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AP Unit II C 1

AP Unit II C 1. Ideal Gases. a) Students should understand the kinetic theory model of an ideal gas, so they can: (1) State the assumptions of the model.

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AP Unit II C 1

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  1. AP Unit II C 1 Ideal Gases

  2. a) Students should understand the kinetic theory model of an ideal gas, so they can: • (1) State the assumptions of the model. • (2) State the connection between the temperature and the mean translational kinetic energy, and apply it to determine the mean speed of gas molecules as a function of their mass and the temperature of the gas.

  3. (3) State the relationship between Avogado’s number, Boltzmann’s constant, and the gas constant R, and express the energy of a mole of a monatomic ideal gas as a function of its temperature. • (4) Explain qualitatively how the model explains the pressure of a gas in terms of collisions with the container walls, and explain how the model predicts that, for a fixed volume, pressure must be proportional to temperature.

  4. b ) Students should know how to apply the ideal gas law and thermodynamic principles, so they can: • (1) Relate the pressure and volume of a gas during an isothermal expansion or compression. • (2) Relate the pressure and temperature of a gas during constant-volume heating or cooling, or the volume and temperature constant-pressure heating or cooling.

  5. (3) Calculate the the work performed on or by a gas during an expansion or compression at constant pressure. • (4) Understand the process of adiabatic expansion or compression of a gas. • (5) Identify or sketch on a PV diagram the curves that represent each of the above processes.

  6. The three Gas Laws • Boyle’s Law • Volume is inversely proportional to pressure at constant temperature. • P α 1/V • Charles’ Law • Volume is proportional to ABSOLUTE temperature at constant pressure (isobaric) • VαT • Gay-Lussac Pressure Law • Pressure is proportional to absolute temperature at constant volume • P α T

  7. Universal Gas Law • From the previous three laws we come up with the combined gas law: • P1 V1 / T1 = P2 V2 / T2 = constant = nR • where n is the number of moles and R is the universal Gas constant : 8.31 J/mol K • or • PV = nRT

  8. Kinetic Theory Model • Assumptions: • 1. Molecules are point objects. Separation is large compared to their dimensions • 2. Move randomly • 3. Undergo elastic collisions • 4. They exert negligible forces on each other except during collisions • 5. The gas is a pure substance

  9. Molecule in a box Molecule mass m d Force on the walls of the box F = rate of change of momentum = 2mvx /Δt vx d d d

  10. Δt = 2d/vx • Force = 2mvx/(2d/vx) = mvx2/d • The average velocity for all N molecules is called the root mean square velocity v (v bar) • (the usual average velocity would just be zero because molecules would be going in opposite directions)

  11. Thus force in x direction becomes Fx = m/d(Nvx2) • v2 = vx2 +vy2 +vz2 = 3vx2 • vx = v2/3 so on one wall F = N/3 (mv2/d) • Pressure = F/A = F/d2 = 1/3 (N/d3) mv2 • Since Kinetic Energy = ½ mv2 we can write P = 2/3 (N/V) (1/2 m v2)

  12. Since temperature is proportional to kinetic energy Then PV = 2/3 N (1/2mv2) or • PV = NkBT • where kB is the Boltzmann’s Constant and has the value of 1.38 x 10-23J/K • or PV = nRt where n is the number of moles (n=N/NA=mass/mass of one mole) • and R is the universal gas constant =8.31j/mol · K • Kinetic energy (1/2 mv2) is UK = (3/2) nRt • root mean square velocity vrms = (3RT/M)

  13. 1. • What is the pressure of 4 moles of gas of volume 2 m3 at a temperature of 273 K? • What is the pressure of 8 grams of oxygen gas at a temperature of 325 K? • What is the temperature of 2 moles of nitrogen at a pressure of 1 x 105 Pa.

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