Physics 1501: Lecture 36, Pg 1 Physics 1501: Lecture 36 Today’s Agenda l Announcements çHomework #12 (Dec. 9): 2 lowest dropped çMidterm 2 … in class.

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2 Physics 1501: Lecture 36, Pg 1 Physics 1501: Lecture 36 Today’s Agenda l Announcements çHomework #12 (Dec. 9): 2 lowest dropped çMidterm 2 … in class Wednesday l Honors’ students: Wednesdat at 3:30 in my office. l Today’s topics çChap. 17: ideal gas »Kinetic theory »Ideal gas law »Diffusion çChap.18: Heat and Work »Zeroth Law of thermodynamics »First Law of thermodynamics and applications »Work and heat engines

3 Physics 1501: Lecture 36, Pg 2 Kinetic Theory of an Ideal Gas l Pressure is

4 Physics 1501: Lecture 36, Pg 3 l Does a Single Particle Have a Temperature? Each particle in a gas has kinetic energy. On the previous page, we have established the relationship between the average kinetic energy per particle and the temperature of an ideal gas. Is it valid, then, to conclude that a single particle has a temperature? Concept of temperature

5 Physics 1501: Lecture 36, Pg 4 l Air is primarily a mixture of nitrogen N 2 molecules (molecular mass 28.0u) and oxygen O 2 molecules (molecular mass 32.0u). çAssume that each behaves as an ideal gas and determine the rms speeds of the nitrogen and oxygen molecules when the temperature of the air is 293K. çFor nitrogen Example: Example: Speed of Molecules in Air

6 Physics 1501: Lecture 36, Pg 5 Internal energy of a monoatomic ideal gas l The kinetic energy per atom is l Total internal energy of the gas with N atoms

7 Physics 1501: Lecture 36, Pg 6 Kinetic Theory of an Ideal Gas: summary l Assumptions for ideal gas: çNumber of molecules N is large çThey obey Newton’s laws (but move randomly as a whole) çShort-range interactions during elastic collisions çElastic collisions with walls çPure substance: identical molecules l Temperature is a direct measure of average kinetic energy of a molecule l Microscopic model for a gas çGoal: relate T and P to motion of the molecules

8 Physics 1501: Lecture 36, Pg 7 l Theorem of equipartition of energy çEach degree of freedom contributes k B T/2 to the energy of a system (e.g., translation, rotation, or vibration) Kinetic Theory of an Ideal Gas: summary  l Total translational kinetic energy of a system of N molecules çInternal energy of monoatomic gas: U = K ideal = K tot trans l Root-mean-square speed:

9 Physics 1501: Lecture 36, Pg 8 l Consider a fixed volume of ideal gas. When N or T is doubled the pressure increases by a factor of 2. 1) If T is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? b) x2a) x1.4c) x4 2) If N is doubled, what happens to the rate at which a single molecule in the gas has a wall bounce? b) x1.4a) x1c) x2 Lecture 36: ACT 1

10 Physics 1501: Lecture 36, Pg 9 Diffusion l The process in which molecules move from a region of higher concentration to one of lower concentration is called diffusion. çInk droplet in water

11 Physics 1501: Lecture 36, Pg 10 Why is diffusion a slow process ? l A gas molecule has a translational rms speed of hundreds of meters per second at room temperature. At such speed, a molecule could travel across an ordinary room in just a fraction of a second. Yet, it often takes several seconds, and sometimes minutes, for the fragrance of a perfume to reach the other side of the room. Why does it take so long? çMany collisions !

12 Physics 1501: Lecture 36, Pg 11 Comparing heat and molecule diffusion l Both ends are maintained at constant concentration/temperature

13 Physics 1501: Lecture 36, Pg 12 concentration gradient between ends diffusion constant SI Units for the Diffusion Constant: m 2 /s Fick’s law of diffusion l For heat conduction between two side at constant T L ThTh TcTc A Energy flow conductivity temperature gradient between ends l The mass m of solute that diffuses in a time t through a solvent contained in a channel of length L and cross sectional area A is

14 Physics 1501: Lecture 36, Pg 13 Chap. 18: Work & 1 st Law The Laws of Thermodynamics 0) If two objects are in thermal equilibrium with a third, they are in equilibrium with each other. 1) There is a quantity known as internal energy that in an isolated system always remains the same. 2) There is a quantity known as entropy that in a closed system always remains the same (reversible) or increases (irreversible).

