Principles of Physical Chemistry PDF

## PRINCIPLES OF PHYSICAL CHEMISTRY ### Chapter 1 #### Chemistry the central science - Chemistry is at the heart of many changes we see in the world around us. - It accounts for the myriad different properties we see in matter. - To understand how these changes and properties arise, we need to l...

## PRINCIPLES OF PHYSICAL CHEMISTRY ### Chapter 1 #### Chemistry the central science - Chemistry is at the heart of many changes we see in the world around us. - It accounts for the myriad different properties we see in matter. - To understand how these changes and properties arise, we need to look beneath the surfaces of our everyday observations. #### Why study Chemistry? - Chemistry lies near the heart of many matters of public concern. - improvement of health care - conservation of natural resources - protection of the environment - supply of energy needed to keep society running. - Using chemistry, we have discovered and continually improved upon: - pharmaceuticals - fertilizers and pesticides - plastics - solar panels - light-emitting diodes - building materials. - We have also discovered that some chemicals are harmful to our health or the environment. - It is in your best interest to understand the effects, both positive and negative, that chemicals can have, in order to arrive at a balanced outlook regarding their uses. #### What Do Chemists Do? - Some chemists work in a lab, in a research environment. - asking questions - testing hypotheses with experiments. - Other chemists: - work on a computer developing theories or models or predicting reactions. - do field work. - contribute advice on chemistry for projects. - write. - teach. - The career options are extensive. #### Who Are Chemists, and What Do They Do? - People who have degrees in chemistry hold a variety of positions in: - industry - government - academia - Those in industry work as: - laboratory chemists - developing new products (research and development) - analyzing materials (quality control) - assisting customers in using products (sales and service) - managers or company directors - Chemists are important members of the scientific workforce in government (the National Institutes of Health, Department of Energy, and Environmental Protection Agency) and at universities. - A chemistry degree is also good preparation for careers in: - teaching - medicine - biomedical research - information science - environmental work - technical sales - government regulatory agencies - patent law. #### Fundamentally, chemists do three things: 1. make new types of matter: materials, substances, or combinations of substances with desired properties 2. measure the properties of matter 3. develop models that explain and/or predict the properties of matter. #### 1.2 Classification of Matter - The matter can be classified in two different ways: - according to its state - according to its composition #### a. Classification According to its States - Matter exists, at room temperature, in three different states: - solid - liquid - gaseous states. - Adsorption or evolution of heat results in the interconversion of these states. - There are two other states of matter: - plasma state - Bose-Einstein condensate. - Plasma state occurs at a very high temperature of 105 K. - Bose-Einstein condensate occurs at a very low temperature of less than 10-7 K ##### The five states of matter: | State | Description | | ---------------- | ------------------------------------------------- | | Bose-Einstein | (only for low density ionized gases) | | Condensate | | | Solids | | | Liquids | | | Gases | | | Plasmas | | ##### Solids - Solids have a definite shape and a definite volume. - Particles are held close together in an orderly fashion with little freedom of motion. - As a result, a solid does not conform to the shape of its container. - Most everyday objects are solids: rocks, chairs, ice, and anything with a specific shape and size. - The molecules in a solid are close together and connected by intermolecular bonds. - Solids can be amorphous. - Solids can be arranged into crystalline structures or networks. - For instance, soot, graphite, and diamond are all made of elemental carbon, and they are all solids. - What makes them so different? - Soot is amorphous, so the atoms are randomly stuck together. - Graphite forms parallel layers that can slip past each other. - Diamond, however, forms a crystal structure that makes it very strong ##### Liquids - Liquids have a definite volume, but they do not have a definite shape. - Instead, they take the shape of their container to the extent they are indeed "contained". - Something such