Temperature, Thermal Expansion, and the Ideal Gas Law

Transcription

1 Temperature, Thermal Expansion, and the Ideal Gas Law Vocabulary system state microscopic macroscopic thermodynamics state variables atom atomic mass molecular mass atomic mass unit Brownian movement thermometers Celsius Fahrenheit centigrade constant volume gas thermometer Thermal Equilbrium Zeroth Law of Linear Expansion equation of state thermodynamics equilibrium states Boyle s law absolute zero Kelvin scale absolute scale Charles Law Gay- Lussac s Law Mole universal gas constant ideal gas law Avogadro s hypothesis Avogadro s number Boltzmann s constant ideal gas temperature scale triple point 17-1 atomic Theory of Matter for this chapter you need to remember the kinetic theory of gases we will consider a particular system, by which we mean a particular objects or set of objects; everything else in the universe is called the environment we can describe the state (or condition) of a particular system- such as a gas in a container- from wither microscopic or macroscopic point of view microscopic- would involve details of the motion of all the atoms or molecules making up the system, which could be very complicated macroscopic- in terms of quantities that are detectable directly by our senses, such as volume, mass, pressure, and temperature description of processes in terms of macroscopic quantities is the field of thermodynamics the number of macroscopic variables required to describe required to describe the state of system at any time depends on the type of system The quantities needed to describe the system are state variables we will be emphasizing the concept of temperature The idea that matter is made of atoms dates back to the ancient greeks Democritus- if a given substance- say a piece of iron- were cut into smaller bits, eventually a smallest piece of that substance would be obtained which could not be divided further. THis smallest piece is called an atom- which means indivisible (Greeks word) The only real alternative to the atomic theory of matter was the idea that matter is continuos and can be subdivided indefinitely relative masses of atoms and molecules- atomic mass or molecular mass These are based on Carbon 12- it is the unified atomic mass unit (1u= E -27 kg) 1 RoessBoss

2 An important piece of evidence for the atomic theory is the Brownian movementnamed after Robert Brown- was observing pollen grains suspended in water in his microscope concluded that the movement was due to the atoms being in constant motion Einstein examined Brownian motion from a theoretical point of view and was able to calculate from the experimental data the approximate size and mass of atoms and molecules His calculations showed that the diameter of a typical atom is about 1 E -10 m three states of matter- solid, liquid, and gas- macroscopic or large scale properties atoms and molecules must exert attractive forces on each other The attractive forces are electrical in nature if the molecules come too close together the force becomes repulsive the molecules maintain a minimum distance from each other in solid material, the attractive forces are strong enough that the atoms or molecules are held in more or less fixed positions, often in an array as a crystal lattice the atoms or molecules in a solid are in motion- they vibrate about their nearly fixed positions liquid- atoms or molecules are moving more rapidly or the forces between then are weaker so that they are sufficiently free to pass over one another gas-the forces are so weak or the speeds are so high, that the molecules do not even stay close together they move rapidly every which way, filling any containers and occasionally colliding with one another on the average, speeds are sufficiently high in a gas that when two molecules collide, the force of attraction is not strong enough to keep them close together and they fly off in new directions Temperature and Thermometers Temperature is a measure of how hot or cold an object is hot objects are said to have high temperatures and vice versa many properties of matter change with temperature most materials expand with the heat (expansion joints on bridges) The electrical resistance of matter changes with temperature the color of radiated bodies at least at high temperatures changes Instruments designed to measure temperature are thermometers Their operation depends on some property of matter that changes with temperature most thermometers rely on the expansion of material with an increase in temperature Galileo made use of the expansion of gases to design the first thermometer Common thermometers consists of a hollow glass tube filled with mercury or alcohol colored with a red dye The liquid expands more than the glass- so the liquid rises and you can take a reading 2 RoessBoss

3 you can also make a thermometer with a bimetallic strip. The two metals expand at two various rates and so you can use it as a switch to turn AC on or off The most common scale for measuring temperature is the Celsius or centigrade scale In the US we use the Fahrenheit scale The last scale is used primarily for science research which is the Kelvin scale or the absolute scale The conversion for the scales is Equation Box 17-1 Different materials do not expand in quite the same way over a wide temperature range due to this some standard kind of thermometer must be chosen so that these intermediate temperatures can be precisely defined- constant volume gas thermometer is the standard 17-3 Thermal Equilibrium and the Zeroth Law of Thermodynamics We know that if two objects at different temperatures are placed in thermal contact, the two objects will eventually reach the same temperature they will then be said to be in thermal equilibrium they are in thermal equilibrium when the temperatures no longer change between the two bodies Zeroth law of thermodynamics- if two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other note that this postulate came in after the first and second law of thermodynamicsthey figured out afterwards that they needed this postulate -it allows a useful definition of temperature 17-4 Thermal Expansion most substances expand when heated and contract when cooled The amount of which it expands or contracts depends on the material Linear Expansion the change in length L of almost all solids is to a very good approximation is directly proportional to the change in temperature T The change of length is also proportional to the original length of the object 3 RoessBoss

