TJC 12 Temperature and Ideal Gases

Temasek Junior College 1 i Unit 12: Temperature and Ideal Gases 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 kelvin to degrees Celsius: / / 273.15T K T C   (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 6.02 × 10 23 particles, and use the Avogadro constant NA = 6.02 × 10 23 mol-1 as well as the relationship Nk = nR between the Boltzmann constant and the molar gas constant, where n is the amount of substance in moles and N = nNA. (e) state the basic assumptions of the kinetic theory of gases. (f) explain how the random motion of gas particles exerts mechanical pressure and hence derive, using the definition of pressure as force per unit area, the relationship 2 3 1 cNmpV  . (A simple model considering one -dimensional collisions and then extending to three dimensions using 22 3 1 ccx  is sufficient.) (g) 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. kTcm 2 3 2 1 2  ) to new situations or to solve related problems. Name : ________________________________ Class : _________

Temasek Junior College 2 1 Temperature 1.1 Temperature and Heat From the macroscopic perspective between systems, temperature is the physical property that determines the direction of heat flow. Heat then refers to the energy transferred between two systems at different temperatures. Microscopically however, we can define temperature a s a measure of the average kinetic energy of molecules in a body. 2 Temperature Scales LO(a) 2.1 Temperature and thermometers Thermometers are devices used to measure temperature. There are many kinds of thermometers, but their operation always depends on some thermometric property.  Thermometric property: a physical property that increases or decreases continuously with temperature, such as the ones below The fact that substances change state (from solid to liquid, or from liquid to gas) at fixed temperatures is used to define reference temperatures, which are called fixed points. By taking the value of the thermometric property at t wo fixed points, and dividing the range of values into a number of equal units (or degrees) between the two points, we can set up what is known as an empirical scale of temperature for that thermometer. 2.2 Empirical Scales ‘Empirical’ means ‘derived by experiment’. For example, if the fixed points are the melting and boiling points of water, and if we choose to have one hundred equal units between the temperatures corresponding to these fixed points, we can assign to these fixed points, values of 0 degrees and 100 degrees respectively. This is how the empirical centigrade scale of temperature for that thermometer is arrived at. It is important to realise that the choice of a different thermometric substance and thermometric property would lead to a different centigrade scale. Agreement between scales occurs only at the two fixed points. This happens because the property may not vary linearly with temperature. Type of Thermometer Thermometric Property liquid-in-glass thermometer length of mercury in a capillary tube resistance thermometer resistance of platinum wire thermocouple thermometer emf of a copper-constantan thermocouple constant volume gas thermometer pressure of a fixed mass of gas at constant volume

Temasek Junior College 3 This situation, with temperature values depending on the type of thermometer on which they are measured, is clearly unsatisfactory for scientific purposes. 2.3 The Thermodynamic Scale It is found that the differences between empirical scales are small in the case of thermometers based on gases as thermometric substances. In the constant -volume gas thermometer (Fig. 1), the pressure of a fixed volume of gas (measured by the height difference h) is used as the thermometric property. The agreement among thermometers using various gases improves as the pressure is reduced. If the graphs shown in Fig. 2 are extended toward negative temperatures, we find in every case, the pressure is zero when the temperature is 273.15 C. This significant temperature is used as the basis for the absolute temperature scale, which sets 273.15 C as its zero point. This temperature is known as absolute zero on the thermodynamic scale of temperature. Fig. 1 Fig. 2 The thermodynamic temperature scale is theoretical and is independent of the property of any real substance. The unit of thermodynamic temperature is the kelvin (symbol K). Gas 1 Gas 2 Gas 3

Temasek Junior College 4 The two fixed points in the Thermodynamic Temperature Scale are: (a) absolute zero (0 K)  temperature at which the pressure of an ideal gas becomes zero. It is also the temperature at which all substances have a minimum internal energy. (b) triple point of water (273.16 K)  temperature at which the three phases of water (i.e. ice, water and water vapour) coexist in dynamic equilibrium. Reasons for using triple point of water: 1. It is unique, invariant & occurs at only one temperature & pressure. On the pressure- temperature graph, the triple point occurs at a precise temperature and pressure, whereas other phase transitions may occur over a range of temperatures. 2. The conditions for triple point of water can be easily reproduced using a triple point cell. One kelvin is defined to be 1/273.16 of the thermodynamic temperature of the triple point of water. 2.4 The Celsius Scale LO(b) The Celsius scale is a shifted thermodynamic scale. The unit for this scale is degree Celsius, symbol oC (same symbol as for degree Centigrade). The Celsius scale is related to the Thermodynamic scale by the exact equation: Example 1 The temperature of a body at 100 C is increased by  as measured on the Celsius scale. How is this temperature change expressed on the Kelvin scale? A  B  + 100 C  + 273 D  +3 73  /°C = T /K – 273.15 Water Vapour water Ice Layer of water Bulb of gas thermometer triple point cell phase diagram of water P / atm T / °C

Temasek Junior College 5 3 Ideal Gas 3.1 The Ideal Gas Equation LO(c) pV = NkT N = number of particles k = Boltzmann constant = 1.38 x 10-23 J K-1 The above equation is called the equation of state of an ideal gas or simply the ideal gas equation. An ideal gas is one which obeys the equation pV = NkT at all pre ssures p, volumes V and temperatures T, where N is the number of particles and k is the Boltzmann constant. No real gas obeys the ideal gas equation exactly. For approximate calculations, the ideal gas equation can be used with real gases if the gas is well above the temperature at which it will liquefy (ie high temperature compared to boiling point) and the pressure is low. Under these conditions, the separation between the gas molecules is lar