11

Chapter 18Chapter 18Acids and BasesAcids and Bases

22

Items from Chapter 17...Items from Chapter 17...

Reversible ReactionsReversible Reactions - p. 416 - p. 416– In a reversible reaction, the In a reversible reaction, the

reactions occur simultaneously in reactions occur simultaneously in both directionsboth directions

– double arrows used to indicate thisdouble arrows used to indicate this

2SO2SO2(g)2(g) + O + O2(g)2(g) 2SO 2SO3(g)3(g)

– In principle, almost all reactions In principle, almost all reactions are reversible to some extentare reversible to some extent

33

Items from Chapter 17...Items from Chapter 17...

Le Chatelier’s PrincipleLe Chatelier’s Principle - p.421 - p.421– If a stress is applied to a system in If a stress is applied to a system in

dynamic equilibrium, the system dynamic equilibrium, the system changes to relieve the stress.changes to relieve the stress.

– Stresses that upset the equilibrium Stresses that upset the equilibrium in a chemical system include: in a chemical system include: changes in concentration, changes changes in concentration, changes in temperature, and changes in in temperature, and changes in pressurepressure

44

Items from Chapter 17...Items from Chapter 17...Equilibrium ConstantsEquilibrium Constants (K (Keqeq) - p. ) - p.

418418– Chemists generally express Chemists generally express

the position of equilibrium in the position of equilibrium in terms of numerical valuesterms of numerical values

– These values relate to the These values relate to the amountsamounts of reactants and of reactants and products at equilibriumproducts at equilibrium

55

Items from Chapter 17...Items from Chapter 17...

Equilibrium ConstantsEquilibrium Constants - p. 418 - p. 418– consider this reaction:consider this reaction:

aA + bB aA + bB cC + dD cC + dD

– The equilibrium constant (KThe equilibrium constant (Keqeq) is the ) is the ratio of product concentration to the ratio of product concentration to the reactant concentration at reactant concentration at equilibrium, with each concentration equilibrium, with each concentration raised to a power (= the coefficient)raised to a power (= the coefficient)

66

Items from Chapter 17...Items from Chapter 17... Equilibrium ConstantsEquilibrium Constants - p. 418 - p. 418

– consider this reaction:consider this reaction:aA + bB aA + bB cC + dD cC + dD– Thus, the Thus, the ““equilibrium constantequilibrium constant

expression”expression” has the general has the general form:form:

[C][C]cc x [D] x [D]dd

[A][A]aa x [B] x [B]bb

( [ ] = molarity )( [ ] = molarity )

Keq =

77

Items from Chapter 17...Items from Chapter 17... Equilibrium ConstantsEquilibrium Constants - p. 418 - p. 418

– the equilibrium constants provide the equilibrium constants provide valuable information, such as valuable information, such as whether products or reactants are whether products or reactants are favored:favored:

KKeqeq > 1, products favored at > 1, products favored at equilibriumequilibrium

KKeq eq < 1, reactants favored at < 1, reactants favored at equilibriumequilibrium

Sample Problem 17-8, p. 419Sample Problem 17-8, p. 419

88

Section 18.1Section 18.1Describing Acids and BasesDescribing Acids and Bases

OBJECTIVES:OBJECTIVES:– List the properties of acids and List the properties of acids and

bases.bases.

99

Section 18.1Section 18.1Describing Acids and BasesDescribing Acids and Bases

OBJECTIVES:OBJECTIVES:– Name an acid or base, when Name an acid or base, when

given the formula.given the formula.

1010

Properties of acidsProperties of acids Taste sour (don’t try this at home).Taste sour (don’t try this at home). Conduct electricity.Conduct electricity.

– Some are strong, others are weak Some are strong, others are weak electrolytes.electrolytes.

React with metals to form hydrogen React with metals to form hydrogen gas.gas.

Change indicators (blue litmus to red).Change indicators (blue litmus to red). React with hydroxides to form water React with hydroxides to form water

and a salt.and a salt.

1111

Properties of basesProperties of bases

React with acids to form water and a React with acids to form water and a salt.salt.

Taste bitter.Taste bitter. Feel slippery (don’t try this either).Feel slippery (don’t try this either). Can be strong or weak electrolytes.Can be strong or weak electrolytes. Change indicators (red litmus turns Change indicators (red litmus turns

blue).blue).

