Announcements 1. Please bring your laptops to lab this week.

Please download to get full document.

View again

of 22
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Information Report
Category:

Thermodynamics

Published:

Views: 0 | Pages: 22

Extension: PDF | Download: 0

Share
Related documents
Description
Announcements 1. Please bring your laptops to lab this week. 2. Physics seminar this week – Thursday, Nov. 20 at 4 PM – Professor Brian Matthews will…
Transcript
Announcements 1. Please bring your laptops to lab this week. 2. Physics seminar this week – Thursday, Nov. 20 at 4 PM – Professor Brian Matthews will discuss relationships between protein structure and function 3. Schedule – Today, Nov. 18th: Examine Eint especially for ideal gases Discuss ideal gas law Continue discussion of first law of thermodynamics Thursday, Nov. 20th: Review Chapters 15-20 Tuesday, Nov. 25th: Third exam 11/18/2003 PHY 113 -- Lecture 20 1 From The New Yorker Magazine, November 2003 11/18/2003 PHY 113 -- Lecture 20 2 First law of thermodynamics ∆Eint = Q − W Vf W = ∫ PdV Vi For an “ideal gas” we can write an explicit relation for Eint. What we will show: ( ideal gas ) n N Eint = RT = k BT γ-1 γ-1 γ is a parameter which depends on the type of gas (monoatomic, diatomic, etc.) which can be measured as the ratio of two heat capacities: γ=CP/CV. 11/18/2003 PHY 113 -- Lecture 20 3 Thermodynamic statement of conservation of energy – First Law of Thermodynamics ∆Eint = Q - W Work done by system Wif depends on path Heat added to system Qif depends on path “Internal” energy of system ∆Eint = Eint(f)−Εint(i) Î independent on path 11/18/2003 PHY 113 -- Lecture 20 4 How is temperature related to Eint? Consider an ideal gas Î Analytic expressions for physical variables Î Approximates several real situations Ideal Gas Law: PV=nRT temperature (K) volume (m3) gas constant (8.31 J/(mole ⋅K)) pressure (Pa) number of moles 11/18/2003 PHY 113 -- Lecture 20 5 Ideal gas – P-V diagram at constant T PV=nRT Pi Pf Pi Vi = Pf Vf Vi Vf 11/18/2003 PHY 113 -- Lecture 20 6 11/18/2003 PHY 113 -- Lecture 20 7 Microscopic model of ideal gas: Each atom is represented as a tiny hard sphere of mass m with velocity v. Collisions and forces between atoms are neglected. Collisions with the walls of the container are assumed to be elastic. 11/18/2003 PHY 113 -- Lecture 20 8 What we can show is the pressure exerted by the atoms by their collisions with the walls of the container is given by: 2N 1 2 2N P= 2 m v avg = K avg 3V 3V Proof: Force exerted on wall perpendicular to x-axis by an atom which collides with it: ∆pix 2mi vix Fix = − = ∆t ≈ 2d / vix ∆t ∆t -vix 2mi vix mi vix2 ⇒ Fix ≈ = vix 2d / vix d x 2 number of atoms Fix mi vix N P=∑ =∑ = mi v x2 