# ALEKS Notes

CHEM 142

Notes on ALEKS objectives.

### Objective 10

Use this Desmos program to help calculate kinetics order and rate problems.

## Understanding the Organization of the Electromagnetic Spectrum

• Electromagnetic radiation is divided into different types, or bands. Each band includes all radiation within a particular range of frequency of wavelength.
• All the different types of radiation together form the electromagnetic spectrum.
• Visible light is electromagnetic radiation with a wavelength between 400nm and 700nm.
• Roy G. Biv - Red, Orange, Yellow, Green, Blue, Indigo, Violet.
• Red light - longest wavelength; blue or violet light - shortest wavelength; green light - middle wavelength.
• The energy $$E$$ in each photon of any electromagnetic radiation is directly proportional to its frequency $$f$$: $$E = hf$$ ($$h$$ is the Planck constant, $$6.62607\times10^{-34}$$J/Hz).
• Thus, electromagnetic radiation with a higher frequency has a higher photon energy, and electromagnetic radiation with lower frequency has a lower photon energy.

## Deducing n and l from a Subshell Label

$$n$$ - principal quantum number.

$$l$$ - angular momentum quantum number.

$$n$$ and $$l$$ tell you key facts about the electron subshell.

• Electrons in subshells with a higher $$n$$ are farther from the nucleus and more likely to participate in chemical reactions. (Electrons with the highest $$n$$ are valence electrons.)
• The number of orbitals in a subshell is indicated by $$l$$. The number of orbitals can also tell you how many electrons the subshell can hold (each orbital can hold 2 electrons).
• The shape of the orbital is given by $$l$$. An orbital in $$l = 0$$ subshell hasa the sphere, whereas orbitals with higher $$l$$ have more complex shapes. More complex orbitals are not as good as screening outer electrons from the attraction of the nucleus.
• $$n$$ and $$l$$ indicate the energy of electrons in the subshell - energy increases with $$n$$ and $$l$$ (roughly speaking).

Electron subhsells are labelled with a number-letter combination, [n][l]. $$l$$ is written as a letter code:

Letter$$l$$
s0
p1
d2
f3

## Determining the Relative Energy of Electron Subhsells

N+L Rule Method

• The subshell with the higher $$n+l$$ has the higher energy.
• If both subshells have the same $$n+l$$, the subshell with a higher $$n$$ has the higher energy.

## Recognizing s and p Orbitals

• An $$s$$ orbital has the shape of a sphere; it looks the same from any direction.
• A $$p$$ orbital has the shape of a dumbbell, or a pair of eggs. $$p_x$$ orbital - points along the $$x$$-axis, $$p_y$$ orbital - points along the $$y$$-axis, etc.
• Node - plan in which the probability of finding an electron is exactly 0.

