Medical Physics - Temperature & Kinetic Theory of Gases.pdf

Medical Physics Medical Physics Temperature & Kinetic Theory of Gases Temperature Temperature Contents : Temperature 4 Temperature Measurement 6 Ideal Gas Law 17 Temperature Kinetic Theory of Gases Contents : Kinetic Theory Model of an Ideal Gas 34 Vapor Pressure and Humidity 43 Boiling 50 Thermal E...

Medical Physics Medical Physics Temperature & Kinetic Theory of Gases Temperature Temperature Contents : Temperature 4 Temperature Measurement 6 Ideal Gas Law 17 Temperature Kinetic Theory of Gases Contents : Kinetic Theory Model of an Ideal Gas 34 Vapor Pressure and Humidity 43 Boiling 50 Thermal Expansion 52 Thermal Expansion of water 61 Temperature Temperature: We can feel the blazing heat of the summer sun or the biting cold of a winter blizzard. Our bodies are sensitive even to small changes in the temperature of our surroundings. We respond to these changes with several adaptive mechanisms, like sweating or shivering, to maintain a nearly constant internal body temperature. Temperature This sensitivity to the thermal environment is the basis for our concepts of hot and cold, out of which the scientific definition of temperature evolved. Temperature is a quantitative measure of how hot or cold something is. In this class we shall see how various kinds of thermometers are used to measure temperature and how temperature can be interpreted as a measure of molecular kinetic energy. Temperature Temperature Measurement: Thermometers Temperature is measured with a thermometer. Galileo invented the first thermometer, which made use of air’s property of expanding as it is heated. The air’s volume indicated the temperature. Today there are various kinds of thermometers, each appropriate for the range of temperatures and the system to be measured. Temperature For example, in addition to the common mercury thermometer, there are different kinds of thermometers. Each thermometer depends on the existence of some thermometric property of matter. For example, the expansion or contraction of mercury in a fever thermometer correlates with the body’s sensation of hot and cold. Temperature A person who has a high fever will register a higher than normal temperature on a mercury thermometer. The length of the mercury column in the glass stem of the thermometer gives us a quantitative measure of temperature. Temperature Temperature Scales To assign a numerical value to the temperature of a body, we need a temperature scale. The two most common scales in everyday use are the Celsius scale (formerly known as the centigrade scale) and the Fahrenheit scale. In establishing a temperature scale, one could use any kind of thermometer and any thermometric property. Temperature For example, the Celsius scale was based on the expansion of a column of liquid such as mercury in a thin glass tube. The absolute, or Kelvin, scale is now the standard in terms of which all other scales, such as Celsius and Fahrenheit, are defined. Temperature The Kelvin scale is chosen as the standard for important reasons. First, various laws of physics are most simply expressed in terms of this scale. We shall see an example of this in the ideal gas law, described in the next section. Second, zero on the absolute scale has fundamental significance. It is the lowest possible temperature a body can approach. Temperature When the Celsius scale was established, the normal freezing and boiling points of water were assigned respective values of 0℃ (zero degrees Celsius) and 100℃. A mercury thermometer was brought to thermal equilibrium with water at each of these temperatures, and the level of the mercury column was marked as 0℃ for ice water and 100℃ for boiling water. The mercury column between these two marks was then divided into 100 equal intervals, corresponding to temperature intervals of 1 Celsius degree (1℃). Temperature Today the Celsius temperature scale is defined in terms of the Kelvin scale. Celsius temperature 𝑇𝐶 is now defined by the equation The normal freezing point of water open to the air at one atmosphere of pressure is 273.15 𝐾, or 0.00℃. The normal boiling point of water is 373.15 𝐾, or 100.00℃. Temperature This definition of the Celsius scale conforms to the earlier definition based on the freezing and boiling points of water. Temperature intervals on the Celsius and Kelvin scales are the same. For example, the difference in temperature between the boiling point of water and its freezing point is 100 Celsius degrees (℃), or 100 kelvins. Temperature Fahrenheit temperature 𝑇𝐹, measured in degrees Fahrenheit (℉), is defined relative to the Celsius temperature 𝑇𝐶 by the The normal freezing and boiling points of water on the Fahrenheit scale are 32 ℉ and 212 ℉ respectively. The interval between these points is 180 ℉. There are only 100 ℃ between these same points. 