Survey of the Natural Sciences Study Guide for the DAT

General Chemistry

Following the biology section, the next 30 questions come from the field of chemistry. A good strategy for success is to save time on general recall questions so that longer time is allotted to calculation problems. It is important to remember that a general periodic table will be provided for use during the test as well as a basic calculator.

Stoichiometry and General Concepts

Stoichiometry is a method in which chemists take given amounts of substances and knowledge of reactions and relationships to change units and perform chemical calculations with ease.

Percent Composition

Percent composition is the ratio of the mass of the given element divided by the formula mass of the compound, all multiplied by 100. This allows scientists to understand how much of an element, by mass, is made up of a given component.

Empirical Formula

The empirical formula is the simplest whole number ratio of the elements in a compound, meaning that the ratio cannot be simplified further. Sometimes the empirical formula is equal to the molecular formula (discussed below), but not always.

Balancing Equations

Balancing equations is when scientists ensure the same number of each element exists on either side of the reaction arrow. This is done so that when chemical calculations are performed, the correct and appropriate relationships are used to relate moles of one substance to moles of another. An example of the process of balancing equations is included below:

Moles and Molecular Formula

A single mole contains \(6.022 \times 10^{23}\) atoms, a relationship that is vital in chemical calculations. This is called Avagadro’s number. A mole is equal to the mass of the sample over the molar mass.

The molecular formula is the exact number of each element making up a chemical compound. For instance, the molecular formula of hydrochloric acid is \(\text{HCl}\), with \(\text{H}\) representing one atom of hydrogen and \(\text{Cl}\) representing one atom of chlorine.

Molar Mass

The molar mass is calculated by adding the individual masses of components of a chemical. The unit used is grams per mole.

Density

Density is a measure of how much mass of an object is present in a given volume. The most common unit for this value is gram per liter (\(\text{g/L}\)).

Calculations from Balanced Equations

Gallium chloride is formed by the reaction of \(2.6 \; \text{L}\) of a \(1.44 \; \text{M}\) solution of \(\text{HCI}\) according to the following equation:

\[1.44 \; \text{M} = 1.44 \; \text{mol/L}\] \[2.6 \; \text{L} \times (1.44 \; \text{mol/L}) = 3.74 \; \text{mol HCl}\] \[3.74 \; \text{mol HCl} \times \left(\frac{2 \text{ mol} \, \text{GaCl}_3}{6 \text{ mol HCl}}\right) = 1.25 \; \text{mol} \, \text{GaCl}_3\]

Molar mass of \(\text{GaCl}_3 = 176.1 \; \text{g/mol}\)

Mass of \(\text{GaCl}_3 = 176.1 \times 1.25 = 219.5 \; \text{g}\)

Answer:

a. Volume HCl solution \(\rightarrow\) mol HCl \(\rightarrow\) mol GaCl₃

b. \(1.25 \; \text{mol} \, \text{GaCl}_3\), \(2.2 \times 10^2 \; \text{g} \, \text{GaCl}_3\)

Practice problem adapted from: https://openstax.org/books/chemistry-atoms-first-2e/pages/chapter-7

Gases

Gas is one of the three phases of matter. There are four variables used to describe a given gas: pressure (\(P\)), volume (\(V\)), temperature (\(T\)), and number of moles (\(n\)). Generally, chemists will assume a standard temperature and pressure of a given gas is \(273.15\,\text{K}\) and \(1\,\text{atm}\), respectively.

Kinetic Molecular Theory of Gases

The kinetic energy (\(KE\)) of a gas particle can be calculated using this formula:

\[KE = \frac{1}{2}mv^2 = \frac{3}{2}kT\]

where \(m\) is the mass, \(v\) is the velocity, \(k\) is the Boltzmann constant, and \(T\) is absolute temperature.

This theory assumes that the volume of gas particles is negligible, gas atoms are inert and in continuous random motion, and collisions result in no loss or gain of energy.