1212

Names and Formulas of AcidsNames and Formulas of Acids An acid is a chemical that produces An acid is a chemical that produces

hydrogen ions (Hhydrogen ions (H1+1+) when dissolved ) when dissolved in waterin water

Thus, general formula = HX, where X Thus, general formula = HX, where X is a monatomic or polyatomic anionis a monatomic or polyatomic anion

HClHCl(g)(g) named hydrogen chloride named hydrogen chloride

HClHCl(aq)(aq) is named as an acid is named as an acid Name focuses on the anion presentName focuses on the anion present

1313

Naming Acid

Hydrogen _______ide becomes hydro____ic acid

Hydrogen_______ate becomes _________ic acid

Hydrogen_______ite becomes _______ous acid

chlor chlor

1414

Names and Formulas of AcidsNames and Formulas of Acids1.1. When anion ends with -ide, the acid When anion ends with -ide, the acid

starts with starts with hydro-,hydro-, and the stem of the and the stem of the anion has the suffix anion has the suffix -ic-ic followed by the followed by the word word acidacid

2. When anion ends with -ite, the anion 2. When anion ends with -ite, the anion has the suffix has the suffix -ous-ous, then , then acidacid

3. When anion ends with -ate, the anion 3. When anion ends with -ate, the anion suffix is suffix is -ic-ic and then and then acidacid

1515

Names and Formulas of BasesNames and Formulas of BasesA base produces hydroxide ions A base produces hydroxide ions

(OH(OH1-1-) when dissolved ) when dissolved in waterin water..Named the same way as any Named the same way as any

other ionic compoundother ionic compound– name the cation, followed by name the cation, followed by

anionanionTo write the formula: write To write the formula: write

symbols; write charges; then symbols; write charges; then cross (if needed)cross (if needed)

1616

Section 18.2Section 18.2Hydrogen Ions and AcidityHydrogen Ions and Acidity

OBJECTIVES:OBJECTIVES:– Given the hydrogen-ion or Given the hydrogen-ion or

hydroxide-ion concentration, hydroxide-ion concentration, classify a solution as neutral, classify a solution as neutral, acidic, or basic.acidic, or basic.

1717

Section 18.2Section 18.2Hydrogen Ions and AcidityHydrogen Ions and Acidity

OBJECTIVES:OBJECTIVES:– Convert hydrogen-ion Convert hydrogen-ion

concentrations into values of concentrations into values of pH, and hydroxide-ion pH, and hydroxide-ion concentrations into values of concentrations into values of pOH.pOH.

1818

Hydrogen Ions from WaterHydrogen Ions from Water Water ionizes, or falls apart into ions:Water ionizes, or falls apart into ions:

HH22O O H H1+1+ + OH + OH1-1-

Called the “self ionization” of waterCalled the “self ionization” of water

Occurs to a very small extent:Occurs to a very small extent:[H[H1+1+ ] = [OH ] = [OH1-1-] = 1 x 10] = 1 x 10-7 -7 MM

Since they are equal, a Since they are equal, a neutralneutral solution results solution results from waterfrom water

KKww = [H = [H1+1+ ] x [OH ] x [OH1-1-] = 1 x 10] = 1 x 10-14-14 ( (mol/L)mol/L)22

KKww is called the “ion product constant” is called the “ion product constant”

1919

Ion Product ConstantIon Product Constant HH22O O H H++ + OH + OH--

KKww is constant in every aqueous solution: is constant in every aqueous solution: [H[H++] x [OH] x [OH--] = 1 x 10] = 1 x 10-14-14

If [HIf [H++] >1x 10] >1x 10-7-7 then [OH then [OH--] < 1x10] < 1x10-7-7 If [HIf [H++] < 1x10] < 1x10-7-7 then [OH then [OH--] > 1x10] > 1x10-7-7

If we know one,the other can be determinedIf we know one,the other can be determined

If [HIf [H++] > 10] > 10-7-7 , it is , it is acidicacidic and [OH and [OH--] < 10] < 10-7-7 If [HIf [H++] < 10] < 10-7-7 , it is , it is basicbasic and [OH and [OH--] > 10] > 10-7-7

Basic solutions are also called “alkaline”Basic solutions are also called “alkaline”

2020

Logarithms and the pH conceptLogarithms and the pH concept Logarithms are powers of ten.Logarithms are powers of ten.

