Properties of Bulk Matter Sections Covered in the Text: Chapter 16

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1 Properties of Bulk Matter Sections Covered in the Text: Chapter 16 In this note we survey certain concepts that comprise the macroscopic description of matter, that is to say, matter in bulk. These include temperature, pressure and changes in phase. We shall touch on the various states of matter, namely solids, liquids and gases, but we shall focus primarily on gases, the simplest macroscopic systems. We have already discussed certain macroscopic aspects of fluids in Notes 13 and 14. These ideas were organized by scientists who are seen today as the founders of modern chemistry. Matter in Bulk By matter in bulk we mean matter in a quantity to be seen, touched or weighed. At a given temperature and pressure a substance is either a solid, liquid or gas. Water, the best-known substance, is a solid (ice) at a pressure of 1 atm and temperature of 0 C and below. Between 0 C and 100 C water is a liquid, and above 100 C it is a gas (in the form of steam or water vapor). The solid, liquid and gaseous states of a substance are called phases. When a substance melts (or freezes) and boils (or condenses) it is said to undergo a phase change. We shall discuss phase changes in more detail later in this note. The three phases of matter can be described succinctly as follows. Solid A solid is a rigid macroscopic system with a definite shape and volume. It consists of particle-like atoms connected together by molecular bonds. Each atom vibrates about an equilibrium position, but an atom is not free to move around inside the solid. A solid is nearly incompressible, meaning that the atoms are about as close together as they can get. Liquid Like a solid, a liquid is nearly incompressible too. A liquid flows and deforms to fit the shape of its container. The molecules are held together by weak molecular bonds and are therefore free to move around. Gas Each molecule in a gas moves through space as a free, noninteracting particle until, on occasion, it collides with another molecule or with the wall of the container. A gas, like a liquid, is a fluid. It is also highly compressible, meaning that a lot of space exists between molecules. State Variables Matter in bulk is described by so-called state variables. These variables include volume V, pressure p, mass M, mass density ρ, thermal energy E th and temperature T. If any one state variable is changed then the state of the system as a whole is changed. All Matter The fundamental building block of all matter is the atom. An example is atomic hydrogen. However, some states of matter, many gases for example, are best described in terms of a grouping of atoms called a molecule. To form a molecule two or more atoms bind together via a combination of electrostatic and quantum mechanical processes that we shall not go into here. The complexity of molecules varies widely from the simplest, hydrogen gas, which consists of only two atoms of hydrogen, to the complex, like DNA, that typically consists of many thousands of atoms. Atoms and Molecules The idea that matter consists of indivisible entities called atoms dates back 2500 years. The Greek philosopher Democritus ( BC) speculated that repeated subdivision of matter would eventually yield a smallest unit which could not be further subdivided. The earliest quantitative ideas were formulated at the beginning of the 19 th century by Dalton (1800) and others. A Few Definitions From experiments too numerous to go into here we know today that an atom consists of a nucleus made up of protons and neutrons, and electrons moving in stable orbits about the nucleus. A proton has a positive charge +e, an electron a negative charge e and a neutron no charge. The best measurements of e yield the value x Coulombs (C) (to 4 significant digits). An atom is electrically neutral; the number of electrons surrounding the nucleus equals the number of protons in the nucleus. 1 The atomic number Z is the number of protons in the nucleus (or the number of electrons circling the nucleus). The atomic mass number A is the number of protons plus neutrons in the nucleus. The number A is 1 We shall be discussing electric charge in detail starting in Note