Analyze how moving along the $$x$$, \$y$$, and$$z$$dimensions of the diagram changes the probability an electron can be found along the cross-section. ## Interpreting the Radial Probability Distribution of an Orbital The radial probability distribution of an electron orbital tells you the probability an electron in that orbital will be a given distance from the nucleus. Important facts about radial probability distributions: • The most likely distance of the electron from the nucleus is where the radial probability distribution reaches its peak. • The energy of an electron in an orbital is proportional to its distance from the nucleus. • An attractive electrostatic force between the nucleus and the electron exists; pulling an electron further from the nucleus takes energy, which is stored in the position as potential energy. • The degree to which an inner electron screens an outer electron from attraction of the nucleus is proportional to how likely it is that the inner electron is between the outer electron and the nucleus. ## Predicting the Qualitative Features of a Line Spectrum Important facts about spectroscopy (the observation of how electromagnetic radiation (for example visible light) is absorbed, emitted, scattered or reflected by an object or substance): • The state with the lowest possible energy is the ground state; all other states are excited states. • There is a line in the absorption spectrum for each possible transition that absorbs energy. • Relationship between energy of photon and its frequency or wavelength: $$E = h\cdot v \implies E = \frac{hc}{\lambda}$$, where $$E$$ is the photon energy, $$v$$ is its frequency, $$h$$ is the Planck constant, $$c$$ is the speed of light, and $$\lambda$$ is the wavelength. • The frequency of a line int he emission or absorption line spectrum is directly proportional to the energy released or absorbed by a transition. • The wavelength of a line is inversely proportional to the energy released or absorbed by a transition. • The line in the absorption or emission spectrum with the longest wavelength will come from the transition that absorbs or releases the least amount of energy. ## Interconverting the Wavelength and Frequency of Electromagnetic Radiation $v\cdot \lambda = c$ …where $$v$$ is frequency, $$\lambda$$ is wavelength, and $$c$$ is the speed of light. ## Understanding the Meaning of a de Broglie Wavelength Very small objects exhibit particle and wave properties. We can use quantum mechanics to describe their behavior and classical mechanics to describe larger objects, but not both. The de Broglie wavelength of an object is given by $\lambda = \frac{h}{p} = \frac{h}{m\cdot v}$ …where $$\lambda$$ is the de Broglie wavelength, $$h$$ is the Planck constant ($$J\cdot s$$), and $$p$$ is momentum. Momentum can be written as $$m\cdot v$$, where $$m$$ is the object mass (kg) and $$v$$ is the object velocity (m/s). If the de Broglie wavelength of an object is equal to or larger than its size, use quantum mechanics. Otherwise, use classical mechanics. ## Interconverting Wavelength, Frequency, and Photon Energy $E = h \cdot v$ …where $$E$$ is the energy of each photon of the radiation, $$v$$ is the frequency of the radiation, and $$h$$ is the Planck constant. $\lambda = \frac{c}{v}$ …where $$\lambda$$ is the light wavelength, $$c$$ is the speed of light, and $$v$$ is the frequency of the radiation. ## Calculating the Wavelength of a Line in the Spectrum of Hydrogen Fact about spectroscopy: when a system makes a transition from one state to another, it emits or absorbs a photon with energy equal to the difference in energy between the states. To find the wavelength of the line in the emission line spectrum of hydrogen caused by the transition of the electron from an orbital with $$n=n_1$$ to an orbital with $$n=n_2$$: 1. Find emmitted energy: $$\Delta E = E_{n_1} - E_{n_2}$$. 2. Use the Bohr formula to find each energy: $$\Delta E = \left(-\frac{R_y}{n_1^2}\right) - \left(-\frac{R_y}{n_2^2}\right)$$. 3. Substitute the value of the Rydberg energy $$2.17987\times 10^{-18} J$$ for $$R_y$$. 4. Find the energy of the photon $$\Delta E J$$. 