180 Thus Celsius degrees are bigger than Fahrenheit degrees: 1℃ is 100 , 9 or 5 , times 1℉. Temperature Measuring a Fever on the Celsius Scale Normal internal body temperature is 98.6 ℉. A temperature of 106 ℉ is considered a high fever. Find the corresponding temperatures on the Celsius scale. Temperature Ideal Gas Law : An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. P𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑃, V𝑜𝑙𝑢𝑚𝑒 𝑉, 𝐾𝑒𝑙𝑣𝑖𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑇, N𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑎𝑠 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑁 Temperature The pressure of a gas can be changed in several ways. One way to increase the pressure of a gas confined to a fixed volume is to increase the number of gas molecules in the volume. You do this, for example, when you pump air into a bicycle tire or an automobile tire. Another way to change the pressure of a gas is to change its temperature. For example, when the air in an automobile tire heats up, its pressure increases significantly. Temperature A third way to change gas pressure is to change the volume containing the gas; decreasing volume causes an increase in pressure. For low-density gases, there is a simple, universal relationship between the gas 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑃, 𝑣𝑜𝑙𝑢𝑚𝑒 𝑉, 𝐾𝑒𝑙𝑣𝑖𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑇, and 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑎𝑠 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑁. The product of 𝑃 and 𝑉 is proportional to the product of 𝑁 and 𝑇: Temperature This equation is called the ideal gas law. The constant k is known as “Boltzmann’s constant” and is found from experiment to have the value In applying the ideal gas law, temperature must be expressed in kelvins, not in ℃ or ℉. Special cases of the gas law are found when one considers the variation of two of the variables 𝑃, 𝑉, 𝑁, and 𝑇, while the other two variables are held constant. Temperature For example, if 𝑁 and 𝑇 are fixed, the ideal gas law implies that the product 𝑃𝑉 is constant: This result is known as Boyle’s law. Boyle’s law implies that if the volume of a gas is reduced to half its original value the pressure of the gas is doubled. Temperature If 𝑃 and 𝑁 are fixed, the ideal gas law implies that the volume of the gas is directly proportional to its temperature: If V and N are fixed, the ideal gas law implies that The very definition of temperature on the Kelvin scale requires that this relationship be satisfied, at least in the limit of a very low-density gas. Temperature Example 2: The temperature of an ideal gas after compression Temperature Temperature Temperature Example 3: The number of air molecules in a Hot-aie Ballon Temperature Temperature Temperature Mole One can compute Avogadro’s number by dividing the mass of 1 mole of carbon 12 (12 g) by the mass of a single carbon-12 atom, equal to 12 atomic mass units, where the atomic mass unit is related to the gram by the preceding equation. Temperature We may express the number of molecules, 𝑁, of a substance as the product of Avogadro’s number, 𝑁𝐴 , and the number of moles, denoted by 𝑛: Temperature Example 6: Finding the mass of a volume of air Temperature Kinetic Theory of Gases Kinetic Theory of Gases Kinetic Theory; Model of an Ideal Gas: Kinetic theory is an area of physics that was developed in the late nineteenth century. Kinetic theory provides an explanation for the behavior of a macroscopic system in terms of its microscopic components—atoms or molecules, which obey dynamical laws. In this section we shall use kinetic theory to provide an explanation for the pressure of an ideal gas in terms of a molecular model. A container of gas consists of a large number of molecules moving randomly and colliding with the walls of the container. Kinetic Theory of Gases A gas contained in a volume of macroscopic dimensions consists of an enormously large number of molecules. These molecules move in a random, chaotic way throughout the volume of the container. When a molecule strikes a surface, it bounces off, exerting a small force on the surface. Kinetic Theory of Gases At any instant there will be many molecules colliding with the surface. The effect of these collisions is to produce a resultant force, which may be quite large. The resultant force is not steady but fluctuates rapidly, depending on the number of molecules striking the surface at any instant. Kinetic Theory of Gases But, for a surface of macroscopic size, the