Dalton’s Law

Dalton’s law is also known as the law of partial pressures. It states that the total pressure (\(P_T\)) of the gas is equal to the sum of the partial pressures of its parts:

\[P_T = P_A + P_B + \dots\]

Boyle’s Law

Boyle’s law states that, at a constant temperature, the product of pressure and volume are also constant:

\[P_1V_1 = P_2V_2\]

Charles’s Law

In Charles’s law, at constant pressure, the ratio of volume divided by temperature is also constant:

\[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]

Ideal Gas Law

The ideal gas law states that:

\[PV = nRT\]

where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of the substance, \(R\) is the gas constant (\(0.0821 \times \frac{L \times atm}{mol \times K}\)), and \(T\) is temperature.

Liquids and Solids

Liquids and solids are the other two fundamental states of matter. Liquids are diffuse and have the ability to evaporate or turn to gas. Viscosity measures a liquid’s resistance to flow. Liquids are heavily studied for their ability to form solutions with other phases. Solids are defined as crystalline or amorphous and have less ability to move in their structure than liquids do. Solids only move slightly via the vibration of particles.

Intermolecular Forces

Liquids incur internal friction between the particles that compose them, which determines their viscosity. The ability of liquids to be mixed is called miscibility. An emulsion is one that appears to be fully mixed but is instead a mixture of separate and unbonded particles of various liquids that are immiscible (unmixable).

Solids have strong electrostatic interactions that lead them to be unable to have much motion or movement. The strong covalent interactions of metallic solids allow them to have characteristically high melting and boiling points.

Phase Changes

Specific heat is the value of heat, \(Q\), that is gained or lost when the temperature is changed by a certain amount. This can be calculated via the following formula:

\[Q = mc\Delta T = mc(T_f - T_i)\]

where \(m\) is mass, \(c\) is the specific heat of the substance, and \(T\) is the temperature (final minus initial).

Heat of transformation is measured when there is a conversion between solid, liquid, or gas phases. This value is calculated with:

\[Q = mH_L\]

where \(Q\) is the heat gained or lost, \(m\) is the mass of the substance, and \(H_L\) is the latent heat of transformation of the substance.

For the DAT, it will be important to be able to read a phase diagram from which you can interpret the temperature needed to turn a substance to a different given phase. The triple point represents where all three states of matter are in equilibria, meaning they are in a state of coexistence.

Retrieved from: https://openstax.org/books/chemistry-atoms-first-2e/pages/10-4-phase-diagrams

Vapor Pressure

Vapor pressure is the pressure of a gas present at a liquid’s surface. This value increases with temperature. Raoult’s law states that when there are two liquids, A and B, when B is added to A, the vapor pressure of A above the solvent decreases.

Structures

The structure of solids are described by the type of unit cells that compose them. Most solids are either simple cubic, body-centered cubic, or face-centered cubic.

Retrieved from: https://openstax.org/books/chemistry-atoms-first-2e/pages/10-6-lattice-structures-in-crystalline-solids

Polarity

Polarity is the concept that matter will interact with other matter in various ways depending on the nature of the molecules that compose them. In some molecules, the electrons being shared between the composing elements are shared in an unequal way, for example between the hydrogen and the oxygen of water. Because of this there is a polarity of the resulting molecule with a partial positive charge on one end and a partial negative charge on the other end. A nonpolar molecule is one without this variation of charge and is often more inert than polar molecules.

Properties

Colligative properties are properties observed simply on the number of particles present with no consideration given to the nature of those particles. These include freezing point depression, boiling point elevation, and osmotic pressure.

Freezing point depression is when the freezing point of a substance is reduced following the introduction of another substance. The freezing point of water is lowered by \(1.86^{\circ}C\) and via the formula:

\[\Delta T_f = iK_fm\]

where \(\Delta T_f\) is the freezing point depression, \(K_f\) is the proportionality constant for freezing, \(m\) is the molality of the solution, and \(i\) is the van ’t Hoff factor.

Boiling point elevation is measured similarly using the following formula:

\[\Delta T_b = iK_bm\]

where \(K_b\) is the proportionality constant for boiling.

Osmotic pressure is measured with the following formula:

\[\pi = iMRT\]

where \(M\) is the molarity of the solution, \(R\) is the ideal gas constant, \(T\) is the temperature in Kelvin, and \(i\) is the van ’t Hoff factor.

