Unit 6 openstax

Gas Laws

Boyle’s law

Volume of a given number of moles of gas held at constant temperature is inversely proportional to the pressure under which it is measured.

Charles’s law

Volume of a given number of moles of gas is directly proportional to its kelvin temperature when the pressure is held constant.

Amontons’s law (also, Gay-Lussac’s law)

Pressure of a given number of moles of gas is directly proportional to its kelvin temperature when the volume is held constant.

Avogadro’s law

Volume of a gas at constant temperature and pressure is proportional to the number of gas molecules.

Gas Units and Measurements

Atmosphere (atm)

Unit of pressure; 1 atm = 101,325 Pa.

Bar

Unit of pressure; 1 bar = 100,000 Pa.

Pascal (Pa)

SI unit of pressure; 1 Pa = 1 N/m2.

Pounds per square inch (psi)

Unit of pressure common in the US.

Torr

Unit of pressure; 1 torr = 1/760 atm.

Gas Properties and Behavior

Diffusion

Movement of an atom or molecule from a region of relatively high concentration to one of relatively low concentration.

Effusion

Transfer of gaseous atoms or molecules from a container to a vacuum through very small openings.

Graham’s law of effusion

Rates of diffusion and effusion of gases are inversely proportional to the square roots of their molecular masses.

Kinetic molecular theory

Theory based on simple principles and assumptions that effectively explains ideal gas behavior.

Gas Equations and Concepts

Dalton’s law of partial pressures

Total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the component gases.

Ideal gas

Hypothetical gas whose physical properties are perfectly described by the gas laws.

Ideal gas law

Relation between the pressure, volume, amount, and temperature of a gas under conditions derived by combination of the simple gas laws.

Van der Waals equation

Modified version of the ideal gas equation containing additional terms to account for non-ideal gas behavior.

Pressure Measurement and Devices

Barometer

Device used to measure atmospheric pressure.

Manometer

Device used to measure the pressure of a gas trapped in a container.

Basic Concepts

Energy

Capacity to supply heat or do work.

Temperature

Intensive property of matter that is a quantitative measure of “hotness” and “coldness”.

Thermal energy

Kinetic energy associated with the random motion of atoms and molecules.

Work (w)

Energy transfer due to changes in external, macroscopic variables such as pressure and volume; or causing matter to move against an opposing force.

Heat and Enthalpy

Heat (q)

Transfer of thermal energy between two bodies.

Enthalpy (H)

Sum of a system’s internal energy and the mathematical product of its pressure and volume.

Enthalpy change (ΔH)

Heat released or absorbed by a system under constant pressure during a chemical or physical process.

Standard enthalpy of formation (ΔH∘f)

Enthalpy change of a chemical reaction in which 1 mole of a pure substance is formed from its elements in their most stable states under standard state conditions.

Laws and Principles

First law of thermodynamics

Internal energy of a system changes due to heat flow in or out of the system or work done on or by the system.

Hess’s law

If a process can be represented as the sum of several steps, the enthalpy change of the process equals the sum of the enthalpy changes of the steps.

Born-Haber cycle

Thermochemical cycle relating the various energetic steps involved in the formation of an ionic solid from the relevant elements.

Calorimetry and Heat Measurement

Calorimeter

Device used to measure the amount of heat absorbed or released in a chemical or physical process.

Calorimetry

Process of measuring the amount of heat involved in a chemical or physical process.

Standard enthalpy of combustion (ΔH∘c)

Heat released when one mole of a compound undergoes complete combustion under standard conditions.

Energy Units and Properties

Joule (J)

SI unit of energy; 1 J = 1 kg m2/s2 and 4.184 J = 1 cal.

Calorie (cal)

Unit of heat or other energy; the amount of energy required to raise 1 gram of water by 1 degree Celsius.

Nutritional calorie (Calorie)

Unit used for quantifying energy provided by digestion of foods, defined as 1000 cal or 1 kcal.

Pressure Units and Measurements/Definitions

Hydrostatic pressure

Pressure exerted by a fluid due to gravity.

Partial pressure

Pressure exerted by an individual gas in a mixture.

Mole fraction (X)

Concentration unit defined as the ratio of the molar amount of a mixture component to the total number of moles of all mixture components.

Energy Properties and Calculations

Specific heat capacity (c)

Intensive property of a substance that represents the quantity of heat required to raise the temperature of 1 gram of the substance by 1 degree Celsius.