– Know how to use the log buttons on your Know how to use the log buttons on your calculatorcalculator

definition: pH = -log[Hdefinition: pH = -log[H++]]

in neutral pH = -log(1 x 10in neutral pH = -log(1 x 10-7-7) = 7) = 7

in acidic solution [Hin acidic solution [H++] > 10] > 10-7-7

pH < 7 (from 0 to 7 is the acid range)pH < 7 (from 0 to 7 is the acid range)

in base, pH > 7 (7 to 14 is base range)in base, pH > 7 (7 to 14 is base range)

2121

pH and pOHpH and pOH pOH = -log [OHpOH = -log [OH--] ]

[H[H++] x [OH] x [OH--] = 1 x 10] = 1 x 10-14 -14 MM22

pH + pOH = 14pH + pOH = 14

Thus, a solution with a pOH less than Thus, a solution with a pOH less than 7 is basic; with a pOH greater than 7 7 is basic; with a pOH greater than 7 is an acidis an acid

2222

0 1 3 5 7 9 11

13

140135791

113

14

Basic

Neutral

pH

pOH

Acidic

2323

[H+]

100

10-

1

10-

3

10-

5

10-

7

10-

9

10-

11

10-

13

10-

14

[OH-]

Acidic Neutral

Basic

100

10-

1

10-

3

10-

5

10-

7

10-

9

10-

11

10-

13

10-

14

2424

Examples:Examples:

Sample 18-1, p.434Sample 18-1, p.434Sample 18-2, p.435Sample 18-2, p.435Sample 18-3, p.436Sample 18-3, p.436Sample 18-4, p.438Sample 18-4, p.438Sample 18-5, p.439Sample 18-5, p.439

2525

Formula’s to KnowFormula’s to KnowKw = [H3O

+] [OH-] = 1 x 10-14

pH = - log[H30+]

pOH = -log [OH-]

pH + pOH = 14

[H3O+] = antilog (-pH)

[OH-] = antilog (-pOH)

This is from your formula sheet

Acid Equations

Base Equations

pH Scale Equation

Ion Product ConstantIon Product Constant

Note H+ is the same as H3O+

2626

Measuring pHMeasuring pH

Why measure pH?Why measure pH?– Everything from swimming Everything from swimming

pools, soil conditions for plants, pools, soil conditions for plants, medical diagnosis, soaps and medical diagnosis, soaps and shampoos, etc.shampoos, etc.

Sometimes we can use Sometimes we can use indicators, other times we might indicators, other times we might need a pH meterneed a pH meter

2727

Acid-Base IndicatorsAcid-Base Indicators

An indicator is an acid or base that An indicator is an acid or base that undergoes dissociation in a known undergoes dissociation in a known pH range, and has different colors pH range, and has different colors in solution (more later in chapter)in solution (more later in chapter)

HIn HHIn H1+1+ + In + In1-1-

Examples: litmus, phenolphthalein, Examples: litmus, phenolphthalein, bromthymol blue: Fig 18.14, p.445bromthymol blue: Fig 18.14, p.445

2828

Acid-Base IndicatorsAcid-Base Indicators Although useful, there are limitations to Although useful, there are limitations to

indicators:indicators:– usually given for a certain temperature usually given for a certain temperature

(25 (25 ooC), thus may change at different C), thus may change at different temperaturestemperatures

– what if the solution already has color?what if the solution already has color?– ability of human eye to distinguish colorsability of human eye to distinguish colors

2929

Acid-Base IndicatorsAcid-Base IndicatorsA A pH meter ((simulation)may give )may give

more definitive resultsmore definitive results– some are large, others portablesome are large, others portable– works by measuring the voltage works by measuring the voltage

between two electrodesbetween two electrodes– needs to be calibratedneeds to be calibrated– Fig. 18.17, p.446Fig. 18.17, p.446

3030

Section 18.3Section 18.3Acid-Base TheoriesAcid-Base Theories

OBJECTIVES:OBJECTIVES:– Compare and contrast acids Compare and contrast acids

and bases as defined by the and bases as defined by the theories of Arrhenius, theories of Arrhenius, Brønsted-Lowry, and LewisBrønsted-Lowry, and Lewis

3131

Section 18.3Section 18.3Acid-Base TheoriesAcid-Base Theories

OBJECTIVES:OBJECTIVES:– Identify conjugate acid-base Identify conjugate acid-base

pairs in acid-base reactions.pairs in acid-base reactions.

