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Lecture 18 – 1/32 • • • • • • • • • • • Physics Topics Phys 220 Kinematics Forces, Torque, and Work Conservation of Energy Conservation of Momentum (Linear & Angular) Density and Pressure Harmonic Motion Traveling Waves Standing Waves Heat and Temperature Ideal Gas Law Thermodynamics Lecture 18 – 2/32 Molecular Picture of Gas Phys 220 • Gas is made up of many individual molecules • Number of moles n = N / NA NA = Avogadro’s number = 6.0221023 mole-1 NA = number of particles (can be atoms or molecules) per mole 1 mole = amount of substance that contains as many elementary entities as there are atoms in exactly 12 grams of carbon-12 Lecture 18 – 3/32 Atoms, Molecules and Moles Phys 220 • 1 mole = 6.022 1023 molecules (NA = Avogadro’s Number) • NA = Number of atoms or molecules that make a mass equal to the substance's atomic or molecular weight in grams. • 1 u = 1 atomic mass unit = (mass of 12C atom)/12 ▪ Approximately # of neutrons + # of protons ▪ Atomic weight A • 1 u = 1.66 10-27 kg = 1gram/NA • Mass of 1 mole of “stuff” in grams = molecular mass in u ▪ E.g. 1 mole of N2 has mass of 2 14 = 28 grams Lecture 18 – 4/32 Phys 220 Question 4 Which contains more molecules ? A) A mole of water (H2O) B) A mole of oxygen gas (O2) C) Same H2O O2 Lecture 18 – 5/32 Phys 220 Question 5 Which contains more atoms ? A) A mole of water (H2O) B) A mole of oxygen gas (O2) C) Same H2O (3 atoms) O2 (2 atoms) Lecture 18 – 6/32 Phys 220 Question 6 Which weighs more ? A) A mole of water (H2O) B) A mole of oxygen gas (O2) C) Same (M=16+1+1) H2O O2 (M=16+16) Lecture 18 – 7/32 Internal Energy Phys 220 • All objects have “internal energy” (measured in Joules) ▪ random motion of molecules » kinetic energy + energy associated with intermolecular bonds + potential energy ▪ collisions of molecules gives rise to pressure • Amount of internal energy depends on ▪ temperature » related to average kinetic energy per molecule ▪ how many molecules » mass ▪ “specific heat” » related to how many different ways a molecule can move ‒ Translation (the only one for monatomic gas) ‒ Rotation ‒ Vibration » the more ways it can move, the higher the specific heat Lecture 18 – 8/32 Ideal Gas: Experiments • Avogadro’s Law ▪ constant P and T ▪ V is proportional to N (where N is the number of molecules) • Boyle’s Law ▪ constant T ▪ P V = constant • Charles’s Law ▪ constant P ▪ change in V is proportional to a change in T • Gay-Lussac’s Law ▪ constant V ▪ change in P is proportional to a change in T Phys 220 Lecture 18 – 9/32 The Ideal Gas Law • P V = N kB T P = pressure in N/m2 (or Pascals) V = volume in m3 N = number of molecules T = absolute temperature in K k B = Boltzmann’s constant = 1.38 x 10-23 J/K Note: P V has units of N*m or J (energy!) Phys 220 Lecture 18 – 10/32 The Ideal Gas Law Phys 220 • P V = N kBT • Another way to write this N (number of molecules) = n (number of moles) x NA (molecules/mole) P V = N kB T = (n NA) kB T = n (NA kB) T = n R T •PV=nRT n = number of moles R = ideal gas constant = NA kB = 8.31 J/(mol*K) Lecture 18 – 11/32 The Ideal Gas Law Phys 220 PV N kB T • N = number of molecules • kB = Boltzmann’s constant = 1.38 x 10-23 J/K PV n R T • n = number of moles • R = ideal gas constant = 8.31 J/(mol*K) N n NA R kB NA Lecture 18 – 12/32 Phys 220 Exercise You inflate the tires of your car so the pressure is 30 psi, when the air inside the tires is at 20 degrees C. After driving on the highway for a while, the air inside the tires heats up to 38 C. Which number is closest to the new air pressure? A) 16 psi Pf P0 B) 32 psi C) 57 psi Tf T0 Tf Pf P0 T0 Careful, you need to use the temperature in K! 38 273 Pf (30 psi) 32 psi 20 273 Lecture 18 – 13/32 Phys 220 Question What happens to the pressure of the air inside a hot-air balloon when the air is heated? (Assume V is constant) A) Increases B) Same C) Decreases Balloon is still open to atmospheric pressure, so it stays at 1 atm Lecture 18 – 14/32 Phys 220 Question What happens to the buoyant force on the balloon when the air is heated? (Assume V remains constant) A) Increases B) Same FB = r V g r is density of outside air! C) Decreases Lecture 18 – 15/32 Phys 220 Question What happens to the number of air molecules inside the balloon when the air is heated? (Assume V remains constant) A) Increases B) Same C) Decreases PV = NkBT P and V are constant. If T increases N decreases. Lecture 18 – 16/32 Balloon Phys 220 • In terms of the ideal gas law, explain how a hot air balloon works. ▪ Air in the balloon gets hot » Net weight of the balloon plus hot air is less than the same volume of cold air » This results in the balloon rising ▪ Temperature increases, then volume of the gas increases » Density of gas decreases since mass decreases and volume is constant » Density of the hot air in the balloon is less than the density of the surrounding air Note: This is not a pressure effect. It is a density effect. As T increases, density decreases. The