Lattice energy (ΔHlattice)

Energy required to separate one mole of an ionic solid into its component gaseous ions.

Internal energy (U)

Total of all possible kinds of energy present in a substance or substances.

Additional Concepts

Bond energy (also, bond dissociation energy)

Energy required to break a covalent bond in a gaseous substance.

Potential energy

Energy of a particle or system of particles derived from relative position, composition, or condition.

Kinetic energy

Energy of a moving body, in joules, equal to 1/2mv2 (where m = mass and v = velocity).

Standard Conditions and States

Standard conditions of temperature and pressure (STP)

273.15 K (0 °C) and 1 atm (101.325 kPa).

Standard molar volume

Volume of 1 mole of gas at STP, approximately 22.4 L for gases behaving ideally.

Standard state

Set of physical conditions accepted as common reference conditions for reporting thermodynamic properties; 1 bar of pressure, and solutions at 1 molar concentrations, usually at a temperature of 298.15 K.

Other Pressure Units and Devices

Compressibility factor (Z)

Ratio of the experimentally measured molar volume for a gas to its molar volume as computed from the ideal gas equation.

Mean free path

Average distance a molecule travels between collisions.

Bomb calorimeter

Device designed to measure the energy change for processes occurring under conditions of constant volume; commonly used for reactions involving solid and gaseous reactants or products.

Chapter 8 Summary

Gases exert pressure, which is force per unit area. The pressure of a gas may be expressed in the SI unit of pascal or kilopascal, as well as in many other units including torr, atmosphere, and bar. Atmospheric pressure is measured using a barometer; other gas pressures can be measured using one of several types of manometers.

The behavior of gases can be described by several laws based on experimental observations of their properties. The pressure of a given amount of gas is directly proportional to its absolute temperature, provided that the volume does not change (Amontons’s law). The volume of a given gas sample is directly proportional to its absolute temperature at constant pressure (Charles’s law). The volume of a given amount of gas is inversely proportional to its pressure when temperature is held constant (Boyle’s law). Under the same conditions of temperature and pressure, equal volumes of all gases contain the same number of molecules (Avogadro’s law).

The equations describing these laws are special cases of the ideal gas law, PV = nRT, where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its kelvin temperature, and R is the ideal (universal) gas constant.

The ideal gas law can be used to derive a number of convenient equations relating directly measured quantities to properties of interest for gaseous substances and mixtures. Appropriate rearrangement of the ideal gas equation may be made to permit the calculation of gas densities and molar masses. Dalton’s law of partial pressures may be used to relate measured gas pressures for gaseous mixtures to their compositions. Avogadro’s law may be used in stoichiometric computations for chemical reactions involving gaseous reactants or products.

Gaseous atoms and molecules move freely and randomly through space. Diffusion is the process whereby gaseous atoms and molecules are transferred from regions of relatively high concentration to regions of relatively low concentration. Effusion is a similar process in which gaseous species pass from a container to a vacuum through very small orifices. The rates of effusion of gases are inversely proportional to the square roots of their densities or to the square roots of their atoms/molecules’ masses (Graham’s law).

The kinetic molecular theory is a simple but very effective model that effectively explains ideal gas behavior. The theory assumes that gases consist of widely separated molecules of negligible volume that are in constant motion, colliding elastically with one another and the walls of their container with average speeds determined by their absolute temperatures. The individual molecules of a gas exhibit a range of speeds, the distribution of these speeds being dependent on the temperature of the gas and the mass of its molecules.

Gas molecules possess a finite volume and experience forces of attraction for one another. Consequently, gas behavior is not necessarily described well by the ideal gas law. Under conditions of low pressure and high temperature, these factors are negligible, the ideal gas equation is an accurate description of gas behavior, and the gas is said to exhibit ideal behavior. However, at lower temperatures and higher pressures, corrections for molecular volume and molecular attractions are required to account for finite molecular size and attractive forces. The van der Waals equation is a modified version of the ideal gas law that can be used to account for the non-ideal behavior of gases under these conditions.

Chapter 9 Summary

Energy is the capacity to supply heat or do work (applying a force to move matter). Kinetic energy (KE) is the energy of motion; potential energy is energy due to relative position, composition, or condition. When energy is converted from one form into another, energy is neither created nor destroyed (law of conservation of energy or first law of thermodynamics).