2 conventionally written as a leading superscript on the symbol for the atom, for example 1 H for hydrogen. Isotopes are atoms that differ only in the number of neutrons in the nucleus. Atomic hydrogen has 3 isotopes, hydrogen 1 H, deuterium 2 H and tritium 3 H. A selection of elements and their A numbers is listed in Table Table A selection of elements and their atomic mass numbers Element A 1 H Hydrogen 1 7 He Helium 4 12 C Carbon N Nitrogen O Oxygen 16 Atomic and Molecular Mass The mass of an atom or molecule is expressed most fundamentally in the absolute unit kg. The mass of an atom of atomic hydrogen is x kg (expressed to 4 significant digits). This is a very small number. It is convenient to express an atomic (and molecular) mass not in this absolute unit but in a relative unit. The relative unit is based on a scale in which the mass of the isotope of carbon, 12 C, is taken to be exactly 12 units atomic mass units (abbreviated u). That is, m( 12 C) = 12 u. The atomic mass of any other atom is its mass relative to 12 C. For example, careful experiments with hydrogen have established that the mass ratio m( 1 H)/m( 12 C) is /12. Thus the atomic mass of hydrogen is m( 1 H) = u. Some examples of atomic masses are listed in Table the atomic and molecular masses are nearly integers. The fourth column in Table 15-2 contains the atomic or molecular mass rounded to an integer; this number is sufficiently accurate to use in most calculations. NOTE: An element s atomic mass (e.g., for 1 H) is not the same as the atomic number (1 for 1 H). The atomic number, which is the element s position in the periodic table, is the number of protons in the element s nucleus. Mole and Molar Mass In chemistry where substances in everyday amounts are weighed on a balance and mixed it is convenient to use what is called a molar mass. (Note the use of the lower case mole, not Mole.) By definition, one mole of a substance (abbreviated 1 mol), whether solid, liquid or gas, is the amount of substance that contains as many basic particles as there are atoms in 12 g of 12 C. 3 A basic particle may be an atom or a molecule. For example, the basic particle of Helium, which is a monatomic gas, is the helium atom. The basic particle of oxygen gas, which is a diatomic molecule, is the oxygen molecule. The number of basic particles in a mole has been carefully measured. It has the value N A = mol 1. This number, denoted N A, is called Avogadro s number. N A has units; they are number of basic particles per mole. Thus the number of moles n in a substance containing N basic particles is n = N N A. [15-1] Table Examples of atomic and molecular masses. The approximate values can be used in calculations. Type Symbol At. mass (u) Approx (u) atomic 1 H molecular H atomic 12 C atomic N molecular N molecular NH Since N A 12 C atoms have a mass of 12 g, the mass of a single 12 C atom is m = 12(g) = g = kg. The conversion factor between atomic mass units and kilograms is A molecule s molecular mass is the sum of the atomic masses of the atoms forming the molecule. Notice that 2 The author of our textbook uses the lower-case letter m to denote the mass of an atom or molecule, and we shall do likewise The definition of the mole given here may differ from its definition in your chemistry text or in the physics text you used in high school. This is the definition given in our textbook, and to avoid confusion the definition we shall use.

3 1u = m(12 C) 12 = (kg) 12 = kg. Some examples of molar masses are listed in Table Thus the molar mass of a substance is the mass in grams of 1 mol of the substance. We shall denote the molar mass by M mol. The number of moles n in a system of mass M (g) consisting of atoms or molecules with molar mass M mol is given by: n = M(g) M mol. Table A selection of molar masses Substance molar mass (g) molecular hydrogen (H 2 ) atomic nitrogen (N) ammonia (NH 3 ) [15-2] It is important to note that M and M mol in eq[15-2] are expressed in grams not kilograms. Let us consider an example using moles. Example Problem 15-1 Calculating the Number of Moles in a Quantity of Oxygen How many moles are contained in 100 g of oxygen gas? Solution: The basic particle in oxygen gas is the oxygen molecule (O 2 ). The molecular mass of the O 2 molecule is m = 32 u. Converting this to kg, we get the mass of one molecule to be m = 32(u) (kg) 1u = kg. The number of molecules in 100 g = kg is N = M m = 0.100(kg) (kg) = The number of moles is therefore this number divided by Avogadro s number: n = N N A = = 3.13 mol Note 15 Alternatively, we can use eq[15-2] to find n more directly M(g) n = M mol (g.mol 1 ) = 100(g) = (g.mol 1 mol. ) The Periodic Table Electronic properties of elements are summarized in the periodic table of the elements. The table was invented by Mendeleev in He found that by arranging the elements in order of increasing atomic mass, regularly-recurring properties were revealed. By so arranging the 62 elements then known, he predicted that unknown elements would soon be discovered to fill the gaps. With the guidance of his predictions of their chemical and physical properties, three new elements were discovered in the next few years, providing striking confirmation of the general correctness of his ideas. However, a full understanding of the periodic table had to await the development of the quantum mechanics of many electron atoms. We shall leave this aspect of matter to a course in chemistry. Another topic concerning matter in bulk is thermodynamics. Thermodynamics is the study of the flow of heat. The laws of thermodynamics relate heat flow, work, and thermal energy. In order to study the flow of heat, it is necessary to introduce a new physical quantity called temperature. But before doing so we introduce two related concepts, thermal contact and thermal equilibrium. Two objects are said to be in thermal contact if heat can be exchanged between them without work being done. If two objects are placed in thermal contact with each other, and no net heat exchange occurs between them, then they are said to be in thermal equilibrium. 4 We are now in a position to discuss temperature in qualitative terms. 4 We don t yet know precisely what heat is. For the moment we shall have to be content with our intuitive sense of the concept. We shall be discussing heat in detail in Notes 16, 17 and

