Survey of the Natural Sciences Study Guide for the DAT

General Chemistry (continued)

Thermodynamics

Thermodynamics deals with principles governing heat, energy, and work. Thermodynamics makes sense of energy transformation and transfer.

Laws of Thermodynamics

Thermodynamics is guided by the law of conservation of energy, which states that the total energy in a system remains constant. The energy of a system is equal to the heat gained or lost within that system added to the work done by the system or to the system.

Heat is calculated using the following equation:

\[q = mc \Delta T\]

where \(m\) is mass, \(c\) is the specific heat capacity, and \(T\) is temperature.

The entropy of a system always increases according to the second law of thermodynamics.

Enthalpies and Entropies

Enthalpy is the heat given off or absorbed by a system. This value is negative in an exothermic reaction and positive in an endothermic reaction. Entropy is the disorder in a given system. To calculate either value of a reaction, simply subtract the given value for the reactants from that of the products.

Heat Transfer

When heat is a reactant, energy is being absorbed, and the reaction is endothermic. \(\Delta H\) is positive in this scenario. If heat is being released by a reaction, \(\Delta H\) is negative, and the reaction is exothermic.

Hess’s Law

This law says that all enthalpies are additive. From this, the enthalpy of a given reaction can be calculated by adding together enthalpies of the reaction broken down into steps.

Spontaneity

Gibbs free energy is a thermodynamic principle that uses enthalpy and entropy to predict the spontaneity of a given reaction. It is calculated with this formula:

\[\Delta G = \Delta H - T \Delta S\]

where \(T\) is the absolute temperature measured in Kelvin.

If \(\Delta G < 0\), then a reaction will proceed spontaneously. If \(\Delta G > 0\), then the reaction is nonspontaneous. If \(\Delta G = 0\), the reaction is at equilibrium. Gibbs free energy has no bearing on the rate of reaction, simply the spontaneity.

Chemical Kinetics

Chemical kinetics studies the rates of chemical reactions and analyzes what factors impact that rate. Various elements, including concentrations, temperatures, pressures, and catalysts, affect the rates of reactions. The slowest step in a given mechanism is called the rate-determining step.

Rate Laws

\[aA + bB \rightarrow cC + dD\] \[\text{Rate} = k[A]^x [B]^y\]

\(x\) and \(y\) describe the order of the reactants, respectively, while \(k\) is the rate constant.

The overall order of the reaction is equal to the sum of the order of the reactants. To calculate the given order, the rate of one reaction may be divided by the rate of another so long as the concentration of one of the reactants stays constant in the two reactions.

Zero-order reactions have a constant rate, independent of the concentration of the reactants. First-order reactions are dependent on the concentration of only one of the reactants. Second-order reactions are proportional to the concentration of two reactants or one reactant concentration squared.

An increasing temperature, increasing concentration of reactants, and the addition of a catalyst will all increase the rate.

Activation Energy

Activation energy is the minimum energy required for a reaction to proceed. The addition of a catalyst decreases the activation energy, thereby reducing or speeding up the chemical reaction. Catalysts are unchanged throughout the reaction.

Half-Life

In chemical kinetics, the half-life is the time required for a reactant concentration to decrease to half of its original value. For first-order reactions, the half-life is constant: \(t_{1/2} = 0.693/k\). In reactions of other orders, the half-life depends on the initial concentration and the rate constant.

Oxidation-Reduction Reactions

An oxidation-reduction reaction is one that involves the transfer of electrons from one compound to another. These types of reactions can be broken up into an oxidation reaction and reduction reaction to better observe the loss and gain of electrons, respectively.

Balancing Equations

When balancing oxidation-reduction reactions, it is important to balance them with respect to stoichiometry and charge. This is done most easily via half-reactions, one describing the oxidation and one describing the reduction. An example of this is included below:

Determination of Oxidation Numbers

Oxidation numbers tell us how many electrons are in a given species. The sum of all oxidation numbers in a neutral compound is zero. The sum of all oxidation numbers in an ionic compound is equal to the charge of the ion. Oxidation number rules are summarized in the table below:

\[\begin{array}{|l|l|} \hline \textbf{Group 1A Elements} & +1 \\ \hline \textbf{Group IIA} & +2 \\ \hline \textbf{Group VIIA} & -1 \text{ (unless combined with a more electronegative element)} \\ \hline \textbf{Hydrogen} & +1 \\ \hline \textbf{Oxygen} & -2 \\ \hline \end{array}\]

Electrochemical Concepts and Terminology

Oxidation is the loss of electrons, and reduction is the gain of electrons. The species undergoing oxidation is the reducing agent, and the species undergoing reduction is the oxidizing agent. There are many types of redox reactions, including combination, decomposition, and displacement.

Electrochemical Calculations

The cell potential (\(E^0_{\text{cell}}\)) is calculated using the following equation:

\[E^0_{\text{cell}} = E^0_{\text{red}} - E^0_{\text{oxid}}\]

Oxidation occurs at the anode of the electrochemical cell, whereas reduction occurs at the cathode. Half reactions are added similarly to how they are in the aforementioned balancing equations section. From there the electrochemical potential of the entire cell can be calculated.

Atomic and Molecular Structure

Atoms compose every living thing and are made of particles called protons, neutrons, and electrons. Atoms compose elements, which compose compounds. During a given chemical reaction of chemical compounds, no new atoms are made. They are simply rearranged.

Electron Configuration

Electron configurations describe the orbitals present in any given element. These orbitals are further used to determine the shape of molecules, their interactions with other molecules and environments, and where electrons might be located in the molecule. Each electron is further described using a set of quantum numbers.