5. Find the wavelength using $$E = \frac{h \cdot c}{\lambda}$$. $\lambda =\frac{6.62607\cdot \:10^{-34}\cdot \:299792458}{\left|\frac{2.17987\cdot \:10^{-18}}{n_1^2}-\frac{2.17987\cdot \:\:10^{-18}}{n_2^2}\right|}$ General formula for finding the wavelength $$\lambda$$ given $$n_1$$ and $$n_2$$. ## Finding the Minimum Uncertainty in a Position or Velocity Measurement Use Heisenberg’s Uncertainty Principle. $\Delta x \times \Delta p \ge \frac{h}{4\pi}$ Heisenberg’s Uncertainty Principle, given $$\Delta x$$ as the uncertainty of the particle’s position and $$\Delta p$$ as the uncertainty of the particle’s momentum. Recall that $$p=mv$$ to find the momentum. ## Deducing the Allowed Quantum Numbers of an Atomic Electron Quantum NumberRestrictionsNumber of Possible Values Principal, $$n$$$$n>0$$infinite Angular Momentum, $$l$$$$0\le l < n$$$$n$$ Magnetic, $$m_l$$$$-l \le m_l \le l$$$$2l+1$$ Spin, $$m_s$$$$ms = +\frac{1}[2}\text{ or }-\frac{1}[2}$$2 ## Predicting the Relative Length and Energy of Chemical Bonds The length and strength of a covalent chemical bond is determined byt he concentration of negative electric charge between the nuclei. The more concentrated the negative charge, the more powerful the electric forces that pull the nuclei towards oeach other. • For the same atoms, more shared electrons means a stronger and shorter chemical bond. • For the same numbe rof shared electron, smaller atoms mean a stronger and shorter chemical bond. • Reducing the distance between nuclei concentrates the same amount of negative charge in a smaller space. • The size of atoms increases as you go down a group in the Periodic Table. ## Predicting the Relative Ionization Energy of Elements Ionization energy (IE) generally increases as we go across or up the Periodic Table. IE is the energy required to pull an electron away from the attractive electric force of the nucleus. The stronger the force, the more energy required. Two factors determine the strength of an electric force: • The magnitude of the electric charge. The larger the effective nuclear charge, the stronger the attractive force the nucleus exerts. ENC generally increases going across the Periodic Table. • The distance between charges. Greater separation reduces the strength of the electric force between charges. ## Calculating the Capacity of Electron Subshells Each subshell is a group of quantum states in an atom with the same principal quantum number $$n$$ and angular momentum quantum number $$l$$. To find how many electrons a subshell can carry, find how many different state combinations can be formed from $$m_l$$ and $$m_s$$. $$m_l$$ can take on any integer value in $$\left[-l, l\right]$$. $$m_s$$ can take on either $$-\frac{1}{2}$$ or $$\frac{1}{2}$$. Thus, given $$l$$, the electron capacity is $$2\cdot(2\cdot l + 1)$$. ## Ranking the Screening Efficacy of Atomic Orbitals 1. Electrons in the inner shells screen more effectively than electrons in the same shell. 2. Electrons in orbitals with a lower $$l$$ screen more effectively because they spend more time close to the nucleus. Screening effectiveness: $$s>p>d>f$$. 3. Electrons in the same subshell screen each other very little. Effective nuclear charge $$Z^*$$ experienced by a valence electron increases as you move from left to right along a period. ## Predicting the Relative Stability of Ionic Crystals from a Sketch Coulomb’s Law tells us that the strength of electric forces drops quickly as the distance between charges grows. • If two crystals differ only in the magnitude of the charges, the crystal with the bigger charges will have the more negative (lower) total energy. • If two crystals differ only by the spacing between the charges, then the crystal with the more closely-spaced charges will have a more negative (lower) energy. The electron configurations of many transition metals are not exactly what you’d expect. Fortunately, for elements in the first transition series, like chromium and vanadium, there is only one deviation from the predictions of the simple aufbau method, and it is relatively easy to explain: • In the first transition series, only the electron configurations of chromium and copper differ from the predictions of the aufbau method using the or Madelung rules, or the Periodic Table. • For $$Cr$$ and $$Cu$$ atoms, the actual configuration differs from the simple aufbau prediction only by the movement of one $$4s$$ electron to the $$3d$$ subshell. Which electron requires a significantly larger energy to remove than other electrons? It is likely to be the first non-valence electron. ## Deducing the Block of an Element from an Electron Configuration • If an atom has a partially-filled $$p$$ or $$f$$ subshell outside the noble-gas core, the s$$co$$d$$rrespo$$f$$nding element lies in the$$p$$or$$f$$ block, respectively.
• Atoms with partially filled $$s$$ subshells correspond to elements in the $$s$$ block if they have no electrons outside the noble-gas core.
• Atoms with partially filled $$d$$ subshells correspond to elements in the block if they have no electrons outside the noble-gas core, or a completely filled $$f$$ subshell.
• If the atom has no partially-filled subshells outside the noble-gas core, the last filled subshell tells you the block in which the corresponding element lies.
• Be careful: keep in mind that if the atom has $$s$$ and $$d$$ electrons (or $$s$$, $$d$$, and $$f$$ electrons) outside the noble-gas core, the $$d$$ subshell is the last subshell to be filled, even though the $$s$$ subshell may be listed last.

