Thermal Physics 3.2 Modelling a gas

Understanding  Pressure  Equation of state for an ideal gas  Kinetic model of an ideal gas  Mole, molar mass, and the Avogadro constant  Differences between real and ideal gases

Applications and Skills Equation of state for an ideal gas Kinetic model of an ideal gas Boltzmann equation Mole, molar mass, and the Avogadro constant Differences between real and ideal gases

Equations Pressure p = F/A # moles of a gas as ratio of # molecules to Avogadro’s constant n = N/N A Equation of state for an ideal gas pV=nRT Pressure and mean square velocity of an ideal gas

Understand the proof for the formula

Equations Mean kinetic energy of ideal gas molecules E k(mean) = (3/2)k B T = (3/2)(R/N A )T

The Gas Laws (1) Developed independently & experimentally between mid 17 th and start of 19 th centuries Ideal gases defined as those which obey the gas laws under all conditions i.e. no intermolecular interactions between molecules and only exert forces when colliding. Real gases only approximate ideal gases as long as pressures are slightly greater than normal atmospheric pressure Boyle showed that p α 1/V or pV = k(at const temp) pV graphs aka isothermal curves Charles, around 1787 confirmed that all gases expanded by equal amounts when subjected to equal pressure. The volume changed by 1/273 of the volume at zero. At -273 °C volume becomes zero. For a fixed mass at constant pressure, volume directly proportional to absolute temperature V α T (const pressure) V/T = constant (at constant pressure)

The Gas laws (2) Third gas law, for a gas of fixed mass and volume, the pressure is directly proportional to the absolute temperature p α T (const volume) p/T = constant (at const volume) Avogadro stated that the number of particles in a gas at const temp and pressure is directly proportional to the volume of the gas n α V n/V = constant

Gas laws (3) Combining the four equations and four constants gives pV/nT = R or pV = nRT R = 8.31 JK -1 mol -1 when p in pascals, V in m 3, n = # moles of gas

The mole and Avogadro’s constant The mole (mol) measures the amount of substance something has and is one of he seven base SI units. Defined as the amount of substance having the same number of particles as there are neutral atoms in 12 grams of carbon – 12 One mole of gas contains 6.02 x atoms or molecules (Avogadro’s constant N A ). So 2 moles of oxygen gas contains x molecules.

Molar mass Since diatomic gases have two atoms per molecule a mole of a diatomic gas will have 6.02 x molecules but x atoms. One mole of oxygen atoms has approximate mass 16.0 g so a mole of oxygen molecules will have mass 32.0 g i.e. its molar mass. Consider one mole of CO 2 (g) which contains one mole of carbon atoms has mass 12.0 g and one mole of oxygen molecules 32.0 g. The molar mass of CO 2 is 44.0 g mol -1

Example 1 Molar mass of Oxygen is 32 x10 -3 kg mol -1 If I have 20g of Oxygen, how many moles do I have and how many molecules? Molar mass of Oxygen gas is 20 x kg / 32 x10 -3 kg mol -1  mol  mol x 6.02 x molecules  x molecules

Example 2 Calculate the percentage change in the volume of a fixed mass of an ideal gas when its pressure is increased by a factor of 3 and its temperature increases from 40.° C to 100.° C. n is constant so (p 1 V 1 /T 1 ) = (p 2 V 2 /T 2 ) V 2 /V 1 = [(p 1 T 2 )/(p 2 T 1 ) = 373/(3 x 313) ≈0.40 i.e. a 60% reduction in volume of gas

Kinetic Model of Ideal gases- Key assumptions Gas consists of large number of identical tiny particles- molecules in constant random motion Statistical averages of this number can be made Each molecule’s volume is negligible when compared to the volume of the whole gas At any instant as many molecules are moving in one direction as any other direction Molecules undergo perfectly elastic collisions between each other and with the walls of the container; momentum is reversed during collision No intermolecular forces between molecules between collisions i.e. energy is completely kinetic Duration of collision negligible compared with the time between collisions Each molecule produces a force on the wall of the container The forces of individual molecules will average out to produce a uniform pressure throughout the gas- ignoring the effect of gravity

Developing a relation between pressure and density (1) Refer to text p. 107 Figure 7 Consider one molecule which has momentum and collides elastically with the right side of the box of length L ∆p = -2mc x F= ∆p/∆t and ∆t =2L/c x so F=(-mc x 2 /L) is force of box on molecule Newton’s III states molecule exerts an equal and opposite force F=mc x 2 /L on box

Developing a relation between pressure and density (2) N molecules exert a total force F x = (m/L)(c x1 2 + c x2 2 + c x3 2 + … c xN 2 ) The forces average out giving a constant force with so many molecules The mean value of the square of the velocities (c x1 2 + c x2 2 + c x3 2 + … c xN 2 )/N The total force on right hand wall is F x = (m/L)

Developing a relation between pressure and density (3) From Pythagoras,

Molecular interpretation of temperature

Boltzmann Constant

Linking temperature with energy The kinetic theory now links temperature with the microscopic energies of the gas molecules The equation resembles the kinetic energy formula. Adjusting for N molecules gives 3/2 Nk B T This represents the total internal energy of an ideal gas (only considering translational motion of molecules of monoatomic gases)

Alternative equation of state for ideal gas

Real Gases vs Ideal Gases (1) Since an ideal gas obeys the ideal gas laws under all conditions, ideal gases cannot be liquefied In 1863, Andrews experiments showed a deviation from Boyles’ pV curves for CO 2 at high pressures and low temperatures Later experimentation showed that real gases do not behave like ideal gases and that all gases can be liquefied at high pressures and low temperatures

Real Gases vs Ideal gases (2) The ideal gas law describes the behaviour of all gases at relatively low pressures and high temperatures. The ideal gas law fails when the main assumptions of the Kinetic Theory are invalid i.e molecular volumes and intermolecular forces are negligible. Real gases can only be compressed so far indicating that molecules occupy a non negligible volume and weak attractive forces exist between the molecules of real gases just as those between molecules of liquids

Deviation of a real gas from an ideal gas Real Gases vs Ideal gases (3) Source Google images

Real Gases vs Ideal gases (4) Short range repulsive forces act between gas molecules when they approach each other reducing the distance they can effectively move i.e. reducing gas volume below the V used to develop ideal gas law. At slightly greater distances molecules attract each other slightly forming small groups and reducing the effective number of particles. This slightly reduces the pressure. Enter van der Waals who modified the ideal gas law introducing two new constants. No single simple equation of state applying to all gases has been found to this date