The thermal energy of matter is due to the kinetic energies of its constituent atoms or molecules. Temperature is an intensive property of matter reflecting hotness or coldness that increases as the average kinetic energy increases. Heat is the transfer of thermal energy between objects at different temperatures. Chemical and physical processes can absorb heat (endothermic) or release heat (exothermic). The SI unit of energy, heat, and work is the joule (J).

Specific heat and heat capacity are measures of the energy needed to change the temperature of a substance or object. The amount of heat absorbed or released by a substance depends directly on the type of substance, its mass, and the temperature change it undergoes.

Calorimetry is used to measure the amount of thermal energy transferred in a chemical or physical process. This requires careful measurement of the temperature change that occurs during the process and the masses of the system and surroundings. These measured quantities are then used to compute the amount of heat produced or consumed in the process using known mathematical relations.

Calorimeters are designed to minimize energy exchange between their contents and the external environment. They range from simple coffee cup calorimeters used by introductory chemistry students to sophisticated bomb calorimeters used to determine the energy content of food.

If a chemical change is carried out at constant pressure and the only work done is caused by expansion or contraction, q for the change is called the enthalpy change with the symbol ΔH, or ΔH°Δ�° for reactions occurring under standard state conditions at 298 K. The value of ΔH for a reaction in one direction is equal in magnitude, but opposite in sign, to ΔH for the reaction in the opposite direction, and ΔH is directly proportional to the quantity of reactants and products. The standard enthalpy of formation, ΔH∘f,Δ�f°, is the enthalpy change accompanying the formation of 1 mole of a substance from the elements in their most stable states at 1 bar and 298.15 K. If the enthalpies of formation are available for the reactants and products of a reaction, the enthalpy change can be calculated using Hess’s law: If a process can be written as the sum of several stepwise processes, the enthalpy change of the total process equals the sum of the enthalpy changes of the various steps.

The strength of a covalent bond is measured by its bond dissociation energy, that is, the amount of energy required to break that particular bond in a mole of molecules. Multiple bonds are stronger than single bonds between the same atoms. The enthalpy of a reaction can be estimated based on the energy input required to break bonds and the energy released when new bonds are formed. For ionic bonds, the lattice energy is the energy required to separate one mole of a compound into its gas phase ions. Lattice energy increases for ions with higher charges and shorter distances between ions. Lattice energies are often calculated using the Born-Haber cycle, a thermochemical cycle including all of the energetic steps involved in converting elements into an ionic compound.

Key Equations (CH8)

P=FA�=��

p = hρg

PV = nRT

PTotal = PA + PB + PC + … = ƩiPi

PA = XA PTotal

XA=nAnTotal��=��������

rate of diffusion=amount of gas passing through an areaunit of timerate of diffusion=amount of gas passing through an areaunit of time

rate of effusion of gas Arate of effusion of gas B=mB√mA√=MB√MA√rate of effusion of gas Arate of effusion of gas B=����=ℳ�ℳ�

urms=u2¯¯¯¯−−√=u21+u22+u23+u24+…n−−−−−−−−−−−−√�rms=�2¯=�12+�22+�32+�42+…�

KEavg=32RTKEavg=32��

urms=3RTM−−−−√�rms=3���

Z=molarvolume of gas at sameTandPmolar volume of ideal gas at sameTandP=(P×VmR×T)measuredZ=molarvolume of gas at same�and�molar volume of ideal gas at same�and�=(�×���×�)measured

(P+n2aV2)×(V−nb)=nRT

Key Equations (CH9)

q=c×m×ΔT=c×m×(Tfinal−Tinitial)�=�×�×ΔT=�×�×(�final−�initial)

ΔU=q+wΔ�=�+�

ΔH∘reaction=∑n×ΔH∘f(products)−∑n×ΔH∘f(reactants)Δ�reaction°=∑�×Δ�f°(products)−∑�×Δ�f°(reactants)

Bond energy for a diatomic molecule: XY(g)⟶X(g)+Y(g)DX–Y=ΔH°XY(�)⟶X(�)+Y(�)DX–Y=Δ�°

Enthalpy change: ΔH = ƩDbonds broken – ƩDbonds formed

Lattice energy for a solid MX: MX(s)⟶Mn+(g)+Xn−(g)ΔHlatticeMX(�)⟶M�+(�)+X�−(�)Δ�lattice

Lattice energy for an ionic crystal: ΔHlattice=C(Z+)(Z−)Ro