Quantum Theory

Through quantum numbers, scientists aim to describe where any given electron could be found. There are four orbitals: \(s\), \(p\), \(d\), and \(f\). The principal quantum number is \(n\), which represents the energy level of the orbital. The larger the number, the farther away from the nucleus. To describe the shape of the orbital, the angular momentum number, \(\ell\), can have any integer value from 0 to \(n-1\). When \(\ell = 0\), this describes the \(\text{s}\) orbital. Furthermore, \(\ell = 1 = p\) orbital, \(\ell = 2 = d\) orbital, and \(\ell = 3 = f\) orbital.

The magnetic quantum number is \(m_{\ell}\), which can be equal to any integer value \(-\ell\) to \(+\ell\). This number describes how many orbitals of a given type are available per energy level. Last, the spin quantum number, \(m_s\), is either \(-\frac{1}{2}\) or \(+\frac{1}{2}\). According to the Pauli exclusion principle, no two electrons can have the same quantum numbers.

Orbital Types

The energy of atomic orbitals increases as the principal quantum number, \(n\), increases.

The energy of the orbitals increases within a shell in the order \(\text{s} < \text{p} < \text{d} < \text{f}\).

• The \(1\text{s}\) orbital at the bottom of the diagram is the orbital with electrons of the lowest energy.

• The energy increases as we move up to the \(2\text{s}\) and then \(2\text{p}\), \(3\text{s}\), and \(3\text{p}\) orbitals.

• The \(3\text{d}\) orbital is higher in energy than the \(4\text{s}\) orbital.

Electrons in successive atoms on the periodic table tend to fill low-energy orbitals first. As the principal quantum number increases, the size of the orbital increases, and the electrons spend more time farther from the nucleus. Thus, the attraction to the nucleus is weaker and the energy associated with the orbital is higher (less stabilized). The Aufbau principle states that each successive electron will try to fill the lowest energy orbital left.

Adapted from: https://openstax.org/books/chemistry-atoms-first-2e/pages/3-4-electronic-structure-of-atoms-electron-configurations

Lewis-Dot Diagrams

Lewis-Dot diagrams are visual representations of the electrons present in an element or a chemical bond. This allows for a visual representation of how electrons may be shared between two elements and how they may interact with their surrounding environments. Each element is proposed to have up to eight valence electrons. In a diagram, one dot is initially represented on each of the four sides of the element, then a second one is filled in.

Retrieved from: https://openstax.org/books/chemistry-atoms-first-2e/pages/4-4-lewis-symbols-and-structures

Atomic Theory

Niels Bohr developed an understanding of atomic theory through modeling a hydrogen atom. He developed this equation to determine angular momentum (\(L\)):

\[L = nh/2\pi\]

where \(n\) is equal to the principal quantum number and \(h\) is equal to Planck’s constant.

Bohr stated that an electron can become excited and jump to the next orbital, then it will release energy and fall back to its original orbital. Each element has a unique atomic absorption spectra and atomic emission spectra.

Molecular Geometry

The valence shell electron pair repulsion (VSEPR) model states that electrons will repel one another in space until they are as far away as possible from one another, which then dictates the shape of the molecule. To determine the molecular geometry, count the orbitals and go from there, as summarized in the table below. Keep in mind that lone pairs also count as a domain and will comprise the same electron domain geometry but differ in molecular geometry.

Orbitals	Molecular Geometry	Bond Angles
\(\text{sp}\)	linear	\(180^{\circ}\)
\(\text{sp}^2\)	trigonal planar	\(120^{\circ}\)
\(\text{sp}^3\)	tetrahedral	\(109.5^{\circ}\)
\(\text{sp}^3\), with 1 lone pair	trigonal pyramidal
\(\text{sp}^3\), with 2 lone pair	bent
\(\text{sp}^3\text{d}\)	trigonal bipyramidal	\(90^{\circ}, 120^{\circ}\)
\(\text{sp}^3\text{d}\), with 1 lone pair	seesaw
\(\text{sp}^3\text{d}\), with 2 lone pair	T-shaped
\(\text{sp}^3\text{d}\), with 3 lone pair	linear
\(\text{sp}^3\text{d}^2\)	octahedral	\(90^{\circ}\)

Bond Types

Ionic bonds occur when a positive (\(+\)) charge is attracted to a negative (\(-\)) charge. Covalent bonds exist when two elements share electrons to complete their respective outer orbitals. These bonds exist in single, double, and triple bonds, which indicate the number of electrons being shared in the bond relationship.

Furthermore, these types of bonds are broken down into polar bonds and nonpolar bonds. In polar bonds, electrons are shared in a geometrical structure with one very electronegative molecule bonding to another much less electronegative molecule, inducing a slightly positive dipole on one side of the molecule and slightly negative dipole on the other side. In nonpolar bonds, the sharing of electrons is equal and no dipoles are induced, leading to no slight charges. Hydrogen bonding exists when a hydrogen atom bonds with another very electronegative atom to share electrons. Metallic bonds exist when multiple metallic elements share electrons in a cloud.

Subatomic Particles

Each element is assigned an atomic number, which is the number of protons in the nucleus of an atom of that element. Each proton carries a single positive charge. Neutrons are neutral in charge, and the number of neutrons in an element can be calculated by subtracting the atomic number from the mass number. Electrons carry a single negative charge and are found in an electron cloud surrounding the nucleus. If the element is neutral, then the number of electrons and protons are equal. If there is a positive charge, then there has been a loss of an electron. If there is a negative charge, then there has been a gain of electrons.