RI Chap 12 Temperature and Ideal Gases Notes

12 TEMPERATURE & IDEAL GASES H2 Physics 9478 Content Page 12.1 Temperature scales 3 12.2 Equation of state 7 12.3 Kinetic theory of gases 11 12.4 Appendix 15 Learning Outcomes Candidates should be able to: (a) show an understanding that a thermodynamic scale of temperature has an absolute zero and is independent of the property of any particular substance. (b) convert temperatures measured in degrees Celsius to kelvin: / K / °C 273.15T θ= + . (c) recall and use the equation of state for an ideal gas expressed as pV NkT= , where N is the number of particles. (d) state that one mole of any substance contains 236.02 10× particles, and use the Avogadro constant 23 16.02 10 molAN −= × as well as the relationship Nk nR= between the Boltzmann constant and the molar gas constant (n is the amount of substance in moles). (e) state the basic assumptions of the kinetic theory of gases. (f) explain how the random motion of gas particles exert mechanical pressure and hence derive, using the definition of pressure as force per unit area, the relationship 21 3pV Nm c= (a simple model considering one -dimensional collisions and then extending to three dimensions using 22 1 3 xcc = is sufficient). (g) recall and use the relationship that the mean translational kinetic energy of a particle of an ideal gas is (directly) proportional to the thermodynamic temperature ( i.e., 213 22m c kT= ) to solve problems.

Page | 2 Temperature and Ideal Gases – An Overview Concepts such as speed, velocity, force and kinetic energy are carefully defined to make the study of mechanics quantitative. Similarly, there needs to be careful definition of terms like temperature, heat and internal energy, which are used in thermal physics. Understanding thermal physics requires us to approach the concepts from both the macroscopic and microscopic perspectives. Heat and temperature are often used interchangeably by the lay-person. However, these terms have different and specific meanings in physics. Macroscopically, temperature can be defined in terms of its measurement using a thermometer, while heat refers to the energy transferred between two systems due to a temperature difference between them. A thermometer is used to measure temperature and it is calibrated according to a temperature scale. The Kelvin scale has a privileged status, as it is independent of the physical properties of the medium used for temperature measurement. This is unlike the Celsius scale, commonly used in liquid-in-glass thermometers, which is calibrated based on the properties of water. Generally, the physical properties of a substance depend on physical quantities such as temperature, pressure and volume. The condition in which a particular material exists is known as its state, and this could be described by such macroscopic physical quantities. We are particularly interested in the study of gases as their volumes can be varied much more dramatically than for typical solids and liquids. At very low pressures, most gases behave in much the same way, which is encapsulated in the ideal gas model. While the exact properties of real gases are very complex, most gases at room temperature and atmospheric pressure behave approximately like that of ideal gases. The ideal gas equation expresses the relationship between the state variables for an ideal gas. A key focus of this topic is to model the ideal gas with t he kinetic theory of gases , which links the macroscopic properties such as pressure, volume and temperature of gases, with the microscopic properties such as the mass and speed of the gas molecules. To a good approximation, Newtonian mechanics can be used to model the microscopic motion of gas molecules , and concepts from kinematics and dynamics can be applied to analyse the average pressure exerted by the randomly moving molecules. The collective macroscopic properties of a system therefore come from taking suitable averages of physical quantities from single-particle mechanics. A crucial area where the investigation into the behaviour of gases will be the environment. Extensive modelling of the atmosphere and relevant large-scale systems is needed to characterise and deal with climate change. Even the simplest models will involve and incorporate many aspects of physics. Tackling climate change is an existential question for human society – a huge issue that provides plenty of scope for discussion and possible action at global and local scales, at policy and personal levels.

RAFFLES INSTITUTION YEAR 56 PHYSICS DEPARTMENT 3 | Page 12.1 Temperature scales  Temperature Temperature is a fundamental physical quantity associated with the physiological sense of hotness and coldness. An object that is cold will cause different sensation from one that is warm. These sensations are not reliable enough for scientific work because they are subjective and dependent on contrast. Early attempts in finding a way to measure the “degree of hotness” involved fixing certain values to particular temperatures (at which a particular substance changes phase) and then establishing a scale between them e.g., the Fahrenheit and Centigrade scales. The “degree of hotness” of an object can then be expressed as a number on the scale. Later in this chapter, you will learn that temperature is a measure of the average kinetic energy possessed by the atoms or molecules of a substance. The greater the average microscopic kinetic energy of the molecules, the higher the temperature of the substance is.  Thermometer Temperature Scales In order to measure temperature reliably, many temperature scales have been developed. A temperature scale is a system of measuring temperature. Most of them rely on measuring the changes in some physical property of a substance that varies with temperature. This is known as thermometric property. Thermometer Thermometric Substance Thermometric Property liquid-in-glass fix mass of liquid in a capillary tube length of liquid platinum resistance fixed length of platinum wire resistance of wire constant volume gas fixed mass of gas at constant volume pressure of a gas thermocouple two junctions of dissimilar metals e.m.f. between the junctions Extra: A good thermometric property should satisfy the following criteria: 1. It varies continuously and uniquely with temperature i.e., different values of the thermometric property for different temperatures. 2. The value of the thermometric property at any temperature within its working range must be reproducible.

Page | 4 3. The change in the property must be large enough to enable accurate measurements of temperature. If changes in the thermometric property of a substance with temperature is small, then the sensitivity of the thermometer is said to be low. 4. It responds quickly to changes in temperature. The responsiveness of a thermometer is different from its sensitivity; sensitivity tells us how much the quantity changes per unit temperature change, while responsiveness is how quickly the quantity changes to a change in temperature. Example 12.1 A resistance thermometer has a resistance of 25.40 Ω at ice point, 27.34 Ω at steam point and 26.95 Ω at the melting point of a certain solid. (a) Calculate the temperature of the solid on the Celsius scale of the resistance thermometer. (b) State an assumption made in your calculation. Solutions: (a) ( ) ( ) RR RR θθ −= ×− 0 100 0 Using 100 ( ) ( ) 26.95 25.40Temperature of melting point of solid, 100 79.9 C27.34 25.40θ −= ×= °− (b) The resistance of the metal varies linearly with temperature. Please refer to Appendix for details on empirical Celsius scale.  Constant-Volume Gas Thermometer The thermometric property of a gas thermometer is the pressure of a fixed mass of gas at constant volume. The bulb may contain dry air, like