Acids, Bases and Salts

Summary

Acids

Electrical conductivity

Any solution's ability to conduct electricity is defined by is charges ions in it. As a result, a strong acid will produce more charged ions than a weak one, and so it's solution will be a better electrical conductor than a weak acid. The same goes for strong/weak bases.

Acids in daily life

• • Ethanoic acid – found in vinegar and tomato juice

  • Citric acid – found in citrus foods like lemons, oranges and grapefruit

  • Lactic acid – found in sour milk and yoghurt, and in muscle respiration

  • Tartaric acid – found in grapes

  • Tannic acid – found in tea and ant’s body

  • Formic acid – found in bee stings

  • Hydrochloric acid – found in stomach juices

Common laboratory acids

• • Hydrochloric acid (HCl)

  • Sulphuric acid (H₂SO₄)

  • Nitric acid (HNO₃)

Dilute acid: solution containing small amount of acid dissolved in water

Concentrated acids: solution containing large amount of acid dissolved in water

Properties of acids

• • • sour taste

    • hazardous - irritants to skin, causing skin to redden and blister

    • change the color of indicators - turn blue litmus red

    • react with metals to produce hydrogen gas - gas is tested with a burning splint which burns with a 'pop' sound

    • react with carbonates and hydrogencarbonates to produce carbon dioxide - to test this, the gas produced is bubbled into limewater which forms a white precipitate

    • react with metal oxides and hydroxides - reach slowly with warm dilute acid to form salt and water

Storage of acids

• • Acids are stored in claypots, glass or plastic containers as sand, glass and plastic do not react with acids.

  • If it is stored in metal container, metal would react with acids

Uses of acids

• • Sulphuric Acid

    • Used in car batteries

    • Manufacture of ammonium sulphate for fertilisers

    • Manufacture of detergents, paints, dyes, artificial fibres & plastics

  • Hydrochloric acid

    • can remove rust (iron(III) oxide) which dissolves in acids

  • Acids are used in preservation of foods (e.g. ethanoic acid)

Acids and hydrogen ions

• • Acids are covalent compounds and do not behave as acids in the absence of water as water reacts with acids to produce H+ ions, responsible for its acidic properties

    • e.g. Citric acid crystals doesn’t react with metals and doesn’t change colours of indicators; citric acid in water reacts with metals and change turns litmus red.

  • Hydrogen gas is formed by acids as H+(aq) ions are present in acid solutions. This means when a solid/gas acid dissolved in water, they produce H+ ions in it

    • Chemical eqation: HCl(s) ---(water)---> HCl (aq)

    • Ionic Equation: HCl(s) ---(water)---> H⁺ (aq) + Cl^- (aq)

  • However when dissolved in organic solutions, they don’t show acidic properties. When metals react with acids, only the hydrogen ions react with metals, e.g.:

    • Chemical equation: 2Na(s) + 2HCl(aq) → 2NaCl(aq) + H₂(g)

    • Ionic equation: 2Na(s) + 2H⁺(aq) → 2Na+(aq) + H₂(g)

  • Basicity of an acid is maximum number of H⁺ ions produced by a molecule of acid

    • dibasic: can replace two hydrogen atoms

    • tribasic: can replace three hydrogen atoms

Fizzy drinks

• • Soft drink tablets contains solid acid (e.g. citric acid, C₆H₈O₇) & sodium bicarbonate

  • When tablet is added to water, citric acid ionises and the H⁺ produced reacts with sodium bicarbonate to produce carbon dioxide gas, making them fizz

Strong and weak acids

• • Strong Acids - acid that completely ionises in water.

    • Their reactions are irreversible.

    • E.g. H₂SO₄, HNO₃, HCl

    • H₂SO₄ (aq) → 2H⁺ (aq) + SO₄^2- (aq)

      • In above H₂SO₄ has completely been ionized in water, forming 3 kinds of particles:

        • H+ ions

        • SO₄^2- ions

        • H₂O molecules

    • Strong acids react more vigorously with metals than weak acids – hydrogen gas bubbles are produced rapidly

  • Weak acids - acids that partially ionise in water.

    • The remaining molecules remain unchanged as acids.

    • Their reactions are reversible.