as a beaker or a cupped hand or even a puddle determines the shape. - If not "contained" by a formal or informal vessel, the shape is determined by other internal (e.g. intermolecular) and external (e.g. gravity, wind, inertial) forces. - The molecules are close, but not as close as a solid. - The intermolecular bonds are weak, so the molecules are free to slip past each other, flowing smoothly. - A property of liquids is viscosity, the measure of thickness when flowing. - For example, water is not nearly as viscous as molasses. ##### Gases - Gas particles have a great deal of space between them and have high kinetic energy. - If unconfined, the particles of a gas will spread out indefinitely; if confined, the gas will expand to fill its container. - When a gas is put under pressure by reducing the volume of the container, the space between particles is reduced, and the pressure exerted by their collisions increases. - If the volume of the container is held constant, but the temperature of the gas increases, then the pressure will also increase. - Gas particles have enough kinetic energy to overcome intermolecular forces that hold solids and liquids together, thus a gas has no definite volume and no definite shape. ##### Plasma - Plasma is not a common state of matter here on Earth, but may be the most common state of matter in the universe. - Plasma consists of highly charged particles with extremely high kinetic energy. - The noble gases (helium, neon, argon, krypton, xenon and radon) are often used to make glowing signs by using electricity to ionize them to the plasma state. - Stars are essentially superheated balls of plasma. ##### Bose-Einstein Condensates - In 1995, technology enabled scientists to create a new state of matter, the Bose-Einstein condensate (BEC). - Using a combination of lasers and magnets, Eric Cornell and Carl Weiman cooled a sample of rubidium to within a few degrees of absolute zero. - At this extremely low temperature, molecular motion comes very close to stopping altogether. - Since there is almost no kinetic energy being transferred from one atom to another, the atoms begin to clump together. - There are no longer thousands of separate atoms, just one "super atom." - A BEC exists when matter is frozen to extremely low temperatures that are a tiny fraction of a degree above absolute zero. - In this state, the atoms overlap into each other to form a wave. - The BEC is a matter wave. - If the wave was compressed, it would form a singularity. - If enough mass was condensed into the singularity it could turn into a black hole. The occurrence of a black hole while making BEC would not need to be too much of a concern anyway because it would require a tremendous amount of energy to compress mass into the critical point. - BEC is used to: - study quantum mechanics on a macroscopic level. - allow study of the particle/wave paradox. - simulate conditions that might apply in black holes. #### b. Classification According to its Composition - Matter is classified into two broad categories: - pure substances - mixtures. - Mixtures can be separated into pure substances by physical methods. - Pure substances are further divided into categories as elements and compounds. - Mixtures are also classified into types: - homogeneous mixtures - heterogeneous mixtures. ##### Figure 1.3. Classification of matter according to its composition ``` Matter / \ / \ NO YES Is it uniform throughout? / \ / \ Heterogeneous mixture Homogeneous / \ / \ NO YES Does it have a variable composition? / \ / \ Element Compound | | | Homogeneous mixture (solution) ``` #### Compounds - In contrast to elements, compounds are composed of different types of atoms. - More precisely, a compound is a chemical substance that consists of two or more elements. - A pure chemical compound is a chemical substance that is composed of a particular set of molecules or ions that are chemically bonded. - The smallest representative for a compound (which means it retains characteristics of the compound) is called a molecule. - In other words, a molecule is the smallest particle that has any of the properties of a compound. - A molecule is formed when two or more atoms join together chemically. - A compound is a molecule that contains at least two different elements. - All compounds are molecules but not all molecules are compounds. - For example, molecular hydrogen (H2), molecular oxygen (O2) and molecular nitrogen (N2) are not compounds because each is composed of a single element. - Water (H₂O), carbon dioxide (CO2) and methane (CH4) are compounds because each is made from more than one element. - The smallest bit of each of