3232

Svante ArrheniusSvante Arrhenius Swedish chemist Swedish chemist

(1859-1927) - Nobel (1859-1927) - Nobel prize winner in prize winner in chemistry (1903)chemistry (1903)

One of the first One of the first chemists to explain chemists to explain the chemical theory the chemical theory of the behavior of of the behavior of acids and basesacids and bases

3333

1. Arrhenius Definition1. Arrhenius Definition AcidsAcids produce hydrogen ions (H produce hydrogen ions (H1+1+) in aqueous ) in aqueous

solution.solution.

BasesBases produce hydroxide ions (OH produce hydroxide ions (OH1-1-) when ) when dissolved in water.dissolved in water.

Limited to aqueous solutions.Limited to aqueous solutions.

Only one kind of base (hydroxides)Only one kind of base (hydroxides)

NHNH33 (ammonia) would not be an Arrhenius base. (ammonia) would not be an Arrhenius base.

HCl H+ + Cl-H2O

NaOH Na+ + OH-

H2O

3434

Polyprotic AcidsPolyprotic Acids

Some compounds have more than 1 ionizable Some compounds have more than 1 ionizable hydrogen.hydrogen.

HNOHNO3 3 nitric acid – monoproticnitric acid – monoprotic

HH22SOSO44 sulfuric acid - diprotic - 2 H sulfuric acid - diprotic - 2 H++

HH33POPO44 phosphoric acid - triprotic - 3 H phosphoric acid - triprotic - 3 H++

Having more than one ionizable hydrogen does Having more than one ionizable hydrogen does not mean stronger!not mean stronger!

3535

Polyprotic AcidsPolyprotic Acids

However, not all compounds that have However, not all compounds that have hydrogen are acidshydrogen are acids

Also, not all the hydrogen in an acid Also, not all the hydrogen in an acid may be released as ionsmay be released as ions

CHCH33COOH CHCOOH CH33COOCOO-- + H + H++

– only those that have very polar bonds are only those that have very polar bonds are ionizable - this is when the hydrogen is ionizable - this is when the hydrogen is joined to a very electronegative elementjoined to a very electronegative element

NH4+ or H2O

3636

Arrhenius examples...Arrhenius examples...

Consider HClConsider HCl

What about CHWhat about CH44 (methane)?(methane)?

CHCH33COOH COOH (ethanoic acid, or acetic acid) - (ethanoic acid, or acetic acid) -

it has 4 hydrogens like methane does…?it has 4 hydrogens like methane does…?

Table 18.3, p. 442 for basesTable 18.3, p. 442 for bases

3737

What is a ProtonWhat is a Proton A proton is a positively charged subatomic A proton is a positively charged subatomic

particle (no neutrons, no electrons)particle (no neutrons, no electrons)

A hydrogen atom is 1 proton(+) and 1 electron (-)A hydrogen atom is 1 proton(+) and 1 electron (-)

A hydrogen ion or proton (HA hydrogen ion or proton (H++) is a hydrogen atom ) is a hydrogen atom that has lost its electron leaving only the protonthat has lost its electron leaving only the proton

H H H H+(proton)+(proton) + 1 electron + 1 electron

3838

2. Brønsted-Lowry Definitions2. Brønsted-Lowry Definitions Broader definition than ArrheniusBroader definition than Arrhenius

Acid is hydrogen-ion donor (HAcid is hydrogen-ion donor (H+ + or proton); base is hydrogen-or proton); base is hydrogen-ion acceptor.ion acceptor.

Acids and bases always come in pairsAcids and bases always come in pairs

HCl is an acid.HCl is an acid.– When it dissolves in water, it gives it’s proton to water.When it dissolves in water, it gives it’s proton to water.

HCl(g) + HHCl(g) + H22O(l) O(l) HH33OO++ + Cl + Cl--

Water is a base; makes hydronium ion.Water is a base; makes hydronium ion.

3939

Acids and bases come in Acids and bases come in pairs...pairs...