    • E.g. CH₃COOH, H₂CO₃, H₃PO₄

    • H₃PO₄ (aq) ⇌ 3H⁺ (aq) + PO₄^2- (aq)

    • Weak acids react slowly with metals than strong acids – hydrogen gas bubbles are produced slowly.

Concentration vs Strength

• • Comparing 10 mol/dm³ and 0.1 mol/dm³ of hydrochloric acids and 10 mol/dm³ and 0.1 mol/dm³ of ethanoic acids

    • 10 mol/dm³ of ethanoic acid solution is a concentrated solution of weak acid

    • 0.1 mol/dm³ of ethanoic acid solution is a dilute solution of weak acid

    • 10 mol/dm³ of hydrochloric acid solution is a concentrated solution of strong acid

    • 0.1 mol/dm³ of hydrochloric acid solution is a dilute solution of strong acid

Bases and Alkalis

• • Bases are oxides or hydroxides of metals

  • Alkalis are bases which are soluble in water

  • All alkalis produces hydroxide ions (OH^-) when dissolved in water.

  • Hydroxide ions give the properties of alkalis.

  • They don’t behave as acids in absence of water.

  • Alkalis are therefore substances that produce hydroxide ions, OH^- (aq), in water.

Laboratory Alkalis

• • Sodium Hydroxide, NaOH

  • Aqueous Ammonia, NH₄OH

  • Calcium Hydroxide, Ca(OH)₂

Properties of Alkalis

• • have a slippery feel

  • hazardous

  • Dilute alkalis are irritants

  • Concentrated alkalis are corrosive and burn skin (caustic(i.e. burning) alkalis)

  • change the colour of indicators: turn common indicator litmus – red litmus to blue

  • react with acids

    • The reaction is called neutralisation

  • react with ammonium compounds

    • They react with heated solid ammonium compounds to produce ammonia gas

    • (NH₄)₂SO₄ (s) + Ca(OH)₂ (aq) → CaSO₄ (aq) + 2NH₃ (g) + 2H₂O (l)

  • react with solutions of metal ions

    • Barium sulphate, BaSO₄ (aq), contains Ba²⁺ (aq) ions

    • Ca(OH)₂ (aq) + BaSO₄ (aq) → Ba(OH)₂ (s) + CaSO₄ (aq)

    • The solid formed is precipitate – the reaction is called precipitate reaction

Strong and weak alkalis

• • Strong Alkalis: base that completely ionises in water to form OH^- (aq) ions.

    • Their reactions are irreversible.

    • E.g. NaOH, KOH, Ca(OH)₂

    • Ca(OH)₂ (s) → Ca²⁺ (aq) + 2OH^- (aq)

  • Weak Alkalis: base that partially ionise in water.

    • The remaining molecules remain unchanged as base.

    • Their reactions are reversible.

    • E.g. NH₃

    • NH₃ (g) + H₂O (l) ⇌ NH₄⁺ (aq) + OH^- (aq)

Uses of Alkalis

• • Alkalis neutralise acids in teeth (toothpaste) and stomach (indigestion)

  • Soap and detergents contain weak alkalis to dissolve grease

  • Floor and oven cleaners contain NaOH (strong alkalis)

  • Ammonia (mild alkalis) is used in liquids to remove dirt and grease from glass

pH and Indicators

Indicators are substances that has different colours in acidic and alkaline solutions

Common indicators:

• • Litmus

  • Methyl orange

  • Phenolphtalein

pH Scale

A measure of acidity or alkalinity of a solution is known as pH

• • pH 7 is neutral – in pure water

  • solutions of less than pH 7 are acidic.

    • The solutions contain hydrogen ions.

      • The lower the pH, the more acidic the solution is and more hydrogen ions it contains.

  • solutions of more than pH 7 are alkaline.

    • The solution contains hydroxide ions.

      • The higher the pH, the more alkaline the solution and more hydroxide ions it contains.

Measuring pH of a Solution

• Universal indicators

  • • It can be in paper or solution form.

    • Universal paper can be dipped into a solution then pH found is matched with the colour chart.

    • It gives approximate pH value.

• pH meter

  • • A hand-held pH probe is dipped into solution and meter will show the pH digitally or by a scale.

    • Measures pH water in lakes, water, and streams accurately

• pH sensor and computer

  • • A probe is dipped into solution and will be sent to computer through interface used to measure pH of solution.

    • The pH reading is displayed on computer screen.