Solutions

Solutions are composed of solutes that are dissolved into solvents. For every solvent, there is a maximum amount of solute that can be dissolved. When this maximum is reached, the solution is said to be saturated. Any additional solute added to a saturated solution will not dissolve; this is a precipitate.

Polarity

The rule of thumb is that like dissolves like within the realm of solutions. Polar solutes are more likely to form a solution with polar solvents, and nonpolar solutes are more likely to form a solution with nonpolar solvents.

Properties

There are two main types of properties that define solutions: colligative and non-colligative.

Colligative Properties

These properties are impacted by the number of particles of solute dissolved into the solvent, not by the specific identity of the solute or solvent. In other words, the identity of the particles does not impact the property, but the addition or removal of molecules in general does. Solutions have higher boiling points than pure solvents. Similarly, freezing points of solutions are lower than that of pure solvents. The vapor pressure of solutions are lower than that of pure solvents.

Non-colligative Properties

Non-colligative properties, such as concentration, solubility, viscosity, and surface tension, are those that depend on what the exact solute or solvent is. For example, the viscosity of honey is different than that of water, therefore viscosity is a non-colligative property because the identity of the material matters.

Forces

Many forces exist in the formation of solutions. First, there are electrostatic forces that exist between the cations and anions within a solution. Then there is hydrogen bonding, which occurs between the hydrogens of water and the solutes. London dispersion forces and dipole-dipole interactions are also present within solutions as well.

Concentration Calculations

Percent composition by mass is determined by dividing the mass of the solute by the mass of the entire solution and multiplying that by \(100\). A mole fraction is the number of moles of a solution divided by the sum of moles of all components (\(x_i\)). Molarity (M) is the number of moles of solute per liter of solution. Molality (m) is equal to moles of solute per kilogram of solvent.

Acids and Bases

Acid-base chemistry is an important section to master for the DAT exam. You should understand the ionizations of acids and bases as they relate to important equations used to calculate pH, pOH, and respective dissociation constants. Understanding acid base reactions can allow scientists to calculate unknown concentrations of materials as well.

pH

The concentrations of hydrogen (H+) or hydroxide (OH-) can be used to determine the pH of a given solution. Additionally, if the pH or pOH of a solution is known, then the values of the concentration of H+ or OH- can be calculated using the aforementioned equation relationships. These are equations you should memorize:

\[\text{pH} = -\log[\text{H}^+]\] \[\text{pOH} = -\log[\text{OH}^-]\] \[\text{Water dissociation constant} (K_W) = [\text{H}^+][\text{OH}^-] = 10^{-14}\] \[\text{pH} + \text{pOH} = 14\]

Strength

The stronger the acid or base, the more it dissociates or ionizes into water. The following table includes some heavy hitters that should be memorized for the DAT exam:

Acids	Bases
\(\text{HClO}_4\) (perchloric acid)	\(\text{NaOH}\) (sodium hydroxide)
\(\text{HNO}_3\) (nitric acid)	\(\text{KOH}\) (potassium hydroxide)
\(\text{H}_2\text{SO}_4\) (sulfuric acid)	\(\text{Ca(OH})_2\), (calcium hydroxide)
\(\text{HCl}\) (hydrochloric acid)	other soluble hydroxides of group IA and IIA metals

Brønsted-Lowry Reactions

Brønsted-Lowry acids and bases exist in pairs of a conjugate acid and a conjugate base. The conjugate acid is formed when a base gains a proton, whereas the conjugate base is formed when the acid loses an electron.

Calculations

Important Equations

Acid dissociation constant (\(K_a\)):
\(K_a = \frac{[\text{H}_3\text{O}^+][\text{A}^-]}{[\text{HA}]}\)

Base dissociation constant (\(K_B\)):
\(K_B = \frac{[\text{B}^+][\text{OH}^-]}{[\text{BOH}]}\)

Water dissociation constant (\(K_w\)):
\(K_w = K_a \times K_b = 1 \times 10^{-14}\)

Practice Problems:

From the equilibrium concentrations given, calculate \(K_a\) for each of the weak acids and \(K_b\) for each of the weak bases.