15 Physics 1501: Lecture 36, Pg 14 Zeroth Law of Thermodynamics l Thermal equilibrium: when objects in thermal contact cease heat transfer çsame temperature T1T1 T2T2 U1U1 U2U2 = If objects A and B are separately in thermal equilibrium with a third object C, then objects A and B are in thermal equilibrium with each other. A C B

16 Physics 1501: Lecture 36, Pg 15 1 st Law: Work & Heat l Two types of variables çState variables: describe the system (e.g. T, P, V, U). çTransfer variables: describe the process (e.g. Q, W). =0 unless a process occurs  change in state variables. l Work done on gas çW = F d cos  = -F  y = - PA  y = - P  V çvalid only for isobaric processes (P constant) çIf not, use average force or calculus: W = area under PV curve PV diagram

17 Physics 1501: Lecture 36, Pg 16 1 st Law: Work & Heat l Work: çDepends on the path taken in the PV-diagram çSame for Q (heat)

18 Physics 1501: Lecture 36, Pg 17 b) = |W 1 |a) > |W 1 |c) < |W 1 | i f p V 2 1 l Consider the two paths, 1 and 2, connecting points i and f on the pV diagram. çThe magnitude of the work, |W 2 |, done by the system in going from i to f along path 2 is Lecture 36: Act 2 Work

19 Physics 1501: Lecture 36, Pg 18 First Law of Thermodynamics l Isolated system çNo interaction with surroundings çQ = W = 0   U = 0. çU f = U i : internal energy remains constant. l First Law of Thermodynamics  U = Q + W variation of internal energy heat flow “in” (+) or “out” (-) work done “on” the system çIndependent of path in PV-diagram çDepends only on state of the system (P,V,T, …) çEnergy conservation statement  only U changes

20 Physics 1501: Lecture 36, Pg 19 Other Applications l Cyclic process: çProcess that starts and ends at the state çMust have  U = 0  Q = -W. l Adiabatic process: çNo energy transferred through heat  Q = 0. çSo,  U = W. çImportant for »expansion of gas in combustion engines »Liquifaction of gases in cooling systems, etc. l Isobaric process: (P is constant) çWork is simply

21 Physics 1501: Lecture 36, Pg 20 Other Applications (continued) l Isovolumetric process: çConstant volume  W =0. çSo  U = Q  all heat is transferred into internal energy »e.g. heating a “can” (no work done). l Isothermal process: çT is constant çUsing PV=nRT, we find P= nRT/V. çWork becomes : çPV is constant. çPV-diagram: isotherm.

22 Physics 1501: Lecture 36, Pg 21 p V l Identify the nature of paths A, B, C, and D çIsobaric, isothermal, isovolumetric, and adiabatic Lecture 36: Act 3 Processes A B C D T1T1 T2T2 T3T3 T4T4

23 Physics 1501: Lecture 36, Pg 22 Heat Engines l We now try to do more than just raise the temperature of an object by adding heat. We want to add heat to get some work done! l Heat engines: çPurpose: Convert heat into work using a cyclic process çExample: Cycle a piston of gas between hot and cold reservoirs * (Stirling cycle) 1) hold volume fixed, raise temperature by adding heat 2) hold temperature fixed, do work by expansion 3) hold volume fixed, lower temperature by draining heat 4) hold temperature fixed, compress back to original V

24 Physics 1501: Lecture 36, Pg 23 Heat Engines l Example: the Stirling cycle Gas T=T H Gas T=T H Gas T=T C Gas T=T C V P TCTC THTH VaVa VbVb 12 3 4 We can represent this cycle on a P-V diagram: 1 1 2 3 4 * reservoir: large body whose temperature does not change when it absorbs or gives up heat

25 Physics 1501: Lecture 36, Pg 24 l Identify whether çHeat is ADDED or REMOVED from the gas çWork is done BY or ON the gas for each step of the Stirling cycle: V P TCTC THTH VaVa VbVb 12 3 4 ADDED REMOVED BY ON 1 HEAT WORK step ADDED REMOVED BY ON 2 ADDED REMOVED BY ON 3 ADDED REMOVED BY ON 4 Heat Engines