4 Temperature We live with the idea of temperature every day. We know that temperature has something to do with hotness or coldness. We know that we can measure the hotness or coldness with a device called a thermometer. The idea of temperature can be understood by considering a number of everyday situations. Consider two objects that are not in thermal equilibrium. If they are placed in thermal contact, then heat flows between them until they reach thermal equilibrium. If two objects A and B are not in thermal contact, then how can we tell whether they are in thermal equilibrium with each other? We can tell by doing the following: 1 Place a thermometer C in thermal contact with A, and wait for thermal equilibrium to occur (i.e. wait until the reading of the thermometer is steady). 2 Now place thermometer C in thermal contact with B, and wait for thermal equilibrium to occur. If the two readings of the thermometer are the same, then A and B are in thermal equilibrium with each other. These observations are expressed by the zeroth law of thermodynamics. Zeroth law of thermodynamics The zeroth law of thermodynamics can be stated in these words: If two objects A and B are separately in thermal equilibrium with object C, then they are in thermal equilibrium with each other. In order to establish a quantitative scale of temperature, it is necessary to adopt some particular kind of thermometer. Thermometers A thermometer measures the temperature of a system by displaying the change in some physical property, such as 15-4 a) length of a solid b) volume of a liquid c) pressure of a gas at constant volume d) volume of a gas at constant pressure e) electrical resistance of a conductor f) colour of a very hot object To understand the working of a thermometer based on a) or b) we need to know something about how a solid responds to a change in temperature. In other words, we need to know something about the thermal expansion/contraction of the substance. Thermal expansion Thermal expansion is a macroscopic property. It is a consequence of the temperature-induced change in the average separation between atoms. There are two kinds of thermal expansion: linear expansion and volume expansion. Linear expansion is commonly described by an expression of the form: ΔL L 0 = αδt, where α is the coefficient of linear expansion and L 0 is the length of the solid at STP. 5 Linear expansion only applies to solids. For solids α 10 5 T 1. Volume expansion is commonly described by an expression of the form ΔV V 0 = βδt where β is the coefficient of volume expansion and V 0 is the volume at STP. Volume expansion applies to solids, liquids and gases. For liquids β 10 4 T 1 while for gases β 10 4 T 1. Temperature Scales The most commonly-used scale of temperature in the world today is the Celsius scale. On this scale the freezing point of water at 1 atm pressure is set at 0, and the boiling point of water is set at 100. Between these two points are 100 Celsius degrees. A temperature of 10 degrees Celsius is written 10 C. The Fahrenheit scale (that actually predates the Celsius scale) is still in use in the U.S. On the Fahrenheit scale the freezing point of water is set at 32 and the boiling point of water is set at 212, making for 180 Fahrenheit degrees between the two points. Thus a temperature of T C C and T F F are related by T F = 9 5 T C [15-2] 5 STP stands for standard temperature and pressure, or 0 C and 1 atm. STP was introduced in Note 13.