Ionic Equations

Ionic equation is equation involving ions in aqueous solution, showing formation and changes of ions during the reaction

Rule to make ionic equations:

• • Only formulae of ions that change is included; ions don’t change = omitted

  • Only aqueous solutions are written as ions; liquids, solids and gases written in full

Reaction between Metals and Acids

Eg. reaction of sodium with hydrochloric acid

2Na (s) + 2HCl (aq) → 2NaCl (aq) + H₂ (g)

Its ionic equation is written as:

2Na (s) + 2H⁺ (aq) + 2Cl^- (aq) → 2Na⁺ (aq) + 2Cl^- (aq) + H₂ (g)

Since 2 Cl^- (aq) ions don’t change, they’re not involved in reaction.

As ionic equation is used to show changes in reactions, we omit Cl^- (aq) ions.

So we’re left with:

2Na (s) + 2H⁺ (aq) → 2Na⁺ (aq) + H₂ (g)

Reaction between soluble ionic compounds and acids

e.g. Reaction of sodium hydrogencarbonate with hydrochloric acid

NaHCO₃ (aq) + HCl (aq) → NaCl (aq) + CO₂ (g) + H₂O (l)

Its ionic equation is:

Na⁺ (aq) + H⁺ (aq) + CO₃^2- (aq) + H⁺ (aq) + Cl^- (aq) → Na⁺ (aq) + Cl^- (aq) + CO₂ (g) + H₂O (l)

Since Na⁺ (aq) and Cl^- (aq) ions don’t change, we omit them, leaving:

H⁺ (aq) + CO₃^2- (aq) + H⁺ (aq) → CO₂ (g) + H₂O (l)

CO₃^2- (aq) + 2H⁺ (aq) → CO₂ (g) + H₂O (l)

Reaction between insoluble ionic compounds and acids

e.g. Reaction between iron(II) oxide and sulphuric acid

FeO (s) + H₂SO₄ (aq) → FeSO₄ (aq) + H₂O (g)

Its ionic equation is:

FeO(s) + 2H+ (aq) + SO₄^2- (aq) → Fe²⁺ (aq) + SO₄^2- (aq) + H₂O (g)

Note: FeO is written in full as it is solid (although it is an ionic compound)

Since SO₄^2- (aq) ions don’t change, we omit SO₄^2- ions, leaving:

FeO (s) + 2H⁺ (aq) → Fe²⁺ (aq) + H₂O (g)

E.g. Reaction between calcium carbonate and hydrochloric acid

CaCO₃ (s) + 2HCl (aq) → CaCl₂ (aq) + CO₂ (g) + H₂O (l)

Its ionic equation is:

CaCO₃ (s) + 2H⁺ (aq) + 2Cl^- (aq) → Ca²⁺ (aq) + 2Cl^- (aq) + CO₂ (g) + H₂O (l)

Since 2 Cl^- (aq) ions don’t change, we omit Cl^- ions, leaving:

CaCO₃ (s) + 2H⁺ (aq) → Ca²⁺ (aq) + CO₂ (g) + H₂O (l)

Reaction producing precipitate

E.g. Reaction between calcium hydroxide and barium sulphate

Ca(OH)₂ (aq) + BaSO₄ (aq) → Ba(OH)₂ (s) + CaSO₄ (aq)

Its ionic equation is written as:

Ca²⁺ (aq) + 2OH^- (aq) + Ba²⁺ (aq) + SO₄^2- (aq) → Ba(OH)₂ (s) + Ca²⁺ (aq) + SO₄^2- (aq)

Since Ca²⁺ (aq) and SO₄^2- (aq) ions don’t change, we omit them, leaving:

Ba²⁺ (aq) + 2OH^- (aq) → Ba(OH)₂ (s)

Displacement reactions

E.g. Reactions between magnesium with zinc sulphate

Mg (s) + ZnSO₄ (aq) → MgSO₄ (aq) + Zn (s)

Its ionic equation is written as:

Mg (s) + Zn²⁺ (aq) + SO₄^2- (aq) → Mg²⁺ (aq) + SO₄^2- (aq) + Zn (s)

Since SO₄^2- (aq) ions don’t change, we omit them, leaving:

Mg (s) + Zn²⁺ (aq) → Mg²⁺ (aq) + Zn (s)

Neutralization

• • Neutralization is the reaction between acid and base to form salt and water only.

