8. Temperature and Ideal Gas

Temperature and Ideal Gas Temperature: A measure of “hotness” of object. It indicates the direction of thermal energy flow – thermal energy flows from a region of higher temperature to a region of lower temperature. Thermal Equilibrium:  No net heat flow between the two objects in thermal contact  Two objects’ temperatures are the same Convert kelvin to degree Celsius: T / K = T / °C + 273.15 Gas Laws Boyle’s Law Charles’s Law Pressure Law When temperature is constant, pressure increases as volume decreases. 1p V When pressure is constant, volume increases as temperature increases. VT When volume is constant, pressure increases as temperature increases. PT Ideal Gas Equation pV nRT or pV NkT  Reflects Boyle’s, Charles’s and Pressure Laws  An ideal gas is one that obeys the ideal gas equation (pV = nRT) at all values of temperature (T), pressure (p) and volume (V). [n is the number of moles of gas; R is the molar gas constant]  Ideal Gas does not exist in reality. Most gases behave more like an ideal gas at high temperature and low pressure.  n is the number of moles of gas. One mole of gas contains 6.02 x 1023 particles. Avogradro number, NA = 6.02 x 1023 mol-1. Hence, n = N/NA Kinetic Theory of Gases Assumptions of Kinetic Theory of Gases: 1. A container with volume V contains a very large number of identical atoms/molecules, each with mass m. 2. The volume of the atoms/molecules is negligible as compared to the volume occupied by the gas (i.e. volume V). 3. The atoms/molecules are in constant motion randomly in agreement with Newton’s Law of Motion. 4. The atoms/molecules exert no forces on each other or on the walls of the container except during collisions. 5. The collisions among atoms/molecules or with the walls of the container are elastic. How molecular movement causes pressure:  When a molecule hits the wall of the container elastically, there is a change in momentum of the molecule over time.  By Newton’s 1st and 2nd law, there is a net force on the molecule.  By Newton’s 3rd law, there is a force on the wall by the molecule.  Hence there is a pressure exerted by the gas which exert a force over the area of the wall.

Derivation of 21pV Nm c3  Consider an ideal gas particle of mass m and speed U, hence momentum mU colliding perpendicularly with a wall of a container. By assumptions 5, the final momentum of the particle is -mU. p mU mU 2mU      By assumption 3, there is a change in momentum of the particle resulting in a force by the particle on the wall. By Newton’s 3 rd law of motion, there is a force on the wall by the particle.  The particle is hitting on the wall constantly by moving back and forth the container of length L. Therefore, the frequency of collision is 1U t 2L   By Newton’s 2nd law, the magnitude of the force is   2dp 1 mUFp dt t L       Therefore, pressure by one particle is 22F mU mUp A LA V    By assumption 1, due to large numbers of particles with various velocity U, summing all mean square values of the velocities in the x-direction, the pressure in x-direction is,   2 2 2 2 1 2 Nm U U ... U Nm U p VV      In 3-dimensions, by the random motion of particles, the mean square speeds in each of the three directions are the same, the mean square speed in 3D is 22c 3 U     22 1Uc 3  Therefore, 21pV Nm c3 Mean Kinetic Energy of a molecule of an Ideal Gas: 21pV Nm c NKT3 21 1 13 Nm c 3 NKT2 3 2     213m c KT22  thus the mean kinetic energy of a molecule of an ideal gas is proportional to the thermodynamic temperature.

Pressure in Kinetic Theory of Gas:  The higher the frequency of collision, the larger the pressure  The higher the change in momentum of the particles (e.g. higher temperature, hence higher speed), the larger the pressure Volume in Kinetic Theory of Gas:  Higher volume suggests larger average spacing between the particles