A A conjugate baseconjugate base is the remainder of is the remainder of the original acid, after it donates it’s the original acid, after it donates it’s hydrogen ionhydrogen ion

A A conjugate acidconjugate acid is the particle formed is the particle formed when the original base gains a when the original base gains a hydrogen ionhydrogen ion

Indicators are weak acids or bases that Indicators are weak acids or bases that have a different color from their original have a different color from their original acid and baseacid and base

4040

Acids and bases come in pairs...Acids and bases come in pairs...

General equation is: General equation is:

HAHA(aq)(aq) + H + H22OO(l) (l) H H33OO++(aq)(aq) + A + A--

(aq)(aq)

Acid Base Conjugate acid Conjugate baseAcid Base Conjugate acid Conjugate base

NHNH33 + H + H22O NHO NH441+1+ + OH + OH1-1-

base acid c.a. c.b.base acid c.a. c.b.

4141

HCl + HHCl + H22O HO H33OO1+1+ + Cl + Cl1-1-

acid base c.a. c.b.acid base c.a. c.b.

Amphoteric - acts as acid or base Amphoteric - acts as acid or base (H(H220, HSO0, HSO44

--, H, H22POPO44 --) must be able to give ) must be able to give a Ha H++ or accept an H or accept an H++

4242

3. Lewis Acids and Bases3. Lewis Acids and Bases Gilbert Lewis focused on the donation or Gilbert Lewis focused on the donation or

acceptance of a pair of electrons during a acceptance of a pair of electrons during a reactionreaction

Lewis AcidLewis Acid - electron deficient - electron deficient

Lewis BaseLewis Base - electron rich - electron rich

Most general of all 3 definitions; acids don’t Most general of all 3 definitions; acids don’t even need hydrogen!even need hydrogen!

Sample Problem 18-6, p.447Sample Problem 18-6, p.447

4343

Lewis Acids/BasesLewis Acids/Bases Several categories of substances can be considered Lewis Several categories of substances can be considered Lewis

acids: Electron Pooracids: Electron Poor

1) positive ions1) positive ions2) having less than a full octet in the valence shell2) having less than a full octet in the valence shell3) polar double bonds (one end)3) polar double bonds (one end)4) expandable valence shells 4) expandable valence shells

Several categories of substances can be considered Lewis Several categories of substances can be considered Lewis bases: Electron Richbases: Electron Rich

1) negative ions1) negative ions2) one of more unshared pairs in the valence shell2) one of more unshared pairs in the valence shell3) polar double bonds (the other end)3) polar double bonds (the other end)4) the presence of a double bond4) the presence of a double bond

4444

Section 18.4Section 18.4Strengths of Acids and BasesStrengths of Acids and Bases

OBJECTIVES:OBJECTIVES:– Define strong acids and weak Define strong acids and weak

acids.acids.

4545

Section 18.4Section 18.4Strengths of Acids and BasesStrengths of Acids and Bases

OBJECTIVES:OBJECTIVES:– Calculate an acid dissociation Calculate an acid dissociation

constant (Kconstant (Kaa) from ) from concentration and pH concentration and pH measurements.measurements.

4646

Section 18.4Section 18.4Strengths of Acids and BasesStrengths of Acids and Bases

OBJECTIVES:OBJECTIVES:– Arrange acids by strength Arrange acids by strength

according to their acid according to their acid dissociation constants (Kdissociation constants (Kaa). ).

4747

Section 18.4Section 18.4Strengths of Acids and BasesStrengths of Acids and Bases

OBJECTIVES:OBJECTIVES:– Arrange bases by strength Arrange bases by strength

according to their base according to their base dissociation constants (Kdissociation constants (Kbb).).

4848

StrengthStrength Strong acids and bases are strong and bases are strong electrolytes

– They fall apart (ionize) completely.They fall apart (ionize) completely.– Weak acids don’t completely ionize.Weak acids don’t completely ionize.

StrengthStrength different from different from concentrationconcentration

Strong-forms many ions when dissolvedStrong-forms many ions when dissolved

Mg(OH)Mg(OH)22 is a strong base- it falls completely is a strong base- it falls completely apart when dissolved. apart when dissolved. – But, not much dissolves- not concentratedBut, not much dissolves- not concentrated

4949

Measuring strengthMeasuring strength

Ionization is reversible.Ionization is reversible.

