DHS 11 Temperature & Ideal Gases (Lecture Notes & Tutorial)

Dunman High School (Senior High Physics) 1 Topic 8 Temperature and Ideal Gases Guiding Questions:  What is the difference between temperature and heat? What is internal energy?  How can we define a simple model of a gas that is valid for real gases under certain assumptions?  How can we relate microscopic behaviour of gas molecules to the macroscopic properties of a gas? Content • Thermal equilibrium • Temperature scales • Equation of state • Kinetic theory of gases • Kinetic energy of a molecule Learning Outcomes Candidates should be able to: (a) show an understanding that regions of equal temperature are in thermal equilibrium. (b) explain how empirical evidence leads to the gas laws and to the idea of an absolute scale of temperature (i.e. the thermodynamic scale that is independent of the property of any particular substance and has an absolute zero). (c) convert temperatures measured in degrees Celsius to kelvin: T / K = T / °C + 273.15. (d) recall and use the equation of state for an ideal gas expressed as pV = nRT, where n is the amount of gas in moles. (e) state that one mole of any substance contains 6.02 x 1023 particles and use the Avogadro number NA = 6.02 x 1023 mol-1. (f) state the basic assumptions of the kinetic theory of gases. (g) explain how molecular movement causes the pressure exerted by a gas and hence derive the relationship 21 3pV Nm c   , where N is the number of gas molecules (a simple model considering the one-dimensional collisions and then extending to three dimensions using 2 2 1 3 xc c    is sufficient). (h) recall and apply the relationship that the mean kinetic energy of a molecule of an ideal gas is proportional to the thermodynamic temperature (i.e. 21 3 2 2m c kT  ) to new situations or to solve related problems.

Dunman High School (Senior High Physics) 2 Introduction Heat and temperature are often used interchangeably in everyday language. However, these terms have different and specific meanings in physics. Macroscopically, temperature can be defined in terms of its measurement, while heat refers to the energy transferred between two systems at different temperatures. Linking these two ideas is the zeroth law of thermodynamics, which states that when two objects at different temperature are placed in thermal contact, there will be energy exchange between them until thermal equilibrium is reached. The zeroth law allows the use of thermometers to measure temperature. When thermal equilibrium is achieved, the thermometer reflects its own temperature as well as the temperature of the other body that it is in thermal contact with. (a) show an understanding that regions of equal temperature are in thermal equilibrium. Temperature is a measure of degree of hotness of an object. Thermal energy moves from objects at a higher temperature to objects at a lower temperature. Note: temperature does NOT measure the amount of thermal energy in an object. It only indicates in which direction thermal energy will flow unaided. Temperature is a fundamental quantity (i.e. it cannot be defined from other physical quantities) that measures the degree of hotness of a body as indicated on a calibrated scale. When two system s have different temperatures, a net thermal energy is transferred from the system at a higher temperature to a system at a lower temperature. Heat is energy that flows from a higher -temperature object to a lower -temperature object because of differences in temperature. Two systems are in thermal contact with each other if there is a mechanism for the transfer of thermal energy (i.e. by conduction, convection or radiation) . Thermal contact may not necessarily mean physical contact only as thermal energy can be transmitted even through a vacuum by radiation (e.g. in space!). When two objects in thermal contact has no net heat transfer between them, they are said to be in thermal equilibrium and are at the same temperature. The zeroth law of thermodynamics states that if objects A and B are separately in thermal equilibrium with object C, then object A and B are in thermal equilibrium with each other. Object C can function as a measuring device known as a thermometer, and the temperature can be measured on a suitable temperature scale. Watch this video about Heat & Temperature!

Dunman High School (Senior High Physics) 3 (b) explain how empirical evidence leads to the gas laws and to the idea of an absolute scale of temperature (i.e. the thermodynamic scale that is independent of the property of any particular substance and has an absolute zero). Thermometers are devices that are used to measure the temperature of a system. All thermometers are based on the principle that some physical property (known as thermometric property) of a system changes as the system’s temperature changes. Empirical temperature scale is a scale of temperature that is based on the variation with temperature of a property of a substance, assuming that the property varies linearly with temperature. How do we set up an empirical temperature scale? Step 1: Choose an appropriate thermometric property. We choose a physical property that varies with temperature. This property must be reproducible. Examples of Thermometric Quantities are: • Volume of a fixed mass of liquid: Liquid-in-glass thermometer  Resistance of a metal: Platinum resistance thermometer • Pressure of a fixed mass of gas at constant volume: Constant volume gas thermometer (see thermometer on p 6) • EMF produced between junctions of dissimilar metals that are at different temperatures: Thermocouple thermometer

Dunman High School (Senior High Physics) 4 Step 2: Select two fixed temperature points. We then choose an upper point and a lower point with which we will calibrate our thermometer from. For example, on the Celsius temperature scale, these two fixed points are the ice point of water, which is assigned to be 0 oC, and the steam point of water, which is assigned to be 100 oC. Step 3: Calibrate the thermometer. Lastly, we calibrate the thermometer by placing it in systems of the lower and upper fixed points. The values of the thermometric quantity at these temperatures are recorded. The value at the lower fixed point is X0 and that of the upper fixed point is X100. We then assume a linear relationship between these two points. Illustration: Using similar triangles or gradient formula, the following relationship can be obtained: In general, for all thermometers, X Value of thermometric quantity at temp  X0 Value of thermometric quantity at ice point X100 Value of thermometric quantity at steam point 100 0100 0   LL LL 100 0100 0   XX XX 0 L0 L L100  100 Temp / oC Length of mercury column

Dunman High School (Senior High Physics) 5 Example 1 At ice-point, the mercury column of an unmarked thermometer has a height of 2.0 cm. At boiling point, the mercury column has a height of 24.0 cm. What is the temperature when the mercury column reads 18.0 cm? (Ans: 72.7 oC) Example 2 A resistance thermometer gives a resistance of 20 Ω when the temperature is known to be -10 ˚C. When the temperature is 110 ˚C, the resistance thermometer has a resistance of 500 Ω. What is the temperature when the resistance is 360 Ω. (Ans: 75 oC) The problem with empirical scales Although d ifferent thermometers agree with one another at the fixed points, they ma y disagree at other temperatures due to their possibly non-linear variation with temperature.

Dunman High School (Senior High Physics) 6 Constant volume gas thermometer It turned out that f