• All $$d$$ and $$f$$ block elements are metals.
• All $$s$$ block elements except hydrogen and helium are metals.
• All $$p$$ block elements with five or six $$p$$ electrons (Groups 7A and 8A) are nonmetals.
• All $$p$$ block elements with only one $$p$$ electron (Group 3A) are metals or metalloids.

## Identifying the Electron Added or Removed to Form an Ion from an Electron Configuration

• If an electron is added to the atom, go to the unfilled orbital with the lowest energy (lowest unoccupied atomic orbital, LUAO).
• If an electron is removed from the atom, it will almost always be the electron in the highest energy occupied orbital - this minimizes the energy needed to remove it.
• Atoms or ions get smaller as you go from left to right along a period of the Periodic Table and as you go up a group of the Periodic Table.
• Anions are larger than a neutral atom of the same element.
• Cations are smaller than a neutral atom of the same element.

## Predicting the Relative Lattic Energy from an Electron Configurations

We can use Coloumb’s law to estimate the strength of the attractive forces holding each ionic compound together, and therefore the size of the lattice energy.

Two factors attractive forces between cations and anions can be stronger (and therefore maximize lattice energy):

• Compound is made from ions with charges of greater magnitude. (more important).

## Understanding the Exceptional Electron Configurations in the First Transition Series

• Use Aufbau method, but make sure that one electron has been moved from the $$ns$$ to $$(n-1)d$$ subshell for the element just before the halfway point.

## Understanding the Definitions of Ionization Energy and Electron Affinity

• Ionization reaction: a reaction in which a neutral atom loses an outer electron. Always absorb energy - ionization energy. $$Na(g) \to Na^+(g) + e^-$$.
• Electron attachment reaction: a reaction in which a neutral atom gains an outer electron. Normally release energy - electron affinity. $$F(g) + e^- \to F^-(g)$$.

## Interpreting a Born-Haber Cycle

Born-Haber cycle: a thermodynamic cycle that lets us write the formation enthalpy of an ionic compound as a sum of its lattice enthalpy and other measurable properties.

• Lattice enthalpy: the heat released in the fifth step of the Born-Haber cycle, where cations and anions combine.
• Enthalpy of formation: heat of the formation reaction $$M(g) + \frac{1}[2}X_2(g) \to MX(s)$$.

• An increase in electron affinity of $$X$$ will lower the $$M(g) + X^{2-}(g)$$ level and all following levels.
• A decrease in $$X-X$$ bond enthalpy will lower the $$M(g) + X(g)$$ level and all following levels.
• A decrease in the ionization enthalpy of $$M$$ will lower the $$M^{2+}(g) + e^ - + X(g)$$ level and all following levels.
• A decrease in the heat of sublimation of $$M$$ will lower the $$2M(s) + \frac{1}{2} X_2(g)$$ level and all following levels.
• If the final level moves down, $$MX$$ is more stable.

## Writing Lewis Structures for Diatomic Molecules

1. Decide how the atoms are arranged.
2. Count the number of valence electrons.
3. Draw a pair of electrons between each pair of bonded atoms.
4. Attempt to assign the rest of the electrons as lone pairs. Obey the Octet Rule.
5. Consider adding double or triple bonds if the Octet Rule is not yet satisfied.

## Writing Lewis Structures for an Expanded Valence Shell Central Atom

When writing Lewis Structures with leftover sets, we must recognize that the octet rule has exceptions.

Valence-Shell Expansion Exception: Atoms of elements in Period and higher can sometimes fit more than pairs of valence electrons around them.

Assign the remaining pair of valence electrons as a lnoe pair in the central atom.