HAHA HH++ + A + A- -

This makes an equilibriumThis makes an equilibrium

Acid dissociation constant = KAcid dissociation constant = Kaa

5050

Ka = [HKa = [H++ ][A ][A-- ] ] [HA][HA]

(water is constant so we can ignore it)(water is constant so we can ignore it)

Stronger acid = more products (ions), thus Stronger acid = more products (ions), thus a larger Ka a larger Ka

(Table 18.4, p.448) (Table 18.5 p. 450)(Table 18.4, p.448) (Table 18.5 p. 450)

HAHA H H++ + A + A--

5151

What about bases?What about bases? Strong bases dissociate completely.Strong bases dissociate completely.

B + HB + H22OO BH BH++ + OH + OH--

Base dissociation constant = KBase dissociation constant = Kbb

5252

Kb = [BHKb = [BH++][OH][OH--]][B][B]

(we ignore the water)(we ignore the water)

Stronger base = more dissociated, thus a larger Stronger base = more dissociated, thus a larger Kb.Kb.

B + HB + H22OO BH BH++ + OH- + OH-

5353

Strength vs. ConcentrationStrength vs. Concentration The words The words concentratedconcentrated and and dilutedilute tell tell

how much of an acid or base is how much of an acid or base is dissolved in solution - refers to the dissolved in solution - refers to the number of moles of acid or base in a number of moles of acid or base in a given volumegiven volume

The words The words strongstrong and and weakweak refer to the refer to the extent of ionization of an acid or baseextent of ionization of an acid or base

Is concentrated weak acid possible?Is concentrated weak acid possible?

5454

Strong AcidsStrong Acids

HClOHClO44

HIHI HBrHBr HClHCl HNOHNO33

HH22SOSO44

HA HA H H++ + A + A--

100% Ionization 100% Ionization occursoccurs

5555

Strong BasesStrong Bases

NaOHNaOH Mg(OH)Mg(OH)22

KOHKOH Ca(OH)Ca(OH)22

BOH BOH BB++ + OH + OH--

100% Ionization 100% Ionization occursoccurs

5656

Concentrated Acid or BaseConcentrated Acid or Base

Based on molarityBased on molarity mol/Lmol/L Dilute solution = 0.5 Dilute solution = 0.5

mol/Lmol/L Concentrated solution Concentrated solution

= 18 mol/L= 18 mol/L

5757

PracticePractice

Write the expression for HNOWrite the expression for HNO22

Write the KWrite the Kbb for NH for NH33

Sample 18.7, p. 452Sample 18.7, p. 452Carefully study Key Terms and Carefully study Key Terms and

equations, p. 453equations, p. 453Be sure to take all the self-tests Be sure to take all the self-tests

that are available!that are available!

5858

18.9 Calculating Dissociation Constants18.9 Calculating Dissociation Constants

A 0.100 M solution of carbonic acid is only partially ionized.

Using a measure of pH, the [H+] is calculated as 1.34 x 10 -3 M.

What is the acid dissociation constant of carbonic acid?

5959

solutionsolution

H2CO3 <-> CO3 2- - + 2H 1+

Given – Initial concentration of Carbonic Acid is 0.100 M- Equilibrium concentration of H + is 1.34 x 10 -3 M

Solution – I need to use the…..

ICE TABLE

6060

[Initial][Initial]

[Change][Change]

[Equilibrium][Equilibrium]

0.1000.100 0 0

-6.7x 10 -6.7x 10 -4-4 +6.7 x 10 +6.7 x 10 -4-4 +1.34 x 10 +1.34 x 10 -3-3

0.099330.09933 6.7 x 10 6.7 x 10 -4-4 1.34 x 10 1.34 x 10 -3-3

H2CO3 <-> CO3 2- + 2H 1+

1.34 x 10 1.34 x 10 -3 -3 HH 1+1+ 1 mol CO3 2- 6.7 x 10 6.7 x 10 -4 -4 mol CO3 2-

2 mol H 1+

6161

1H2CO3 <-> 1CO3 2- + 2H 1+

Ka = ProductsReactants

Ka = [CO3 2-]1

[H2CO3]1

[H+]2

Ka = [6.7 x 10 6.7 x 10 -4-4]1

[0.09933]1

[1.34 x 10 1.34 x 10 -3-3]]22

Ka = 4.5 x 10 – 6 which indicates that Carbonic acid is a weak acid( less than 1)