5 Both scales have disadvantages, making it desireable to define an absolute scale of temperature. This we do in the next section. Absolute Zero and Absolute Temperature One of the most important scientific thermometers is the constant volume gas thermometer (Figure 15-1). The working of the thermometer is based on the fact that the absolute pressure of a gas in a sealed bulb container (whose volume cannot change) varies linearly with temperature. A pressure gauge is provided for measuring the pressure. The bulb is immersed in the system whose temperature is to be measured. Note 15 2 Whatever gas is used in the bulb the corresponding straight-line graph extrapolates to the same temperature at zero pressure, namely T 0 = 273 C. These facts form the basis of the absolute temperature scale. The temperature at zero pressure is taken to be the zero of the absolute temperature scale. (Zero pressure implies zero movement of gas molecules and therefore zero thermal energy and zero temperature.) On this scale the temperature of 273 C is taken to be zero zero Kelvin, denoted 0 K. The freezing point of water (0 C) is thus 273 K. This means that one Kelvin unit and one Celsius unit are of the same size. On this scale there is no negative temperature. The absolute temperature scale is the SI scale of temperature. The conversion between the Celsius scale and the Kelvin scale is T K = T C [15-3] Figure A constant volume gas thermometer (a) and typical results for three gases (b). Before using the thermometer it is first calibrated by recording the pressure at two reference temperatures, the boiling and freezing points of water. The points are plotted on a graph (Figure 15-1b) and a line drawn between them. The bulb is then brought into contact with the system whose temperature is to be measured. The pressure is measured then the corresponding temperature is read off the graph. With this thermometer you would observe two facts: 1 The gas pressure depends linearly on temperature Phase Changes It is common knowledge that water can exist in three phases, solid (ice), liquid (water) and gas (water vapor or steam). An idealized apparatus for the study of the phase changes of water is sketched in Figure 15-2a. A quantity of ice at an initial temperature of 20 C is placed in a sealed container equipped with a thermometer. The container is then heated (an open flame will do nicely). The heating is done slowly enough to ensure that the thermometer registers a temperature that applies to the whole container. The kind of graph of temperature versus time that would be obtained with this apparatus is shown in Figure 15-2b. The temperature rises from the initial 20 C to 0 C where it remains for some time. During this time the ice melts and the temperature of the icewater mixture remains at 0 C; the solid-liquid mixture is said to be in a phase equilibrium. Once all the ice has melted the liquid water begins to warm. Warming takes place uniformly until the temperature of 100 C is reached, at which temperature boiling occurs. During this time the water is transformed from the liquid to the gas phase. As the water boils the temperature remains at a constant 100 C; the liquid-gas mixture is now in a phase equilibrium. Once all the water has boiled off the system is completely gaseous. Subsequent warming just raises the temperature of the gas (steam). Unhappily, the results in Figure 15-2b apply only if the apparatus is at a pressure of 1 atm. Thus the graph only applies to a location at sea level. To avoid this disadvantage many experiments have been carried 15-5

6 out at various pressures yielding the so-called phase diagrams shown in Figures Figure 15-3a for water shows three regions corresponding to the solid, liquid and gas phases. The boundary lines separating the regions delineate the phase transitions K at the triple point of water. Figure An idealized apparatus for the study of phase changes in water (a) and the temperature-time graph obtained with the apparatus (b). A point of special interest on the diagram for water is the triple point where the phase boundaries meet. The triple point is the one value of temperature and pressure for which all three phases can coexist in phase equilibrium. For water, the triple point occurs at T 3 = 0.01 C and p 3 = atm. The triple point of water is actually used to complete the definition of the Kelvin temperature scale. Recall that the Celsius scale required the boiling and melting points of water as two reference points. These temperatures are not satisfactory because their values depend on pressure. The Kelvin scale requires only one reference point since the low end, zero temperature at zero pressure, is fixed. The triple point of water is taken as the second reference point because it is a fixed, identifiable point (same pressure and temperature) regardless of the location of study. The Kelvin temperature scale is therefore defined to be a linear temperature scale starting from 0 K at absolute zero and passing through the temperature Figure Phase diagrams (not to scale) for water (a) and carbon dioxide (b). Because the temperature at the triple point of water is T 3 = 0.01 C on the Celsius scale, absolute zero on the Celsius scale is T 0 = C. This completes the definition of the absolute temperature scale. A gas is the simplest macroscopic system. A real gas is known to consist of small, hard atoms or mole-cules moving randomly at high speeds which, on occasion, collide with each other and with the walls of the container. No matter how simple a real gas is, it is still useful to model it as a so-called ideal gas. The Idea of an Ideal Gas The potential energy curve of two atoms or molecules in an ideal gas can be represented as in Figure This curve represents the interaction of two hard spheres that have no interaction at all until they come into actual contact, at separation r contact, and then bounce. It is found experimentally that the ideal gas model is a good model for real gases if the following conditions are met: 15-6