## Predicting Bond Angles in Molecules

Linear2$$180^\circ$$
Trigonal Planar3$$120^\circ$$
Tetrahedral4$$109.5^\circ$$

## Naming the Shape of Molecules with One Central Atom and No Octet Rule Exceptions

lg = ligand, an atom or group of atoms bonded to something. eg = electron group (includes both ligands and lone pairs).

 — 2 lgs 3 lgs 4 lgs 2 egs linear — 3 egs bent trigonal planar — 4 egs bent trigonal pyramidal tetrahedral

## Drawing Lewis Structures for Simple Organic Compounds

1. Draw a skeleton using just the carbon atoms.
2. Add heteroatoms (atoms other than carbon and hydrogen - halogens, oxygen, nitrogen).

## Recognizing Exceptions to the Octet Rule

• Valence shell expansion - atoms of elements in period 3 and higher are big enought o fit more than 4 pairs of valence electrons around them, possibly because of access to $$d$$ orbitals.
• Electron-deficient molecules or molecular ions - there simply aren’t enough valence electrons to give every atom a complete cation.
• Electropositive Period 2 exception - some more electropositive elements in Period 2 (beryllium, boron, and aluminum) can form compoudns in whicht he central atom has fewer than 4 pairs of valence electrons.
• Odd electron exception - some molecules have an odd number of electrons, meaning it is necessary to put a single electron somewhere.

## Predicting Deviations from Ideal Bond Angles

1. Lone pairs are fatter than bonding electron groups. This repels other electron groups more strongly, forcing them to cluster together more.
• If lone pairs are placed perfectly symmetrically, they can cancel each toher out.
2. Double and triple bonds are fatter than single bonds.
• Molecules that ‘seem’ to have certain bonds may not truly ‘have’ them because of the involvement of resonance structures.

## Identifying a Molecule from its Electrostatic Potential Map

Consider size, shape, and chemical identity.

• Hydrogen atoms are smaller than any other atom. It is also less electronegative than most nonmetals, so it usually has a positive partial charge.
• Fluorine and oxygen are the most electronegative nonmetals; often have negative partial charges.
• Most nonmetal atoms can have positive or negative partial charges, depending on what they are bonded to.
• Identical atoms in symmetirc molecules share valence electrons equally.

## Predicting the products of a neutralization reaction

• Recognize strong acids and bases.
• Chemical formula beginning with a hydrogen is likely an acid.
• Ionic compounds that release hydroxide anions when it dissolves is a strong base.
• The products of a neutralization reaction are always water and a salt, made from the cations of the base and anions left over.

## Writing net ionic equations

Any ionic compounds dissolved in aqueous solutions are not present as whole formula units, but rather as separated ions.

## Predicting the products of a single displacement reaction

• Single displacement reaction: metal becomes an anion and swaps places with a hydrogen anion.

## Identifying Precipitation, Combustion, and Acid-Base Reactions

• Precipitation reactions take place in a solution. All of the reactants are and some of the products may be soluble, but at least one fot eh products is insoluble and precipitates from the solution.
• Combustion reactions occur when a molecular compound reacts with oxygen gas to form carbon dioxide and water.
• Acid-base reactions occur when a hydrogen cation is transferred from one compound to another. Many reactions take place in an aqueous solution and have water as one of the products.
• Double discplacement reaction - a chemical reaction in which two elements displace each other in two compounds. Also called metathesis reactions.

## Recognizing Reduction and Oxidation

• An atom is oxidized when its oxidation state goes up. This means it controls fewer valence electrons.
• An atom is reduced when its oxidation state goes down. This means it controls more valence electrons.

## Writing and Balancing Complex Half-Reactions in Acidic Solution

1. Write the bare half reaction that has the reactant and product you’re given.
2. Balance all elements except oxygen and hydrogen.
3. Balance oxygen by adding water to whatever side has too few oxygen atoms.
4. Balance hydrogen by adding $$H^+$$ cations to whichever side has too few hydrogen atoms. The reaction takes place in an acidic solution, so there are many $$H^+$$ cations to take part in the reaction.
5. Balance the charge by adding electrons to whichever side has too little negative charge or too much positive charge.