  • From ionic equation, we know that the reaction only involves H⁺ ions from acids with OH^- ions from alkali to form water .

E.g. NaOH + H₂SO₄ forms Na₂SO₄ + H₂O

H₂SO₄ (aq) + NaOH (aq) --> Na₂SO₄ (aq) + H₂O (g)

Ionic equation is:

H⁺ (aq) + OH^- (aq)→ H₂O (g)

Plants don’t grow well in acidic soil. Quicklime (calcium hydroxide) is added to neutralise the acidity of soil according to equation:

Acid (aq) + Ca(OH)₂ (aq) --> Ca(acid anion) (aq) + H₂O (g)

Reaction between Base and Ammonium Salts

E.g. Reaction between NaOH and NH₄OH

NaOH (aq) + NH₄Cl (aq) --> NaCl (aq) + NH₃ (g) + H₂O (g)

Ionic equation:

NH₄⁺ (aq) + OH^-(aq) → NH₃ (g) + H₂O (g)

Oxides

Preparation of Salts

Preparation of insoluble salts - ionic precipitation

• • Insoluble salts, e.g. BaSO₄, CaSO₄, PbSO₄, PbCl₂, AgCl and most carbonates

  • involves mixing a solution that contains its positive ions with another solution that contains its negative ions

E.g. Preparation of BaSO₄

• • First BaCl, since it contains wanted barium ion, is reacted with H₂SO₄, since it contains wanted sulphate ion, to produce solid BaSO₄ & aqueous KCl.

  • BaSO₄ then separated from KCl by filtration, leaving filtrate KCl & BaSO₄ left on filter paper.

  • Salt is washed with water to completely remove KCl & filter paper is squeezed with another filter paper to dry BaSO₄.

Preparation of soluble salts

1. By Metal Hydroxide and Acid

• • This method is suitable for soluble metal hydroxides called alkalis. - titration

KOH (aq) + HCl (aq) --> KCl (aq) + H₂O (l)

• • 25.0 cm³ acid, as standard solution, is placed in conical flask using pipette.

  • Add a few drops of indicator & titrate with alkali from burette until indicator changes colour, showing all acid has just reacted.

  • Volume of alkali added is measured.

  • Prepare new 25.0 cm³ acid again with pipette & add same volume of alkali as before to prevent excess alkali/acid because both reactant & product are aqueous.

  • Next, the product is evaporated to dryness to obtain the salt.

2. By reacting metal with acid

• • This preparation is suitable for the more reactive metals like Mg, Zn, Fe, Al (but not Na, K and Ca).

  • In general, the metal is added to the acid until there is no further reaction. (when no more bubbles of hydrogen gas are produced)

  • Excess metal is then filtered out, and the clear filtrate is then evaporated until crystals form.

E.g. Reacting Zn with H₂SO₄ to prepare ZnSO₄

Zn (s) + H₂SO₄ (aq) --> ZnSO₄ (aq) + H₂O(l)

• • Zn is added to H₂SO₄ until in excess to ensure no more H₂SO₄ is present.

  • Then the mixture is filtered off to separate Zn from ZnSO₄.

  • The filtrate (ZnSO₄) is then placed in evaporating dish to evaporate most of water then it’s cooled after ZnSO₄ crystals are formed.

  • The crystals then filtered and squeezed between filter papers to dry.

3. By reacting metal oxide with acid

• • Nearly all the metal oxides react with acids, but most require warming,

  • This method is especially suitable for those metals which do not react with dilute acids.

  • eg, copper metal has no reaction with dilute acids but copper(II) oxide, if warmed with dilute acids, forms salts

E.g. Reacting CuO with Acids

CuO (s) + H₂SO₄ (aq) --> CuSO₄ (aq) + H₂O (l)

• • Add excess copper(II) oxide to the warm sulphuric acid so that all the acid is neutralised. The unreacted oxide is then removed by filtering.

  • The filtrate is a blue solution of copper(II) sulphate.

  • The crystals are obtained by concentrating the solution by evaporation, and then leaving it to cool.

  • The crystals formed can be removed by filtration.

  • As copper(II) sulphate crystals contain water of crystallisation, it is important not to evaporate the solution to dryness.