4 Equation Box 17-2 where is the proportionality constant and is called the linear expansion for the particular material and has the units ( C)^-1 The equation can also be written as Equation Box 17-3 L is the length after heating or cooling However if the temperature range is not too great, the variation can usually be ignored Volume Expansion The change in volume of a material which undergoes a temperature change is given by a relation similar to 17-2 Equation Box 17-4 ß is the coefficient of volume expansion- units are the same as the other coefficient in linear expansion ß itself varies substantially with temperature for gases Atomic Theory of Expansion Thermal expansion is due to the non-symmetry of the potential energy function Anomalous Behavior of water below 4 C Water is unusual it expands when cooled from 4 C to 0 C This enables organisms to survive the winter months As it expands the water becomes more dense at 4 C and then decreases in density as it moves to 0 C 4 RoessBoss

5 17-5 Thermal Stresses If temperature should change, large compressive or tensile stresses called thermal stresses will occur magnitude of such stresses can be calculated using the concept of elastic modulus To calculate the internal stress we can think of this process in two steps The rod expands (or contracts) by an amount L then a force is applied to compress (or expand) the material back to its original length Equation Box 17-5 E is the Young s modulus for the material Equation Box 17-6 to calculate the internal stress F/A we then set L equal to L in the equation above and find 17-6 The gas Laws and absolute temperature Those equations do not work with gases based on their methods of expansion The volume of a gas depends very much on the pressure and the temperature so you have to determine the relationship between volume, pressure, and temperature- equation of state if the state of a system is changed, we will always wait until the pressure and temperature have reached the same value throughout Thus we consider only the equilibrium states of a system- when the variables that describe it (such as temperature and pressure) are the same throughout the system and are not changing in time For a given quantity of gas it is found experimentally that to a good approximationthe volume of gas is inversely proportional to the pressure applied to it when the temperature is kept constant 5 RoessBoss

6 Equation Box 17-7 P is the absolute pressure (gauge pressure) Boyle s Law- names after Robert Boyle- stated that relationship It can also be written as PV= constant at constant temperature, if either the pressure or volume of the gas is allowed to vary, the other variable also changes so that the product PV remains constant Temperature also affects the volume of a gas but a quantitative relationship between V and T was not found until more than a century after Boyle s work Frenchman Jacques Charles found that when the pressure is not too high and is kept constant, the volume of a gas increases with temperature at a nearly constant rate However all gases liquefy at low temperatures the lowest temperature is absolute zero- it is C it forms the basis of a temperature scale known as absolute or Kelvin scale it used extensively in scientific work The relationship between Kelvin and Celsius is Equation Box 17-8 The volume of a given amount of gas is directly proportional to the absolute temperature when the pressure is kept constant- Charles Law Equation Box 17-9 The third Law is known as Gay-Lussac s Law- at constant volume, the pressure of a gas is directly proportional to the absolute temperature 6 RoessBoss

7 Equation Box These are really only approximation that are accurate for real gases only as long as the pressure and density of the gas are not too high and the gas is not too close to condensation The term law applied to these three relationships has become traditional however we are stuck with that usage 17-7 The Ideal Gas Law The gas laws of Boyle, Charles, and Gay-Lussac were obtained by means of a technique that is very useful in science; namely to hold one or more variables constant in order to see clearly the effects of changing only one of the variables We therefore place this into a single law However we must also include the relationship of the amount of gas present one mole is defined as the amount of substance that contains as many atoms or molecules as there are in grams of carbon 12 1 mol is the number of grams of a substance numerically equal to the molecular mass of the substance The mole is the official unit in the SI system the number of moles, n in a given sample of a pure substance is equal to its mass in grams divided by its molecular mass specified as grams per mole We can now write the proportion as an equation Equation Box This is the Ideal gas Law where n represents the number of moles R is the constant of proportionality- universal gas constant because its value is found experimentally to be the same for all gases The value of R is 7 RoessBoss

8 Equation Box The ideal gas law or the equation of state for an ideal gas is the main equation We use ideal because real gases do not follow this equation precisely, particularly at high pressure (and density) or when the gas is near the liquefaction point (=boiling point) however at pressures less than an atmosphere or so, and when T is not close to the liquefaction point of the gas is quite accurate and useful for real gases 17-8 Problem Solving with the Ideal Gas Law The ideal gas law is extremely useful tool Always remember when using the ideal gas law that temperatures must be given in Kelvins and that the pressure must always be absolute pressure not gauge pressure Also check the units of the other variables to ensure that they match the R In some situations you can just look at them as a relationship between the two P, T, and V and you will not need the R Read your problems carefully to ensure that you have started with the right idea for solving the problem 17-9 Ideal Gas Law in Terms of Molecules Avogadro s number The fact that the gas constant R has the same value for all gases is remarkable reflection of simplicity in nature Amedeo Avogadro- stated that equal volumes of gas at the same pressure and temperature contain equal numbers of molecules This is sometimes called Avogadro s hypothesis That this is consistent with R being the same for all gases the number of molecules in a mole is known as Avogadro s number Na Although Avogadro conceived the notion, he was not able to actually determine the value of Na precise measurements were not made until the twentieth century Na= 6.02 E23 = Avogadro s number Since the total number of molecules N in a gas is equal to the number per mole times the number of moles (N=nNa), the ideal gas law, can be written in terms of the number of molecules present 8 RoessBoss

9 Equation Box This is the ideal gas law (in terms of molecules) k= R/Na is called Boltzmann s constant and has the value Equation Box RoessBoss