the following factors will affect the rate of a reaction: (a) $K_{\mathrm{eq}}$ (b) $\Delta G$ (e) $E_{\mathrm{a}}$ (c) Temperature (f) $\Delta S$

What would be the maximum cost of a CFL such that the total cost (investment plus operating) would be the same for both CFL and incandescent 60-W bulbs? Assume the cost of the incandescent bulb is 25 cents and that electricity costs 10 cents/kWh. Calculate the cost for 1000 hours, as in the cost effectiveness of CFL example.

(a) The velocity of sound in a gas is proportional to the square root of the absolute temperature T. Use this fact, and the result of the previous problem, to show that when a thermal gradient exists in the vertical direction (z) sound waves will be turned initially with a radius of curvature $R=\frac{2 T}{d T / d z}$ (b) On a still day, the temperature of the atmosphere is found to decrease more or less linearly with height. Sketch the paths of "rays" of sound emitted from a source suspended high in the atmosphere. Assuming that the velocity of sound at ground level is 1110 ft/sec, estimate the horizontal distance at which an airplane flying at 15,000 ft firs becomes audible to an observer on the ground, if the temperature decreases by $1^{\circ} \mathrm{C}$ per 500-ft increase in altitude.

The area of a triangle is 96 square inches. Its height is 12 inches. Find the base.

$$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \ B=\left[\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right], \ C=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]$$

$$D=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right], \ E=\left[\begin{array}{lll} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 1 & 3 \end{array}\right]$$

,

$$F=\left[\begin{array}{lll} 1 & 2 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 1 \end{array}\right]$$

Determine whether the given matrix is singular or nonsingular. If a matrix M is singular, give all solutions of $M \mathbf{x}=\theta$. AB.

Let A = $\mathbb{Z}/4\mathbb{Z}$ and let G be the cyclic group of order 2 acting trivially on A. (a) Prove that $\left|C^{2}(G, A)\right|=2^{8}$. (b) Use the coboundary condition to show that $\left|B^{2}(G, A)\right|=2^{3}$. (c) Use the cocycle condition to show that $\left|Z^{2}(G, A)\right| \leq 2^{4}$. (d) Show that $\left|H^{2}(G, A)\right|=2$ by exhibiting two inequivalent extensions of G by A and their corresponding cocycles.

In the following exercise, use a calculator to draw the graph of the piecewise-defined function and study the graph to evaluate the given limits.

$$g(x)=\left\{\begin{array}{ll} x^{3}-1, & x \leq 0 \\ 1, & x>0 \end{array}\right.$$

(a) $\lim _{x \rightarrow 0^{-}} g(x)$ (b) $\lim _{x \rightarrow 0^{+}} g(x)$

Consider identical spherical conducting space ships in deep space where gravitational fields from other bodies are negligible compared to the gravitational attraction between the ships. Construct a problem in which you place identical excess charges on the space ships to exactly counter their gravitational attraction. Calculate the amount of excess charge needed. Examine whether that charge depends on the distance between the centers of the ships, the masses of the ships, or any other factors. Discuss whether this would be an easy, difficult, or even impossible thing to do in practice.

Recall the discussion on spacecraft from the chapter opener. The following problems consider a rocket launch from Earth’s surface. The force of gravity on the rocket is given by $F(d)=-m k / d^{2}$, where m is the mass of the rocket, d is the distance of the rocket from the center of Earth, and k is a constant. Determine the value and units of k given that the mass of the rocket on Earth is 3 million kg.

Water expands significantly when it freezes: a volume increase of about 9% occurs. As a result of this expansion and because of the formation and growth of crystals as water freezes, anywhere from 10% to 30% of biological cells are burst when animal or plant material is frozen. Discuss the implications of this cell damage for the prospect of preserving human bodies by freezing so that they can be thawed at some future date when it is hoped that all diseases are curable.

The middle-C hammer of a piano hits two strings, producing beats of 1.50 Hz. One of the strings is tuned to 260.00 Hz. What frequencies could the other string have?

Let $\mathcal{O}$ be the ring of integers in the algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$. (a) Show that the infinite sequence of ideals in

$$\mathcal{O}(2) \subseteq(\sqrt{2}) \subseteq(\sqrt[4]{2}) \subseteq(\sqrt[8]{2}) \subseteq \cdots$$

is strictly increasing, and so $\mathcal{O}$ is not Noetherian. (b) Show that $\mathcal{O}$ has Krull dimension 1. [Use Theorem 26.] (c) Let K be a number field and let I be any ideal in $\mathcal{O}_{K}$. Show that there is some finite extension L of K such that I becomes principal when extended to $\mathcal{O}_{L}$, i.e., the ideal $I O_{L}$ is principal (where L depends on I)-you may use the theorem that the class group of K is a finite group. (d) Prove that $\mathcal{O}$ is a Bezout Domain.

A matrix A and an eigenvector v are given. Find the corresponding eigenvalue directly by solving $A \mathbf{v}=\lambda \mathbf{v}$.

$$A=\left[\begin{array}{rrrr} 1 & 1 & 1 & 0 \\ -1 & -1 & 0 & -1 \\ -1 & 1 & 0 & 1 \\ 0 & -1 & -1 & 0 \end{array}\right]$$

$$\mathbf{v}=\left[\begin{array}{r} 0 \\ 1 \\ -1 \\ 0 \end{array}\right]$$

The ion $HTe^{-}$ is an amphiprotic species; it can act as either an acid or a base. (a) What is $K_{\mathrm{a}}$ for the acid reaction of $HTe^{-}$ with H2O? (b) What is $K_{\mathrm{b}}$ for the reaction in which $HTe^{-}$ functions as a base in water? (c) Demonstrate whether or not the second ionization of $\mathrm{H}_{2} \mathrm{Te}$ can be neglected in the calculation of [$HTe^{-}$] in a 0.10 M solution of $\mathrm{H}_{2} \mathrm{Te}$.