4. By Reacting Metal Carbonate with Acid

• • Similar to that involving metal oxide and acid, but this time, no heat is required.

  • The carbonate fizzes and gives of carbon dioxide gas.

  • Excess carbonate must be added to ensure that all the acid is neutralised

  • The solution is then filtered to remove unreacted carbonate and evaporate to concentrate the solution for crystallisation

E.g. Reacting CaCO₃ with Acids

K₂CO₃ (s) + H₂SO₄ (aq) --> K₂SO₄ (aq) + CO₂ (g) + H₂O (l)

• • The same process is used as reaction of acid with metal, just that carbon dioxide is produced.

  • Carbon dioxide can be tested by bubbling it into limewater which will turn limewater colourless to milky.

Properties and uses of Ammonia

• • produced from nitrogen reacted with hydrogen

  • For producing: fertilisers, nitric acid, nylon, dyes, cleaners and dry cell

• The Manufacture of Ammonia: The Haber Process

  • • The Process: Nitrogen and hydrogen are mixed together in ratio 1:3, where nitrogen is obtained from air and hydrogen is obtained from natural gas, and passed over iron catalyst.

    • Since the reaction is reversible so H₂ and O₂, reproduced from decomposition of produced NH₃, are passed over the catalyst again to produce ammonia.

Conditions for Manufacturing Ammonia

to have high yield of ammonia we should have:

• • Higher pressure

  • Lower temperature

But in practice, we use lower pressure of 200 atm and higher temperature of 450^oC. This is because:

• • Using low temperature is too slow to reach equilibrium

  • Using high pressure involves safety risk and higher cost

Ammonia as fertilizers

Plants need nitrogen as one of component for growth and ammonium fertilizers contain nitrogen for that.

% content of nitrogen in ammonium fertilizers

E.g. Ammonium sulphate, (NH4)2SO4, and urea, (NH2)2CO, are 2 kinds of fertilizers. Deduce, in terms of nitrogen content, which of these fertilizers best for plants.

% mass = (total mass of element / total mass of compound) x 100

(NH₄)₂SO₄ = (28 / 132) x 100 = 21.2% of N

(NH₂)₂CO = (28 / 60) x 100 = 46.7% of N

Therefore, (NH₂)₂CO is a better fertilizer since it contains more nitrogen.

Problems with Ammonia

• • Eutrophication is the increase in organic content of water when fertilizers leach into soil and washed into rivers and streams.

  • When excess fertilizers washed away by rain, nitrate ions in it gets into rivers and helps aquatic plants like algae to grow swiftly.

  • When too much algae, water turns murky and sunlight would not penetrate into water to help their growth which in turn lead to deaths of algae.

  • Decay of this organic matter uses up oxygen, hence killing aquatic animals.

  • Then even more algae dies and even more animals die

  • Water pollution results from runoff of fertilizer use, leaching from septic tanks, sewage and erosion of natural deposits.

  • ions from nitrogen in soil leaches down the soil into groundwater due to its solubility.

  • Since groundwater is our drink source, when humans drink this water, they will get seriously ill and babies may suffer breathlessness to death

MCQ Questions

1. Which of the following gases cannot be removed from the exhaust gases of a petrol powered car by its catalytic converter?

a. carbon dioxide

b. carbon monoxide

c. hydrocarbons

d. nitrogen dioxide

2. An excess of dilute sulphuric acid reacts with both aqueous barium hydroxide and aqueous barium chloride. In what way are the two reactions the same?

a. a gas is produced

b. an insoluble salt is produced

c. the final pH is 7

d. water is produced

3. In separate experiments, an excess of aqueous sodium hydroxide or aqueous ammonia was gradually added to a solution X. In both experiments, a precipitate was obtained which dissolved in an excess of the added reagent. What could X contain?

a. copper(II) nitrate

b. iron(II) nitrate

c. iron(III) nitrate

d. zinc nitrate

4. Which of the following is a property of ethanoic acid but is not a property of sulphuric acid?

a. it reacts with copper(II) oxide

b. it reacts with sodium carbonate

c. it reacts with magnesium

d. it burns in air

5. Which salt could be prepared by a method involving crystallization as the final stage?

a. barium sulphate

b. calcium carbonate

c. silver chloride

d. sodium nitrate

6. An element X forms a hydroxide which dissolves both in acids and in alkalis. What could X be?

a. aluminium

b. calcium

c. copper

d. iron

7. Under suitable conditions, hydrochloric acid reacts with each of the following substances. Which reaction gives a colorless solution only?