7 constant. This means we can write eq[15-4] as pv = nrt. Note 15 [15-5] Eq[15-5] is called the ideal gas law. Remarkably, all gases yield the same graph and the same value of R. For a gas in a sealed container the number of moles n is constant. Thus we can write from eq[15-5]: Figure The idealized potential energy diagram for two atoms or molecules in an ideal gas. pv T = nr = const. [15-6] This means that for any two states i and f of the gas, 1 the density is low 2 the temperature is well above the condensation point The Ideal Gas Law Recall that the macroscopic state of a system is described by the state variables volume V, number of moles n, temperature T and pressure p. Numerous experiments have shown that for any gas, whether monatomic, diatomic or polyatomic, a plot of pv vs nt yields the kind of graph shown in Figure The pressure p must be in units of Pa and temperature T must be in units of Kelvin. The graph is a straight line, with pv and nt being related by pv = (const)nt. [15-4] p f V f T f = p i V i T i. [15-7] The ideal gas law written in terms of the number N of molecules in the gas is pv = nrt = N N A RT = N R N A T. [15-8] The factor R/N A is known as Boltzmann s constant and is denoted k B : k B = R N A = J.K 1. Thus the ideal gas law can be written in terms of k B : pv = Nk B T. [15-9] Eqs[15-5] and [15-9] are equivalent. Figure A graph of pv vs nt for an ideal gas. The slope of the graph, found experimentally, is const = 8.31 J.mol 1 K 1. The constant, denoted R, is called the universal gas Ideal Gas Processes The state of an ideal gas changes in a process in which p changes, V changes or T changes. These changes are most easily represented on a so-called pv diagram (Figures 15-6). Each state of the gas is represented as a point on the diagram (Figure 15-6a). A process in which a gas changes state is represented by a path or trajectory on the pv diagram (Figure 15-6b). Different processes take a gas from one state to another via different trajectories (Figure 15-6c). The ideal gas law applies only to gases in thermal equilibrium. Thus the kind of processes we shall consider here are those which keep the gas in thermal equilibrium as it is taking place. These are processes 15-7

8 that take place very slowly and are therefore called quasistatic processes. A quasistatic process is also a reversible process. We now consider three of the simplest processes in turn, a constant volume process, a constant pressure process, and a constant temperature process. process appears on a pv diagram as a vertical line. Figure Illustration of an isochoric process. Example Problem 15-2 The Constant Volume Gas Thermometer: An Isochoric Device 15-8 Figure A pv diagram of an ideal gas Constant Volume Process A constant volume process is called an isochoric process. In such a process V f = V i. [15-10] An idealized representation of an isochoric process involving a gas in a container with an unchangeable volume is shown in Figures If the gas is heated its pressure rises but its volume remains the same. The path between any two states of the gas is shown in Figure 15-7b. If the gas were cooled a similar process would occur in reverse. Thus an isochoric A constant volume gas thermometer is placed in contact with a reference cell containing water at the triple point. After reaching equilibrium, the gas pressure is recorded as kpa. The thermometer is then placed in contact with a sample of unknown temperature. After the thermometer reaches a new equilibrium, the gas pressure is kpa. What is the temperature of the sample? Solution: This is an example of an isochoric process. The temperature of the triple point of water is T 1 = 0.01 C = K. The ideal gas law gives p 2 V 2 T 2 = p 1 V 1 T 1, which, for V 1 = V 2 yields for the new temperature