balloon then floats due to Archimedes principle. The pressure remains constant. Lecture 18 – 17/32 Ideal Gas • Model of particles in an ideal gas Phys 220 Lecture 18 – 18/32 Kinetic Theory Phys 220 • Helps to explain gas laws by applying Newton’s laws to the microscopic molecular motions p=2mv (momentum transfer for one collision) Lecture 18 – 19/32 Maxwell-Boltzmann Distribution Phys 220 • How many molecules have speeds in a certain range? Lecture 18 – 20/32 Phys 220 Kinetic Theory • The relationship between energy and temperature (for a monatomic ideal gas) Force is caused by collisions and is given by: p x 2 m vx p 2 m vx F t t Total Force for N molecules depends on the number reaching the +x wall: N d traveled N vx t N x, reaching wall 6 L 6 L 2 m vx N vx t N m vx2 Ftotal t 6 L 3L L Pressure is given by: F N m vx2 N m vx2 P A 3A L 3V N m vx2 PV 3 Remember: PV N kB T N m vx2 N kB T 3 Kinetic Energy: 1 3 2 KE N m vx N k B T 2 2 Lecture 18 – 21/32 Phys 220 Kinetic Theory • For N molecules: • For one molecule: 1 3 2 KE N m vx N k B T 2 2 1 3 2 KE m vx k B T 2 2 • Solving for the rms velocity of one molecule in a monatomic gas: vrms 3 kB T m • Internal Energy (U) ▪ The kinetic energy is held “internally” by the gas molecules 3 3 U N kB T n R T 2 2 Only for a monatomic gas! Lecture 18 – 22/32 Internal Energy Phys 220 • Energy of all molecules including ▪ Translational and rotational kinetic energy of Only molecules due to their individual random motions. translational ▪ Vibrational energy (both kinetic and potential) of kinetic energy molecules and of atoms within molecules due to considered for a random vibrations about their equilibrium points. monatomic gas! ▪ Potential energy due to interactions between the atoms and molecules of the systems. ▪ Chemical and nuclear energy (kinetic and potential energy associated with the binding of atoms to form molecules, the binding of electrons to nuclei to form atoms, and the binding of protons and neutrons to form nuclei). • DOES NOT INCLUDE ▪ Macroscopic motion of object ▪ Potential energy due to interactions with other objects Lecture 18 – 23/32 Example Phys 220 • What is the rms speed of a nitrogen N2 molecule in this classroom? vrms 3 kB T m This equation is for the velocity of one molecule in a gas! m is the mass of one molecule of N2 gas: 2 (14.01 g) 4.65 1023 g 4.65 1026 kg NA 1.66 10-27 kg m 2 (14.01 u ) 4.65 1026 kg 1u m vrms 3 kB T 3 (1.38 1023 J/K ) (293 K) m 4.65 1026 kg Temperature must be in Kelvin. (Room temperature is about 20 °C) v = 510 m/s = 1150 mph! Lecture 18 – 24/32 Phys 220 Question Suppose you want the rms (root-mean-square) speed of molecules in a sample of gas to double. By what factor should you increase the temperature of the gas? A) 2 B) √2 C) 4 1 3 2 KE m vx k B T 2 2 vrms 3 kB T m Lecture 18 – 25/32 Phys 220 Diffusion 2 r t D • Diffusion is the motion of a particle in a random zig-zag path as it collides with other particles in a gas • D is the Diffusion Constant and D depends on ▪ v (typical particle speed) ▪ l (mean free path) » Mean free path is the average distance between collisions » depends on ρ (density) Lecture 18 – 26/32 Ideal Gas Law PV N kB T NA = 6.02 x 1023 mol-1 N = number of molecules kB = Boltzmann’s constant = 1.38 x 10-23 J/K PV n R T n = number of moles R = ideal gas constant = 8.31 J/(mol*K) Pressure in Pa Volume in m3 Temperature in K Phys 220 Room Temperature is 20°C = 293 K N n NA R kB NA 9 TF TC 32 5 TK TC 273.15 Lecture 18 – 27/32 • • • • • • • • • • • Physics Topics Phys 220 Kinematics Forces, Torque, and Work Conservation of Energy Conservation of Momentum (Linear & Angular) Density and Pressure Harmonic Motion Traveling Waves Standing Waves Heat and Temperature Ideal Gas Law Thermodynamics Lecture 18 – 28/32 First Law of Thermodynamics Phys 220 • The change in the internal energy of a system (∆U) is equal to the heat flow into the system (Q) minus the work done by the system (W) ∆U = Q – W Increase in internal energy of system Heat flow into system Work done by system U Q W Lecture 18 – 29/32 First Law of Thermodynamics • Energy Conservation U Q W Q U W Phys 220 Lecture 18 – 30/32 Signs Example Phys 220 You are heating some soup in a pan on the stove. To keep it from burning, you also stir the soup. Apply the 1st law of thermodynamics to the soup. What is the sign of A) Q Positive, heat flows into soup B) W Negative, stirring does work on soup C) U Positive, soup gets warmer Lecture 18 – 31/32 Work Done by a System Phys 220 M M • W > 0 if V > 0 y W = F d cos q= P A d = = P A y = P V ▪ expanding system does positive work • W < 0 if V < 0 ▪ contracting system does negative work • W = 0 if V = 0 ▪ system with constant volume does no work W P V Lecture 18 – 32/32 • • • • • Work on P-V Diagram Phys 220 Pressure vs Volume = PV Diagram The initial state is a point on the diagram The process is the line or curve W P V The final state is another point Work = P ∆V which is the area under the curve Positive Work Negative Work