these substances would be referred to as a molecule. - For example, a single molecule of molecular hydrogen is made from two atoms of hydrogen while a single molecule of water is made from two atoms of hydrogen and one atom of oxygen. - When the compound is formed, altogether a new substance is formed and the properties of which are quite different from its reacting elements. - Compounds can be separated into elements using chemical methods but not the physical methods. - A chemical compound can be either atoms bonded together in molecules or crystals in which atoms, molecules or ions form a crystalline lattice. - Compounds made primarily of carbon and hydrogen atoms are called organic compounds, and all others are called inorganic compounds. - Compounds containing bonds between carbon and a metal are called organometallic compounds. - Compounds have properties different from the elements that created them. - For example, Water is composed of hydrogen and oxygen. Hydrogen is an explosive gas and oxygen is a gas that fuels fire. - Water has completely different properties, being a liquid that is used to extinguish fires. #### Figure 1.4. (a) atoms of an element, (b) molecules of an element, (c) molecules of a compound, consisting of more than one element, and (d) A mixture of atoms of an element and molecules of an element and a compound. ``` a O O -O -O- O O b -O- O- O- O- -O- O - O- O- C -O- H- H - O - D -O-O- O-O- -O- H - H- O- ``` #### Mixtures - A mixture is a combination of two or more substances in which each substance retains its chemical identity. - Mixtures can be solids, liquids, or gases. - Some familiar examples are mixed nuts, 14-carat gold, apple juice, milk, and air. - Mixtures do not have a universal constant composition. - For example, samples of air collected in different locations will differ in composition because of differences in altitude, pollution, and other factors. - Various brands of apple juice may differ in composition because of the use of different varieties of apples, or there may be differences in processing and packaging, and so on. - Mixtures can broadly be classified as: - heterogeneous - homogeneous mixtures. - They take the form of: - alloys - solutions - suspensions - colloids. - All mixtures have two parts: - the "dispersing medium" - the "dispersed phase" - Generally speaking, the dispersed phase is in the smaller amount and is spread throughout the dispersing medium. - In most cases, the dispersed phase is quite small in amount compared to the amount of the dispersing medium. ##### Heterogeneous Mixtures > A heterogeneous mixture is a mixture of two or more chemical substances (elements or compounds), where the different components can be visually distinguished and easily separated by physical means. - Examples include: - Mixtures of sand and water - Mixtures of sand and iron filings - A conglomerate rock - Water and oil - A salad - Trail mix - Mixtures of gold powder and silver powder ##### Homogenous Mixtures > A homogeneous mixture is a mixture of two or more chemical substances (elements or compounds), where the different components cannot be visually distinguished. The composition of homogeneous mixtures is constant. - Often separating the components of a homogeneous mixture is more challenging than separating the components of a heterogeneous mixture. - Mixtures, whether homogeneous or heterogeneous, can be separated by physical means into pure components without changing the identities of the components. - For example, sugar can be recovered from a water solution by evaporating the solution to dryness. - Condensing the vapor will give us back the water component. - To separate the Sulphur-iron mixture, we can use a magnet to remove the iron filings from the Sulphur, because sand is not attracted to the magnet. #### 1.3 Properties of Matter - The properties of matter can be categorized as: - physical - chemical ##### A physical property of a substance is one that can be observed and measured without changing the identity and composition of the substance. - These properties include: - color - odor - density - melting point - boiling point - hardness - For example, we can determine the melting point of ice by heating a block of ice and measuring the temperature at which the ice is converted to water. - Liquid water differs from ice in appearance but not in composition; both liquid water and ice are H2O. - Melting is a physical change. - One in which the state of matter changes, but the identity of the matter does not change. - We can recover the original ice by cooling the water until it freezes. - Therefore, the melting point of a substance is a