number of molecules involved is so large that fluctuations in the net force are negligibly small. Thus the molecules exert a pressure on the container walls that is effectively constant. Kinetic Theory of Gases Kinetic Interpretation of Temperature In this section we shall derive the ideal gas law, using the more realistic model of molecules moving randomly in all directions. This derivation, first presented by James Clerk Maxwell in 1859, shows that the average kinetic energy of an ideal gas 𝟏 molecule equals 𝟐 kT. A molecule of mass 𝑚 traveling at speed 𝑣 has kinetic 𝟏 energy 𝐾= 𝟐 𝑚𝑣 2. Kinetic Theory of Gases Denoting the average values of 𝐾 and 𝑣2 by K and 𝑣2 , we may express Maxwell’s result as: This equation shows that the temperature of an ideal gas is a measure of the average kinetic energy of its molecules. The phenomenon of thermal equilibrium is easy to understand. Two systems of gas that are initially at different temperatures have different values of average kinetic energy per molecule. Kinetic Theory of Gases Kinetic Theory of Gases When the systems are placed in thermal contact, the system with the higher temperature will lose energy as the system with the lower temperature gains energy. This process continues until the average molecular kinetic energies and hence also the temperatures of the two systems are the same. Thus thermal equilibrium is simply a consequence of the equal sharing of kinetic energy among the molecules of both systems. Kinetic Theory of Gases Although this equation is derived only for an ideal gas, it applies to any system, including liquids and solids. The average translational kinetic energy of the molecules in a 𝟑 body at absolute temperature 𝑇 equals 𝑘𝑇. 𝟐 Kinetic Theory of Gases Vapor Pressure and Humidity : The molecules in a liquid have a distribution of velocities, similar to the Maxwell-Boltzmann distribution of molecular velocities in a gas. Although intermolecular forces bind most of the molecules close together in the liquid, some of the molecules move fast enough to leave the surface of the liquid, like a rocket with a velocity greater than escape velocity leaving the earth. Since the molecules that evaporate are those with the greatest velocity and kinetic energy, the average kinetic energy of the molecules remaining in the liquid decreases, and so the temperature of the liquid decreases. Kinetic Theory of Gases Kinetic Theory of Gases You can feel the cooling effect of evaporation when you step out of a shower and water evaporates from your skin. The effect is more dramatic when the air is very dry. If you step out of a swimming pool in the desert, even though the air may be quite hot, you can be chilled by water evaporating from your skin. Kinetic Theory of Gases To better understand the process of evaporation, consider a liquid in a closed container with a piston (Fig. a). If the piston is raised, evaporation begins. Vapor fills the space above the liquid (Fig. b) and creates a pressure, called vapor pressure. As more and more molecules enter the vapor, some molecules begin to go from the vapor back into the liquid. Kinetic Theory of Gases Initially more molecules leave the liquid than enter it, and both the density and pressure of the vapor increases. There is soon reached an equilibrium state, in which as many molecules enter the liquid as leave it. We say the vapor is then “saturated,” since there can be no further increase in the number of molecules in the vapor (Fig. c). In this equilibrium state, vapor pressure reaches its maximum value, called saturated vapor pressure. Kinetic Theory of Gases If the piston is raised higher, more molecules enter the vapor phase until the vapor pressure again reaches the same saturated vapor pressure. As the temperature of a liquid increases, more of its molecules have sufficient kinetic energy to escape the liquid. Thus, as temperature increases, the rate at which molecules leave the surface of the liquid increases; that is, the rate of evaporation increases. Kinetic Theory of Gases An equilibrium state is not reached until the density of the vapor phase increases enough that the rate at which molecules reenter the liquid matches the new higher rate at which molecules leave the liquid (Fig. d). The higher-density equilibrium state is one of higher pressure. Thus, as temperature increases, saturated vapor pressure increases. Kinetic Theory of Gases Boiling: If the temperature of a liquid is raised enough so that the liquid’s vapor pressure equals the pressure of the surrounding air, the liquid begins to boil. That is, bubbles of vapor form within the bulk liquid. These vapor bubbles push outward against the liquid, which is at approximately