(a)

\[\text{NH}_3: [\text{OH}^-] = 3.1 \times 10^{-3} \; M\] \[[\text{NH}_4^+] = 3.1 \times 10^{-3} \; M\] \[[\text{NH}_3] = 0.533 \; M\]

(b)

\[\text{HNO}_2: [\text{H}_3\text{O}^+] = 0.011 \; M\] \[[\text{NO}_2^-] = 0.0438 \; M\] \[[\text{HNO}_2] = 1.07 \; M\]

Problem adapted from: https://openstax.org/books/chemistry-atoms-first-2e/pages/14-exercises

(a) \(K_b = ([3.1 \times 10^{-3} \; M][3.1 \times 10^{-3}])/[0.533] = 1.8 \times 10^{-5}\)

(b) \(K_a = ([0.011 \; M][0.0438 \; M])/[1.07 \; M] = 4.5 \times 10^{-4}\)

Chemical Equilibria

A reversible reaction is impacted by the amount and concentration of reactants and products that are involved, and they do not go to completion. At equilibrium the concentrations of A and B are constant, and the reaction may proceed in either direction.

\[aA + bB \leftarrow \rightarrow cC + dD\] \[\textbf{equilibrium constant} ({K_{eq}}) = \frac{[A]^a[B]^b}{[C]^c[D]^d}\]

Molecular

Molecular chemical equilibrium occurs in a reversible chemical reaction when reactants and products are equal and the rates of the forward and reverse reactions are the same as dictated by the equilibrium constant.

Acid/Base

Equilibrium can be applied to acid-base reactions as well, described by the dissociation constants for acids (\(K_a\)) and bases (\(K_b\)). These constants describe the ability of an acid or a base to accept or donate a proton. The pH of acid-base reactions can be understood and, furthermore, calculated using those dissociation constants.

Precipitation

\(K_{sp}\) is the equilibrium constant for a solid dissolved in an aqueous solution. \(Q_{sp}\) is the quotient solubility product. This is used to calculate a ratio of ion concentrations at any point in a reaction other than equilibrium, which is thereby used to determine if the reaction is at equilibrium or not and if a precipitate has formed according to the relationships described below:

\[A_mB_n(s) \leftarrow \rightarrow mA \; (aq) + nB \; (aq)\] \[K_{sp} = [A]^m[B]^n \text{ in saturated solution}\]

At a given temperature, a salt has a given \(K_{sp}\).

When \(Q_{sp} = K_{sp}\), the solution is saturated.

When \(Q_{sp} > K_{sp}\), a precipitate will form until equilibrium is once again reached.

When \(Q_{sp} < K_{sp}\), no precipitate will form as the reaction proceeds in a forward direction.

Calculations

The reaction quotient is calculated as follows, with \(A\), \(B\), \(C\), and \(D\) representing reactants, and it is used to determine if a reaction is at equilibrium:

\[\textbf{reaction quotient} (Q) = \frac{[A]^a[B]^b}{[C]^c[D]^d}\] \[\begin{array}{|l|l|} \hline Q < K_{eq} & \begin{array}{l} \text{Concentration of reactants is greater than at equilibrium} \\ \text{Reaction proceeds in the forward direction} \end{array} \\ \hline Q = K_{eq} & \text{Reactants and products are at equilibrium} \\ \hline Q > K_{eq} & \begin{array}{l} \text{Concentration of products is greater than at equilibrium} \\ \text{Reaction proceeds in the reverse direction} \end{array} \\ \hline \end{array}\]

Le Chatelier’s Principle

This principle states that when changes are made in a system, the equilibrium will shift to counteract that change. This box explains how this principle works:

Change	Direction of Equilibrium Shift
increase in concentration of reactant	favors product formation
increase in product concentration	favors reactant formation
increase in temperature	favors endothermic reaction
decrease in temperature	favors exothermic reaction
increase in pressure	favors side with fewer molecules
decrease in pressure	favors side with more molecules