Calculate the percent by mass of the element whose symbol occurs first in the following compounds' formulas. a. $\mathrm{C}_6 \mathrm{H}_6(l)$ b. $\mathrm{Na}_2 \mathrm{SO}_4(s)$ c. $\operatorname{CS}_2(l)$ d. $\mathrm{AlCl}_3(s)$ e. $\mathrm{Cu}_2 \mathrm{O}(s)$ f. $\mathrm{CuO}(s)$ g. $\mathrm{Co}_2 \mathrm{O}_3(s)$ h. $\mathrm{C}_6 \mathrm{H}_{12} \mathrm{O}_6(s)$

Predict the sign of $\Delta S^{\circ}$ for each of the following reactions. a $\mathrm{CaCO}_3(s) \longrightarrow \mathrm{CaO}(s)+\mathrm{CO}_2(\mathrm{~g})$ b $\mathrm{CS}_2(\mathrm{l}) \longrightarrow \mathrm{CS}_2(\mathrm{~g})$ c $2 \mathrm{Hg}(l)+\mathrm{O}_2(\mathrm{~g}) \longrightarrow 2 \mathrm{HgO}(s)$ d $2 \mathrm{Na}_2 \mathrm{O}_2(s)+2 \mathrm{H}_2 \mathrm{O}(l) \longrightarrow 4 \mathrm{NaOH}(a q)+\mathrm{O}_2(g).$

Complete each of the following bombardment reactions:

a. $^{1}_{0}\text{n}+^{9}_{4}\text{Be}\rightarrow?$

b. $?+^{32}_{16}\text{S}\rightarrow+^{32}_{15}\text{P}$

c. $^{1}_{0}\text{n}+?\rightarrow+^{24}_{11}\text{Na}+^{4}_{2}\text{He}$

d. $^{4}_{2}\text{He}+^{27}_{13}\text{Al}\rightarrow?+^{1}_{0}\text{n}$

Respond with true or false to each of the following assertions. Be prepared to justify your answer.

$\int\left[f(x) g^{\prime}(x)+g(x) f^{\prime}(x)\right] d x=f(x) g(x)+C$.

In this problem, find the solution of the given initial value problem.

$$y^{\prime \prime}+4 y=3 \sin (2 t), \quad y(0)=0, \quad y^{\prime}(0)=-1$$

Ruff degradation of two D pentoses, A and B, gave two new sugars, C and D. Oxidation of C with $\mathrm{HNO}_3$ gave meso-2,3-dihydroxybutanedioic (tartaric) acid, that of D resulted in an optically active acid. Oxidation of either A or B with $\mathrm{HNO}_3$ furnished an optically active aldaric acid. What are compounds A, B, C, and D?

At the conclusion of the height versus shoe length experiment, did you accept, reject, or modify your original hypothesis? Explain.

Distinguish among meteors, meteoroids, and meteorites.

In this problem, use the results of the previous problem to find the inverse Laplace transform of the given function.

$$F(s)=\dfrac{e^2 e^{-4 s}}{2 s-1}$$

A 0.100 M solution of chloroacetic acid (ClCH$_2$COOH) is 11.0% ionized. Using this information, calculate each of the following: [ClCH$_2$COO$^-$], [H$^+$], [ClCH$_2$COOH], and the Ka for chloroacetic acid.

A sample of water contains $23.0 \mathrm{~g} \mathrm{~L}^{-1}$ of $\mathrm{Na}^{+}(\mathrm{aq})$, $10.0 \mathrm{~g} \mathrm{~L}^{-1}$ of $\mathrm{Ca}^{2+}(\mathrm{aq}), 40.2 \mathrm{~g} \mathrm{~L}^{-1} \mathrm{CO}_3{ }^{2-}(\mathrm{aq})$, and $9.6 \mathrm{~g} \mathrm{~L}^{-1} \mathrm{SO}_4^{2-}(\mathrm{aq})$. What is the $\mathrm{pH}$ of the solution if the only other ions present are $\mathrm{H}_3 \mathrm{O}^{+}$and $\mathrm{OH}^{-}$?

Predict the sign of $\Delta S^{\circ}$, if possible, for each of the following reactions. If you cannot predict the sign for any reaction, state why. a. $\mathrm{N}_2 \mathrm{H}_4(l) \longrightarrow \mathrm{N}_2(g)+2 \mathrm{H}_2 \mathrm{O}(g)$ b. $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}(l)+3 \mathrm{O}_2(g) \longrightarrow 2 \mathrm{CO}_2(g)+3 \mathrm{H}_2 \mathrm{O}(l)$ c. $\mathrm{P}_4(s)+5 \mathrm{O}_2(g) \longrightarrow \mathrm{P}_4 \mathrm{O}_{10}(s)$ d. $2 \mathrm{NaHCO}_3(s) \longrightarrow \mathrm{Na}_2 \mathrm{CO}_3(s)+\mathrm{H}_2 \mathrm{O}(g)+\mathrm{CO}_2(g)$

In Exercise, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine.

$$\sin ^2 x \cos ^2 x$$

Classify each of the following sets as finite, countably infinite, or uncountable. (a) $\left\{(x, y) \in Z^2 \mid x y=1\right\}$