9 p T 2 = T 2 1 = (273.16K) 65.12(kPa) p (kPa) = K = C. The gas pressure inside the cylinder is p = p atmos + mg A. Note 15 [15-12] Keep in mind here that pressures are expressed in Pa, temperatures in K. Constant Pressure Process A constant pressure process is called an isobaric process. In such a process p f = p i. [15-11] An idealized apparatus for producing an isobaric process is shown in Figure A cylinder of gas has a tight-fitting piston of cross sectional area A on which is placed a mass m. As the temperature is raised the volume increases but the pressure remains unchanged. If the temperature is lowered the volume decreases but still the pressure remains unchanged. This pressure is constant so long as m remains constant. Thus an isobaric process appears on a pv diagram as a horizontal line. Example Problem 15-3 An Isobaric Process A gas occupying 50 cm 3 at 50 C is cooled at constant pressure until the temperature is 10 C. What is its final volume? Solution: This is a constant pressure, isobaric process. Using the ideal gas law as in Example Problem 15-2 but this time putting p 2 = p 1 we have for the final volume: T V 2 = V 2 1 = (50cm 3 ( )K ) = 43.8 cm3 T 1 ( )K The volume decreases as it must if the temperature decreases. Constant Temperature Process A constant temperature process is called an isothermal process. In such a process T f = T i. [15-13] Using the ideal gas law we can put eq[15-13] into the form: p f V f nr = T f = T i = pv i i nr or p f V f = p i V i. [15-14] Figure Illustration of an isobaric process. An idealized apparatus for studying an isothermal process is shown in Figure The container of gas equipped with a movable closefitting piston is held in thermal equilibrium with a constant temperature bath. If the piston is moved downwards (slowly) or upwards (slowly) then the gas and the bath remain at thermal equilibrium and therefore at the same temperature. 15-9

10 Example Problem 15-4 An Isothermal Process A gas cylinder of the type shown in Figure 15-9a containing 200 cm 3 of air at 1.0 atm pressure initially floats on the surface of a swimming pool in which the water is at 15 C. The cylinder is then slowly pulled underwater to a depth of 3.0 m. What is the volume of the gas cylinder at this depth? Solution: The water in the swimming pool acts as a constant temperature bath. This process is an isothermal one. The pressure of the gas, initially 1.0 atm, increases to a value p water = p 0 + ρgd The ideal gas law gives, for T 2 = T 1, V 2 = V 1 p 1 p 2 = V 1 p 0 p 0 + ρgd Plugging in numbers we obtain V 2 = 155 cm 3. Figure Illustration of an isothermal process. In an isothermal process both p and V change. Since T is constant we have p = nrt V = const V. [15-15] As a result of the pressure under water the volume of the cylinder is reduced from 200 cm 3 to 155 cm 3. Isochoric, isobaric and isothermal processes are special cases. An arbitrary process may consist of a number of these cases followed one after the other. This is called a multistep process. Such a multistep process is illustrated in Figure We shall treat it as an example problem. This means that an isothermal process at some temperature T follows an hyperbolic path on a pv diagram (as shown in Figures 15-9b and c). Each curve of the family of hyperbolas (one for each temperature) is called an isotherm. Figure An example of a multistep process

11 Example Problem 15-5 An Example of a Multistep Process A gas at 2.0 atm pressure and a temperature of 200 C is first expanded isothermally until its volume has doubled. It then undergoes an isobaric compression until it returns to its original volume. First show this process on a pv diagram. Then find the final temperature and pressure. Solution: This process is shown on the pv diagram of Figure The gas starts in state 1 at pressure p 1 = 2.0 atm and volume V 1. As the gas expands isothermally it moves downward along an isotherm until it reaches volume V 2 = 2V 1. The pressure decreases during this process to a lower value p 2. The gas is then compressed at constant pressure p 2 until its final volume V 3 equals its original volume V 1. State 3 is on an isotherm closer to the origin, so we expect to find T 3 < T 1. Since from eq[15-15], p 1 = nrt 1 /V 1 and p 2 = nrt 2 /V 2 we have Thus p 2 = nrt 2 V 1 = V 1 = 1 p 1 V 2 nrt 1 V 2 2. p 2 = 1 2 p 1 =1.0 atm. During the isobaric process we have V 3 = V 1 = V 2 /2 and so T 3 = V 3 T 2 = V /2 2 = 1 V 2 V 2 2 T 2 Thus, indeed, T 3 < T 1. = K = 36.5 C 15-11

12 To Be Mastered Definitions: atomic mass unit, mole, Avogadro s number, molar mass Definitions: Thermal contact, thermal equilibrium Statement: zeroth law of thermodynamics Definitions: Celsius temperature scale, Fahrenheit temperature scale, kelvin temperature scale Definitions: coefficient of linear expansion, coefficient of volume expansion Physics of: Ideal gas, the ideal gas law Physics of: Ideal gas processes: isochoric, isobaric, isothermal Physics of: pv diagram of an ideal gas Typical Quiz/Test/Exam Questions 15-12