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The Physical Basis of Chemistry

Warren S. Warren

Chapter 2

Essentials of Calculus for Chemical Applications - all with Video Answers

Educators

Chapter Questions

01:20

Problem 1

Find $d y / d x$ if $y=6-2 x-x^2$. Use two different approaches-first evaluate this derivative explicitly, as in Equation 2.3; then use the relations listed in Equations 2.4, 2.11 and 2.12.

Meredith Murphy
Meredith Murphy
Numerade Educator
01:30

Problem 2

Find the value of $x$ which maximizes the function $y=6-2 x-x^2$.

Erika Bustos
Erika Bustos
Numerade Educator
01:52

Problem 3

Use the rules for differentiation in this chapter to find $d f(x) / d x$ for the following functions:
(a) $f(x)=\sin ^2 x$
(b) $f(x)=\ln (6 x)$

Lauren Shelton
Lauren Shelton
Numerade Educator
01:52

Problem 4

Use the rules for differentiation in this chapter to find $d f(x) / d x$ for the following functions:
(a) $f(x)=(\cos x)(\sin x)$
(b) $f(x)=e^{-6 x}$

Lauren Shelton
Lauren Shelton
Numerade Educator
06:09

Problem 5

Find the first nonzero term in the Taylor series expansion for the function in $(1+x)$. Use this expansion to evaluate $\ln 1.01$, and compare your answer to the exact value.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
07:31

Problem 6

Find the error made by the approximation $1 /(1+x)^n \approx 1-n x(n x \ll 1)$ for $x=0.01$ and $n=2$, and for $x=0.01$ and $n=50$.

Charles Carter
Charles Carter
Numerade Educator
01:29

Problem 7

Find the full Taylor series expansion for the function $\ln (1+x)$

David Nguyen
David Nguyen
Numerade Educator

Problem 8

Use the first term in the Taylor series (Equation 2.18 to prove Equation 2.24.

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04:08

Problem 9

Evaluate the following integrals without using an integral table:
(a) $\int_{x=0}^{x=\pi / 2} \sin x d x$
(b) $\int_{x=0}^{x=1} e^{2 x} d x$

Israel Hernandez
Israel Hernandez
Numerade Educator
04:08

Problem 10

Evaluate the following integrals without using an integral table:
(a) $\int_{x=0}^{x=\pi / 2} \sin x d x$
(b) $\int_{x=0}^{x=1} e^{2 x} d x$

Israel Hernandez
Israel Hernandez
Numerade Educator
02:22

Problem 11

Use Equation B-18 from the table in Appendix B

$$
\left(\int_{\psi=-\infty}^{x=\infty} e^{-x^2 / 2 \sigma^2} d x=\sigma \sqrt{2 \pi}\right),
$$

plus the fact that the function $e^{-x^2 / 2 \sigma^2}$ is symmetric about $x=0$, to evaluate the integral $\int_{x=0}^{x=\infty} e^{-a x^2} d x$ (note the different lower limit).

Vikash Ranjan
Vikash Ranjan
Numerade Educator
04:08

Problem 12

Use Appendix B and Equation 2.27 to evaluate the following integrals:
(a) $\int_{x=0}^{x=2 \pi} \sin ^2 x d x$
(b) $\int_{x=-\infty}^{x=\infty} 2 \exp \left(-x^2 / 8\right) d x$

Israel Hernandez
Israel Hernandez
Numerade Educator
03:20

Problem 13

Many chemical species undergo dimerization reactions. For example, two molecules of butadiene, $\mathrm{C}_4 \mathrm{H}_6$, can combine to form the dimer $\mathrm{C}_8 \mathrm{H}_{12}$. Often such reactions go almost to completion, because the product is more stable than the reactant. Starting with a sample of pure butadiene at time $t=0$, the concentration of butadiene at a later time $t\left(\left[\mathrm{C}_4 \mathrm{H}_6\right](t)\right)$ is given by the expression:

$$
\frac{1}{\left[\mathrm{C}_4 \mathrm{H}_6\right](t)}=\frac{1}{\left[\mathrm{C}_4 \mathrm{H}_6\right](t=0)}+k t
$$

(a) Find an expression for the rate of change of butadiene concentration $\frac{d\left[\mathrm{C}_4 \mathrm{H}_6(t)\right.}{d t}$. The correct expression only contains $\left[\mathrm{C}_4 \mathrm{H}_6\right](t)$ and $k$.
(b) How long does it take for the concentration of butadiene to fall to half of its initial value?

WH
Wilhamena Hobbs
Numerade Educator
07:04

Problem 14

Some chemical reactions obey what is called a "zero-order rate law"-the rate of the reaction is independent of concentration, for a limited time. A typical example might be an enzyme which has a limited number of "active sites," but which binds the reactant so tightly that all the sites are filled if any significant concentration of the reactant is present in solution. Writing the concentration of the reactant as $[A]$, this means that $d[A] / d t=-k$.
(a) Derive an expression for $[A](t)$. Your expression should include the concentration at time $t=0$ and the rate constant $k$.
(b) How long does it take for the concentration of the reactant to fall to half its initial value?

AA
Arwa Ali
Numerade Educator
02:00

Problem 15

The Environmental Protection Agency has established a guideline for radon concentration in air of 4 picocuries per liter. One curie is defined as $3.7 \times 10^{10}$ disintegrations per second, so this means one liter of air can have no more than $\left(4 \times 10^{-12}\right)\left(3.7 \times 10^{10}\right)=.148$ radon disintegrations per second. For the isotope of radon most commonly found in basements ( $\left.{ }^{222} \mathrm{Rn}\right)$ the half-life $t_{1 / 2}$ is 3.82 days. Use Equations 1.23 and 2.16 to determine how many radon atoms are in one liter of air which just meets the EPA guidelines, and to determine the concentration of radon in this air (one liter of air at 298 K and atmospheric pressure contains about $2.4 \times 10^{22}$ molecules).

David Collins
David Collins
Numerade Educator
02:02

Problem 16

The unit "curie" used in the last problem is named after Pierre and Marie Curie, who did pioneering experiments with radium in the nineteenth century. One curie $\left(3.7 \times 10^{10}\right.$ disintegrations per second) is the decay rate of one gram of radium, atomic mass $226 \mathrm{~g} \cdot \mathrm{~mol}^{-1}$. What is the half-life of radium-226?

Hunza Gilgit
Hunza Gilgit
Numerade Educator