Temperature and Ideal Gases JPJC Notes

1 JURONG PIONEER JUNIOR COLLEGE 9749 H2 PHYSICS TEMPERATURE AND IDEAL GASES 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 236.02 10 particles and use the Avogadro number 236.02 10AN = 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 one -dimensional collisions and then extending to three dimensions using 22 1 3 xcc = 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. 213 22m c kT= ) to new situations or to solve related problems.

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 concept of thermal equilibrium. The properties of matter depend on temperature, pressure, volume, etc. The condition in which a particular material exists is known as its state, and this could be described by such macroscopic physical quantities, including its mass, which is a measure of the amount of substance. Most gases at room temperature and atmospheric pressure behave approximately like ideal gases. The ideal gas equation expresses the relationship between state variables for an ideal gas. The kinetic theory of gases is a simple e xample of a link between macroscopic properties (pressure, volume, temperature) with microscopic properties (mass and speed of randomly - moving individual molecules that make up a gas). To a good approximation, we can use Newtonian mechanics to model the mi croscopic motion of gas molecules, and we can apply concepts from kinematics and dynamics to analyse the average pressure exerted by the randomly-moving molecules. 1 Thermal equilibrium 1.1 Temperature ▪ Temperature is a property of a body or system. It measures the degree of hotness or coldness of a body as indicated on a calibrated scale. It determines the way heat is transferred from one body to another. (a) Candidates should be able to show an understanding that regions of equal temperature are in thermal equilibrium. 1.2 Thermal equilibrium ▪ When two objects A and B are placed in thermal contact, heat flows from the hotter object to the colder object B, until they reached thermal equilibrium. ▪ At thermal equilibrium, both objects A and B are at the same temperature. Fig. 1.1 Thermal equilibrium ▪ Hence, two objects are said to be in thermal equilibrium if there is no net heat exchange when they are placed in thermal contact. hot cold objects A and B in thermal contact temperature rises temperature drops A B same temperature objects A and B in thermal equilibrium A B

3 ▪ Note that thermal contact does not necessarily mean physical contact between two objects. Two systems are in thermal contact as long as there is a mechanism for the transfer of heat (thermal energy). ▪ The method for comparing temperatures is encapsulated in the zeroth law of thermodynamics (not in syllabus), which states that if two objects A and B are each in thermal equilibrium with a third object C, then A and B are also in thermal equilibrium with each other. 2 Temperature scales (b) Candidates should be able to 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). 2.1 Gas laws 2.1.1 Boyle’s law ▪ For a given quantity of gas, it is found experimentally that the volume of a gas is inversely proportional to the pressure of the gas when the temperature is kept constant. ▪ This relation is known as Boyle’s law, named after Robert Boyle. ▪ Mathematically, the law can be expressed as 1V p , or 1 1 2 2pV p V= where 1p and 1V are the initial pressure and volume of the gas, and 2p and 2V are the final values after a change of pressure and volume carried out at constant temperature. ▪ A graph of p against V shows a curve for a fixed temperature, where the curve is known as an isotherm. On the other hand, a graph of p against 1 V shows a straight line passing through the origin. Fig. 2.1 Boyle’s law p V 0 p 0

4 2.1.2 Charles’ law ▪ For a given quantity of gas, it is found experimentally that when the pressure of a gas is kept constant, the volume of the gas increases with temperature at a nearly constant rate. ▪ When the straight line graph is projected to lower temperatures, it crosses the axis at about −273 °C. For any gas, a straight line is obtained and always project back to −273 °C at zero volume. Fig. 2.2 Charles’ law ▪ It could be argued that −273 °C is the lowest temperature possible for any gas. The exact value is determined to be −273.15 °C, which is equivalent to the absolute zero of temperature (0 K). ▪ If the Celsius temperatures are converted to thermodynamic temperatures, a straight line passing through the origin would be obtained. Hence, the volume of a gas is directly proportional to the temperature of the gas when the pressure is kept constant. ▪ This relation is known as Charles’ law, named after Jacques Charles. ▪ Mathematically, the law can be expressed as VT , or 12 12 VV TT= where 1V and 1T are the initial volume and temperature of the gas, and 2V and 2T are the final values after a change of volume and temperature carried out at constant pressure. 2.1.3 Gay-Lussac’s law ▪ For a given quantity of gas, it is found experimentally that the pressure of a gas has a linear relationship with the temperature of the gas when the volume is kept constant. ▪ This relation is known as Gay-Lussac’s law, named after Joseph Gay-Lussac. V T / K 0 V θ / °C 0 −273

5 ▪ Mathematically, the law can be expressed as pT , or 12 12 pp TT= where 1p and 1T are the initial pressure and temperature of the gas, and 2p and 2T are the final values after a change of pressure and temperature carried out at constant volume. ▪ A graph of p against T shows a straight line passing through the origin, when projected at lower temperatures. For any gas , a straight line always extrapolates back to 0 K at zero pressure. Fig. 2.3 Gay-Lussac’s law ▪ Similar to the graph shown in Fig. 2.2 , −273 .15 °C is the lowest temperature possible for any gas. 2.2 Absolute scale of temperature ▪ The k elvin scale is an absolute scale of temperature and is also called the thermodynamic temperature scale. It is a theoretical scale that is independent of the thermometric properties of any substance, that is, it doe