physical property. - Similarly, when we say that nitrogen dioxide gas is brown, we are referring to the physical property of color. - Some properties, such as temperature and melting point, are intensive properties. ##### Intensive properties do not depend on the amount of sample being examined and are particularly useful in chemistry because many intensive properties can be used to identify substances. ##### Extensive properties depend on the amount of sample. - Two examples are: - mass - volume. ##### Chemical properties describe the way a substance may change, or react, to form other substances. - A common chemical property is flammability, the ability of a substance to burn in the presence of oxygen. #### Physical and Chemical Changes - The changes substances undergo are either: - physical - chemical ##### During a physical change, a substance changes its physical appearance but not its composition. - It is the same substance before and after the change. - Adding or removing energy from matter causes a physical change as matter moves from one state to another. - For example, adding thermal energy (heat) to liquid water causes it to become steam or vapor (a gas). - Removing energy from liquid water causes it to become ice (a solid). ##### Physical changes can also be caused by: - motion - pressure - The evaporation of water is a physical change. - When water evaporates, it changes from the liquid state to the gas state, but it is still composed of water molecules. - All changes of state (for example, from liquid to gas or from liquid to solid) are physical changes. ##### Figure 1.5. (a) The three physical states of water-water vapor, liquid water, and ice. The red arrows show that the three states of matter interconvert. (b) All changes of state. ``` a Ice Water vapor Liquid water <Recombination Ionization > Plasma Gas Vaporization Condensation Liquid < Deposition Sublimation > Solid <Freezing Melting > Enthalpy of system ``` ##### Melting and freezing - When heat is applied to a solid, its particles begin to vibrate faster and move farther apart. - When the substance reaches a certain combination of temperature and pressure, its melting point, the solid will begin to melt and turn into a liquid. - When two states of matter, such as solid and liquid, are at the equilibrium temperature and pressure, additional heat added into the system will not cause the overall temperature of the substance to increase until the entire sample reaches the same physical state. - For example, when you put ice into a glass of water and leave it out at room temperature, the ice and water will eventually come to the same temperature. - As the ice melts from heat coming from the water, it will remain at zero degrees Celsius until the entire ice cube melts before continuing to warm. - When heat is removed from a liquid, its particles slow down and begin to settle in one location within the substance. - When the substance reaches a cool enough temperature at a certain pressure, the freezing point, the liquid becomes a solid. - Most liquids contract as they freeze. - Water, however, expands when it freezes into ice, causing the molecules to push farther apart and decrease the density, which is why ice floats on top of water. - Adding additional substances, such as salt in water, can alter both the melting and freezing points. - For example, adding salt to snow will decrease the temperature that water freezes on roads, making it safer for drivers. - There is also a point, known as the triple point, where solids, liquids and gases all exist simultaneously. - Water, for example, exists in all three states at a temperature of 273.16 Kelvin and a pressure of 611.2 pascals. ##### Sublimation - When a solid is converted directly into a gas without going through a liquid phase, the process is known as sublimation. - This may occur either when the temperature of the sample is rapidly increased beyond the boiling point (flash vaporization) or when a substance is "freeze-dried" by cooling it under vacuum conditions so that the water in the substance undergoes sublimation and is removed from the sample. - A few volatile substances will undergo sublimation at room temperature and pressure, such as frozen carbon dioxide, or dry ice. ##### Vaporization - Vaporization is the conversion of a liquid to a gas and can occur through either evaporation or boiling. - Because the particles of a liquid are in constant motion, they frequently collide with each other. Each collision also causes energy to be transferred, and when enough energy is transferred to particles near the surface they may be knocked completely away from the sample as