Eg: $$MnO^-_4 + 8H^+ + 5e^- \to Mn^{2+} + 4H_2O$$

## Writing and Balancing Complex Half-Reactions in Basic Solution

1. Write the bare half reaction that has the reactant and product you are given.
2. Complete and balance as if it had taken place in an acidic solution.
3. Cancel the $$H^+$$ cations by adding an equal number of hydroxide anions to both sides of the half-reaction.
4. Combine pairs of $$H^+$$ cations and $$OH^-$$ anions into neutral $$H_2O$$ molecules.
5. Cancel out $$H_2O$$ if it ends up on both sides of the equation.

## Using Charles’ Law

The volume of gas at a constant pressure is directly proportional to its absolute temperature. Use absolute temperature (Kelvins), rather than absolute temperature. Add 273 to Celsius.

## Solving Applications of Boyle’s Law

When gas is expanded or compressed at a constant temperature, the pressure multiplied byt eh volume stays constant: $$P_i V_i = P_f V_f$$.

## Understanding How Average Molecular Speed Scales With Temperature and Molar Mass

Kinetic theory of gases: average kinetic energy $$E_k$$ of the atoms or molecules in a gas is proportional to the absolute temperature $$T$$ of the gas.

$E_k = \frac{3}{2} k_B T$ $E_k = \frac{1}{2} mv^2$ $V = \sqrt{\frac{3k_B T}{m}}$

$$k_B$$ - the Boltzmann constant.

The average speed of the atoms in a gas is proportional to the square root of the absolute temperature of the gas and inversely proportional to the mass of the atoms.

Lower temperature, slower speed. Smaller atomic weight, faster speed.

## Calculating Mole Fraction in a Gas Mixture

Mole fraction - fraction of the total moles of the mixture that are moles of a component.

## Calculating Average Molecular Speed

$v = \sqrt{\frac{3k_B T}{m}}$

## Using Relative Effusion Rates to Find an Unknown Molar Mass

$$\frac{r_1}{r_2} = \sqrt{\frac{A_2}{A_1}}$$

$$r_1$$ and $$r_2$$ - rate of effusion of the two gases, $$A_1$$ and $$A_2$$ - molar masses.

## Using an Integrated Rate Law for a First-Order Reaction

• Calculate change in concentration of a reactant ovr tim by using integrated rate law.
• Rate law: $$\text{rate} = k\left[ A\right]$$.
• Integrated rate law: $$\ln\left(\frac{\left[A\right]}{\left[A\right]_0}\right) = -kt$$.

## Writing a Plausible Missing Step for a Simple Reaction Mechanism

• Two steps must add up to an overall reaction.
• Infer the product and write reactants to accomodate.

## Writing the Rate Law Implied by a Simple Mechanism with an Initial Slow Step

• Chemical equation of overall chemical reaction is sum of individual steps in the mechanism.
• The step of the mechanism that is slower determines the rate of the reaction.

## Using First and Second Order Integrated Rate Laws

Rate LawIntegrated Rate Law
rate = $$k[A]$$$$\ln\frac{[A]}{[A]_0} = -kt$$
rate = $$k[A]^2$$$$\frac{1}{[A]} - \frac{1}{[A]_0} = kt$$

## Expressing the Concentration of an Intermediate in Terms of the Concentration of Reactants

• You can write the rate law of an elementary reaction from its balanced chemical equation.
• At equilibrium, the rate of the forward reaction must equal the rate of the reverse reaction.

## Finding Half Life and Rate Constant From a Graph of Concentration Versus Time

$k = \frac{\ln(2)}{t_\frac{1}{2}}$

## Writing the Rate Law Implied by a Simple Mechanism

If first step slower than second, rate constant $$k$$ for overall chemical reaction is $$k = k_1$$. Otherwise, rate constant is $$k = \frac{k_2 k_1}{k_{-1}}$$.