the same pressure as the air. Water boils at a temperature of 100℃ when the surrounding air is at a pressure of 1.00 𝑎𝑡𝑚 because the saturated vapor pressure of air at 100℃ is 1.00 𝑎𝑡𝑚. Kinetic Theory of Gases If the surrounding air is at a lower pressure, water will boil at a lower temperature. For example, on a mountain at an elevation of 3000 𝑚, where atmospheric pressure is only 0.7 𝑎𝑡𝑚, water boils at a temperature of 90℃, since its saturated vapor pressure at that temperature equals 0.7 𝑎𝑡𝑚. Kinetic Theory of Gases Thermal Expansion : Nearly all solids and liquids expand as they are heated. The fractional increase in volume, ∆𝑉/𝑉, is often found to be directly proportional to the increase in temperature, ∆𝑇. The constant of proportionality is called the “volume coefficient of expansion,” denoted by 𝛽. Thus The change in volume Δ𝑉 is proportional to the original volume 𝑉, as well as to the temperature change Δ𝑇. Kinetic Theory of Gases Thermal Expansion : Thus, for example, if 1 𝑙𝑖𝑡𝑒𝑟 (1000 𝑐𝑚3) of water is heated from 20℃ to 25℃, its volume increases by only about 1 𝑐𝑚3. But if the water in a swimming pool of volume 1000 𝑚3 is heated over the same temperature interval, the water increases in volume by 1 𝑚3 , or 106 𝑐𝑚3. In both cases the ratio ∆𝑉/𝑉 is 10−3. Kinetic Theory of Gases Thermal Expansion : In the case of solids, the volume expansion is accomplished by an increase in all linear dimensions. As a solid is heated, the distance between any two points in the solid increases. The fractional increase in length is normally the same in all directions. Kinetic Theory of Gases Thermal Expansion : Thus, if a block of metal expands thermally by 0.1% in length, the block’s height and width will each also increase by 0.1%. The increase in size is like a photographic enlargement the Figure. As a washer is heated from temperature 𝑇 to temperature 𝑇 + ∆𝑇, all its linear dimensions get bigger. Kinetic Theory of Gases Thermal Expansion : The actual expansion, however, is much smaller than indicated here. The expansion shown here is 20%, which is approximately 100 times greater than the expansion of aluminum heated 100℃. Kinetic Theory of Gases Thermal Expansion : Instead of using a volume coefficient of expansion for solids, we normally use a linear expansion coefficient 𝛼, which is a measure of the fractional change in the linear dimensions of the solid. For a temperature change ∆𝑇, a length 𝑙 changes by ∆𝑙, where It is possible to show that the volume coefficient of expansion for a solid equals 3 times its linear coefficient: Kinetic Theory of Gases Coefficients of expansion for various materials are given in Table. For liquids there is no measure of linear expansion, since liquids must conform to the shapes of their containers. Kinetic Theory of Gases Example: Overflow Of An Expanding Liquid: Kinetic Theory of Gases Example: Overflow Of An Expanding Liquid: Thermal Expansion of Water Kinetic Theory of Gases Thermal Expansion of Water: In many cases the simple linear dependence of Δ𝑉 on Δ𝑇 expressed by Eq. 12–20 is valid over all temperature ranges commonly encountered, with a constant value for 𝛽. However, for some substances the variation of volume with temperature is more complicated. Water is such a substance. Figure below shows the density of water as a function of temperature. Kinetic Theory of Gases Notice that at most temperatures the density of water decreases as its temperature increases; that is, water expands as it is heated. But in the temperature range from 0℃ to 4℃ water contracts as it is heated. Kinetic Theory of Gases But in the temperature range from 0℃ to 4℃ water contracts as it is heated. Water is one of the few materials that have this property. This has an important effect on the rate at which lakes freeze. As air temperatures drop, the temperature of the water in a lake drops also, with the cooling occurring first at the surface of the lake. Kinetic Theory of Gases For temperatures above 4℃ this cooling proceeds very efficiently. As water cools, it becomes more dense and sinks to a lower level in the lake, as warmer, less dense water rises to take its place. Kinetic Theory of Gases Thus there is a natural mixing of warmer and colder water, causing rapid cooling of water beneath the surface. However, when the water reaches a uniform temperature of 4℃, the process changes. Cooling of the surface water below 4℃ decreases its density. Thus it stays at the surface, and further cooling of the water beneath proceeds more slowly. The surface of the lake may freeze. But in even the coldest weather, large lakes do not freeze solid. The water at the bottom of the lake remains at 4℃, enabling the marine life there to survive.