free gas particles. - Liquids cool as they evaporate because the energy transferred to surface molecules, which causes their escape, gets carried away with them. - Liquid boils when enough heat is added to a liquid to cause vapor bubbles to form below the surface. - This boiling point is the temperature and pressure at which a liquid becomes a gas. ##### Condensation and Deposition - Condensation occurs when a gas loses energy and comes together to form a liquid. - For example, water vapor condenses into liquid water. - Deposition occurs when a gas transforms directly into a solid, without going through the liquid phase. - Water vapor becomes ice or frost when the air touching a solid, such as a blade of grass, is cooler than the rest of the air ##### Chemical change - In a chemical change (also called a chemical reaction), a substance is transformed into a chemically different substance. - For example, when hydrogen burns in air, it undergoes a chemical change because it combines with oxygen to form water. - The statement "hydrogen gas burns in oxygen gas to form water" describes a chemical property of hydrogen, because to observe this property we must carry out a chemical change-burning in oxygen (combustion). - After a chemical change, the original substance (hydrogen gas in this case) will no longer exist. What remains is a different substance (water, in this case). - We cannot recover the hydrogen gas from the water by means of a physical process, such as boiling or freezing. - Every time we bake cookies, we bring about a chemical change. - When heated, the sodium bicarbonate (baking soda) in cookie dough undergoes a chemical change that produce carbon dioxide gas. ##### Figure 1.6. A chemical change (chemical reaction). ``` H2 O2 H-H O=O Burn H2O / | \ H - O - H ``` #### 1.4 Units of Measurement - Many properties of matter are quantitative, that is, associated with numbers. - When a number represents a measured quantity, the units of that quantity must be specified. - There are two common systems of measurement: - Metric System - SI System ##### (1) Metric System - The metric system, developed in France during the late eighteenth century, is used the system of measurement in most countries. - It is a decimal system of weights and measures originally based on the meter as the unit of length and the kilogram as the unit of mass. ##### (2) SI System - In 1960, an international agreement was reached specifying a particular choice of metric units for use in scientific measurements. - These preferred units are called SI units, after the French Système International d'Unités. - This system has seven base units from which all other units are derived. - With SI units, prefixes are used to indicate decimal fractions or multiples of various units. - For example, the prefix milli-represents a 10-³ fraction, one-thousandth, of a unit: A milligram (mg) is 10-3 gram (g), a millimeter (mm) is 10-3 meter (m), and so forth. ##### Table 1.1. Base SI units | Physical Quantity | Name of unit | Abbreviation | | --------------------- | ------------------- | ------------- | | Length | Meter | m | | Mass | Kilogram | kg | | Temperature | Kelvin | K | | Time | Second | S | | Amount of substance | Mole | mol | | Electric current | Ampere | A | ##### SI units of length - The SI unit of length is the meter (m). - Fractions and multiples of SI units are named by adding appropriate prefixes. - The commonly used metric length units are listed in Table 1.2. ##### Table 1.2. Common metric length units. | Unit | Symbol | Relation | | ----------- | ------ | -------------- | | meter | M | | | Kilometer | Km | 1 km = 10³ m | | decimeter | Dm | 1 dm = 10-¹ m | | centimeter | Cm | 1 cm = 10-2 m | | milimeter | Mm | 1 mm = 10-3 m | | micrometer | µm | 1 µm = 10-6 m | | nanometer | Nm | 1 nm = 10-9 m | | picometer | Pm | 1 pm = 10-12m | | angstrom | A | 1 Å= 10-8 cm =10-10m | - It may be noted that the metric symbols are not changed into plurals. - For example, 5 centimeters of length is written as: - Correct: 5 cm - Incorrect: 5 cm. 5 c.m. 5 cms ##### SI units of volume - The derived SI unit of volume is Cubic meter m³. - The related units of volume which are also used are: - Cubic centimeter cm³ - Cubic decimeter dm³ ##### Figure 1.7. Relationship between length and volume. - A liter is the volume occupied by a cube 10 cm on edge. - That is, 1 L = (10 cm)³ = 1000 cm³ - Also 1 L = 1000 mL - Therefore, 1000 mL = 1000 cm³ - 1 mL = 1 cm³ (cc) - Hence the volume units millilitre (mL) and cubic centimeter (cc) can be used interchangeably. - It may again be stated that metric symbols are not changed into plurals. - Thus, Correct: mL (or ml) Incorrect: mLs (mls), m.l., ml. ##### SI unit of temperature - The series of markings on a thermometer which read temperature is called a temperature scale. - A temperature scale in which 0º is assigned to the freezing point of pure water and 100° to the boiling-point is known as the Celsius scale. - The temperatures are expressed in degrees Celsius (°C). - Room temperature on the celsius scale is taken to be 25° C. - The celsius scale is not a part of the SI system. - Since it is widely used in scientific literature, it is difficult to abandon it. - The SI system uses the Kelvin scale. - A degree on the Kelvin scale has the same magnitude as a degree on the Celsius scale, but zero on the Kelvin scale equals 273.15°C. - The temperature (0 K) is often referred to as the absolute zero. - Celsius and Kelvin temperature are related as: - K = °C + 273.15, °C = K - 273.15 - It may be noted that the unit for temperature on the Kelvin scale is K and not °K. - This notation has been approved by IUPAC and is now used by chemists all over the world. - Thus it may be noted that a degree sign (°) is not used with the Kelvin scale. - On the Fahrenheit scale pure water freezes at 32º and boils at 212º. - Thus 100° Celsius equals 212 - 32 = 180 Fahrenheit degrees. - Celsius and Fahrenheit temperatures are related by the following equations: - ⁵/₉ (°F-32) - ⁹/₅ (°C + 32) - Using these relations, it is easy to convert a temperature reading from Fahrenheit to Celsius and vice versa. ##### Figure 1.8. A comparison of Kelvin, Celsius, and Fahrenheit scales. | Scale | Boiling point of water | Normal body temperature | Average room temperature | Freezing point of water | |--------------|-----------------------|--------------------------|---------------------------|------------------------| | Kelvin | 373 | 310 | 293 | 273 | | Celsius | 100° | 37° | 20° | 0° | | Fahrenehit | 212° | 98.6° | 68° | 32° | ##### Units of pressure - Pressure is defined as the force per unit area exerted on a surface. That is, P = F/A - Thus we can determine the SI unit for pressure as: - Force F: kg ms¯² or N - Area A: m² - Pressure F/A: kg m¯¹s¯² or Nm¯² - The SI unit Nm¯² is named pascal and given the symbol Pa. - The unit of pressure 'Pascal" is not in common use. - Three other units which have been traditionally used are: - atmosphere, symbol atm - torr, symbol Torr - millimeter of mercury or mm Hg - Atmosphere, symbol atm, is defined as the pressure exerted by a column of mercury 760 mm in height at 0°C. - Torr, symbol Torr, is defined as the pressure exerted by a 1 mm column of mercury at 0°C. - Millimeter of mercury or mm Hg, which is the height in millimeters of mercury that the pressure can support. - The relation between atmosphere, torr and pascal is: - 1 atm = 76 cm Hg = 760 mm Hg = 760 torr = 1.01325 × 10⁶ Pa = 101.325 kPa = 1.01325 bar ##### Units of density - One of the physical properties of a solid, a liquid, or a gas is its density (d). - Density is defined as mass per unit volume. - This may be m/V. - By using the base SI units and remembering that the unit for volume is m3, we can derive the SI unit for density. - kg/m³ or kg m¯³. - The other units of density commonly used are: - g cm¯³ or g ml¯¹ for liquid or solid densities - g L¯¹ or g dm¯³ for gas densities - The term specific gravity is the ratio of the density of a substance to the density of a reference substance. - The reference substance for solids and liquids is usually water. - Specific gravity = density of a substance / density of reference substance - Specific gravity, being the ratio of two densities has no units. ### Chapter 2 #### Gases #### 2.1. Introduction - In Chapter One, we have discussed matter and its different states. - The three common states of mater are: - solid - liquid - gas. - In fact, most substances that are liquids or solids under ordinary conditions can also exist in the gaseous state, where they are often referred to as **vapors.** - The substance H2O, for example, can exist as: - liquid water - solid ice - water vapor. - This chapter focuses on gases and various laws. - Despite the randomness of gases, the study of the gaseous state is very simple. - This is because all gases, regardless of their nature, behave identically under the same experimental conditions. - This generality in behavior is usually stated in the form of certain laws collectively called the gas laws. ##### Figure 2.1. Molecular representation of the gaseous, liquid and solid states of water. ``` Gas (vapours) Liquid (water) Solid (ice) ``` #### General characteristics of gases 1. Expansibility - Gases have limitless expansibility. - They expand to fill the entire vessel they are placed in. 2. Compressibility - Gases are easily compressed by application of pressure to a movable piston fitted in the container. 