a. calcium carbonate

b. iron(II) hydroxide

c. potassium hydroxide

d. silver nitrate

8. After acidification with dilute nitric acid, a colorless solution of X reacts with aqueous silver nitrate to give a white precipitate. What could X be?

a. calcium iodide

b. copper(II) chloride

c. lead(II) chloride

d. sodium chloride

9. Two tests were carried out on a colorless liquid X. X turned anhydrous copper(II) sulphate from white to blue. X reacted with calcium, giving hydrogen. What could X be?

1. dilute hydrochloric acid

2. ethanol

3. water

a. 1 only

b. 1 and 2 only

c. 1 and 3 only

d. 1, 2 and 3

10. Which of the following is a characteristic property of alkalis in aqueous solution?

a. they liberate ammonia from ammonium salts

b. they liberate carbon dioxide from carbonates

c. they give hydrogen with any metal

d. they turn universal indicator red

11. A mineral X dissolves in dilute hydrochloric acid, giving off a gas which turns limewater milky. When aqueous ammonia is added to the colorless solution, a white precipitate is formed. The precipitate dissolves in an excess of aqueous ammonia to give a colorless solution. What is X?

a. calcium carbonate

b. copper(II) carbonate

c. zinc carbonate

d. zinc sulphide

12. Aqueous sodium hydroxide reacts with a metal ion producing a colored precipitate. This precipitate changes color on standing. What is the ion present?

a. Al³⁺

b. Cu²⁺

c. Fe²⁺

d. Zn²⁺

13. A bee sting is acidic. Which household substance will neutralize a bee sting?

a. damp bicarbonate of soda pH 8

b damp common salt pH 7

c. lemon juice pH 5

D. vinegar pH 4

14. An excess of sodium hydroxide is added to an aqueous solution of salt X and boiled. Ammonia gas is only given off after aluminium foil is added to the hot solution.

What could be X?

a. ammonium chloride

b. ammonium nitrate

c. sodium chloride

d. sodium nitrate

15. Which ion reacts with aqueous ammonia to give a precipitate that dissolves in an excess of ammonia?

a. Al³⁺

b. Fe²⁺

c. Fe³⁺

d. Zn²⁺

16. A solid element conducts electricity. The element burns in air to form a white solid. This white solid dissolves in water to give an alkaline solution. What is the element?

a. aluminium

b. calcium

c. carbon

d. copper

17. Which calcium compound does not increase the pH of acidic soils?

a. calcium carbonate

b.calcium hydroxide

c. calcium oxide

d. calcium sulphate

18. What is the concentration of hydrogen ions in 0.05 mol/dm³ sulphuric acid?

a. 0.025 g/dm³

b. 0.05 g/dm³

c. 0.10 g/dm³

d. 2.0 g/dm³

19. A solution X forms a white precipitate with dilute sulphuric acid and also with aqueous silver nitrate. What could solution X contain?

a. barium chloride

b. barium nitrate

c. magnesium chloride

d. magnesium sulphate

20. Which of the following is a reaction of dilute hydrochloric acid?

a. ammonium chloride reacts to give ammonia

b. calcium carbonate reacts to give carbon dioxide

c. copper reacts to give hydrogen

d. universal indicator paper turns blue

21. Which compound in solution produces a precipitate with aqueous ammonia that does not dissolve when an excess of ammonia is added?

a. copper(II) sulphate

b. iron(II) chloride

c. potassium hydroxide

d. zinc chloride

22. A white compound produces a mixture of gases when heated. This mixture turns moist Universal Indicator paper red and relights a glowing splint. What does this mixture contain?

a. an acidic gas and hydrogen

b. an acidic gas and oxygen

c. an alkaline gas and hydrogen

d. an alkaline gas and oxygen

23. Which of the following describes a step in the preparation of insoluble barium sulphate from aqueous barium chloride and dilute sulphuric acid?

a. add dilute sulphuric acid until no more gas is produced

b. add the indicator methyl orange

c. collect the precipitate of barium sulphate by filtration

d. evaporate the filtrate until it crystallises

24. Solid R is gradually added to aqueous solution S. The changes in pH are shown in the graph