3. Diffusibility - Gases can diffuse rapidly through each other to form a homogeneous mixture. 4. Pressure - Gases exert pressure on the walls of the container in all directions. 5. Effect of Heat - When a gas, confined in a vessel is heated, its pressure increases. - Upon heating in a vessel fitted with a piston, volume of the gas increases. - The above properties of gases can be easily explained by the Kinetic Molecular Theory which we will consider later in the chapter. #### Factors affecting the behavior of gases - A gas sample can be described in terms of four parameters (measurable properties): - The volume, V of the gas. - Its pressure, P. - Its temperature, T. - The number of moles, n, of gas in the container. #### The Volume, V - The volume of the container is the volume of the gas sample. - It is usually given in: - litre (1 or L) - millilitres (ml or mL) - The SI unit for volume is cubic metre (m³) and the smaller unit is decimeter³ (dm³). #### The pressure, p - The pressure of a gas is defined as the force exerted by the impacts of its molecules per unit surface area in contact. - The pressure of a gas sample can be measured with the help of a mercury manometer. - Similarly, the atmospheric pressure can be determined with a mercury barometer. ##### Figure 2.2. (a) A mercury manometer, and (b) A mercury barometer. - The pressure of air that can support 760 mm Hg column at sea level, is called one atmosphere (1 atm). ``` a Vacuum GAS hmm Pressure = h mm Hg b Vacuum 760 mm Hg 1 Atm pressure ``` #### Temperature, T - The temperature of a gas may be measured in: - Centigrade degrees (°C) - Celsius degrees. - The SI unit of temperature is Kelvin (K) or Absolute degree. - The centigrade degrees can be converted to kelvins by using the equation: - K = °C + 273 - The Kelvin temperature (or absolute temperature) is always used in calculations of other parameters of gases. - Remember that the degree sign (°) is not used with K. #### The Moles of a Gas Sample, n - The number of moles, n, of a sample of a gas in a container can be found by dividing the mass, m, of the sample by the molar mass, M (molecular mass). - The equation is: - moles of gas (n) = mass of gas sample (m) / molecular mass of gas (M) #### 2.2. The gas laws - The volume of a given sample of gas depends on the temperature and pressure applied to it. - Any change in temperature or pressure will affect the volume of the gas. - As results of experimental studies from 17th to 19th century, scientists derived the relationships among the pressure, temperature and volume of a given mass of gas. - These relationships, which describe the general behaviour of gases, are called the **gas laws** which have played a major role in the development of many ideas in chemistry. #### The Pressure-Volume Equation: Boyle's Law - Boyle investigated the pressure-volume relationship of a gas sample at constant temperature and composition. - As a result of his investigation, he concluded that the volume of a given amount of a gas decreases as the total applied pressure is increased. - “**At constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure.**” - If the pressure is doubled, the volume is halved. ##### Figure 2.3. Boyle's Law states that at constant temperature, If the pressure is doubled, the volume is halved. - **The mathematical expression which is consistent with this conclusion is:** ``` V ∝ 1/P (T, n are constant) V = kx 1/P PV = k ``` - If P1, V₁ are the initial pressure and volume of a given sample of gas and P2, V2 the changed pressure and volume, we can write: - P₁V₁ = k = P₂V₂ - P₁V₁ = P₂V₂ - This relationship is useful for the determination of the volume of a gas at any pressure, if its volume at any other pressure is known. ##### Figure 2.4. Graphical representation of Boyle's law. (a) a plot of V versus P for a gas sample is hyperbola; (b) a plot of V versus 1/P is a straight line. ``` a V (litre) P (mm Hg) b V (litre) 1/P (mm Hg) ``` - The Boyle's law can be demonstrated by adding liquid mercury to the open end of a J-tube. - As the pressure is increased by addition of mercury, the volume of the sample of trapped gas decreases. - Gas pressure and volume are inversely related; one increases when the other decreases ##### Figure 2.5. - Demonstration of Boyle's law: - An inflated balloon has a volume of 0.55 L at sea level (1.0 atm) and is allowed to rise to a height of 6.5 km, where the pressure is about 0.40 atm. - Assuming that the temperature remains constant, what is the