d i A i dA V volume average over atoms 11/18/2003 PHY 113 -- Lecture 20 9 Ideal gas law continued: Recall that N 2 1N 2 2 N ⎧ 1R 2 ⎫ P = m v = Macroscopic relation x m v = : PV = nRT = N⎨ m Tv = ⎬Nk BT V 3V 3 V ⎩ 2N ⎭ A 2 1 Microscopic model : PV = N ⎧⎨ m v 2 ⎫⎬ = Nk BT 3 ⎩2 ⎭ Therefore: ⎧ 1 2 ⎫ 3 ⎨ m v ⎬ = k BT ⎩2 ⎭ 2 2 3k BT ⇒ vrms = v = m Also: ⎧1 m v 2 ⎫ = 3 k T ⎨ ⎬ B ⎩ 2 ⎭ 2 ⎧ 1 2 ⎫ 3 ⇒ Eint = N ⎨ m v ⎬ = N k BT ⎩2 ⎭ 2 11/18/2003 PHY 113 -- Lecture 20 10 Big leap! Internal energy of an ideal gas: 1 3 Eint = N ⎧⎨ m v 2 ⎫⎬ = N k BT ⇒ N n k BT = RT ⎩2 ⎭ 2 γ-1 γ-1 derived for monoatomic ideal gas more general relation for polyatomic ideal gas Gas γ (theory) γ (exp) He 5/3 1.67 N2 7/5 1.41 H2O 4/3 1.30 11/18/2003 PHY 113 -- Lecture 20 11 Determination of Q for various processes in an ideal gas: n Eint = RT γ-1 n ∆Eint = R∆T = Q − W γ-1 Example: Isovolumetric process – (V=constant Î W=0) n ∆Eint i → f = R∆Ti → f = Qi → f γ-1 n In terms of “heat capacity”: Qi → f = R∆Ti → f ≡ nCV ∆Ti → f γ-1 R CV = γ-1 11/18/2003 PHY 113 -- Lecture 20 12 Example: Isobaric process (P=constant): n ∆Eint i → f = R∆Ti → f = Qi → f − Wi → f γ-1 In terms of “heat capacity”: R∆Ti → f + Pi (V f − Vi ) = n n Qi → f = R∆Ti → f + nR∆Ti → f ≡ nC P ∆Ti → f γ-1 γ-1 R γR ⇒ CP = +R= γ-1 γ-1 Note: γ = CP/CV 11/18/2003 PHY 113 -- Lecture 20 13 More examples: Isothermal process (T=0) n Eint = RT γ-1 n ∆Eint = R∆T = Q − W γ-1 Î∆T=0 Î ∆Eint = 0 Î Q=W Vf dV Vf ⎛V f ⎞ W = ∫ PdV = nRT ∫ =nRT ln⎜⎜ ⎟⎟ Vi Vi V ⎝ Vi ⎠ 11/18/2003 PHY 113 -- Lecture 20 14 Even more examples: Adiabatic process (Q=0) ∆Eint = −W n R∆T = − P∆V γ-1 PV = nRT ∆PV + P∆V = nR∆T nR∆T = −(γ-1)P∆V = ∆PV + P∆V ∆V ∆P −γ = V P ⎛ V fγ ⎞ ⎛ Pf ⎞ ⎜ ⎟ ⇒ − ln γ = ln⎜⎜ ⎟⎟ ⇒ PiVi γ = Pf V fγ ⎜V ⎟ ⎝ Pi ⎠ ⎝ i ⎠ 11/18/2003 PHY 113 -- Lecture 20 15 PiVi γ Adiabat : P = γ V PiVi Isotherm : P = V 11/18/2003 PHY 113 -- Lecture 20 16 Peer instruction question Suppose that an ideal gas expands adiabatically. Does the temperature (A) Increase (B) Decrease (C) Remain the same PiVi γ = Pf V fγ Ti PiVi = nRTi ⇒ Pi = nR Vi TiVi γ-1 = T f V fγ-1 γ -1 ⎛ Vi ⎞ T f = Ti ⎜⎜ ⎟ ⎟ ⎝V f ⎠ 11/18/2003 PHY 113 -- Lecture 20 17 Review of results from ideal gas analysis in terms of the specific heat ratio γ ≡ CP/CV: n R ∆Eint = R∆T = nCV ∆T ; CV = γ-1 γ-1 γR CP = γ-1 For an isothermal process, ∆Eint = 0 Î Q=W Vf ⎛V f ⎞ ⎛V f ⎞ W = ∫ PdV =nRT ln⎜⎜ ⎟⎟ = PiVi ln⎜⎜ ⎟⎟ Vi ⎝ Vi ⎠ ⎝ Vi ⎠ For an adiabatic process, Q = 0 PiVi γ = Pf V fγ TiVi γ-1 = T f V fγ-1 11/18/2003 PHY 113 -- Lecture 20 18 Extra credit: Show that the work done by an ideal gas which has an initial pressure Pi and initial volume Vi when it expands adiabatically to a volume Vf is given by: γ −1 Vf PiVi ⎜ ⎛⎜ Vi ⎞⎟ ⎞⎟ ⎛ W = ∫ PdV = 1− ⎜ ⎟ γ − 1⎜ ⎝ V f ⎠ ⎟ Vi ⎝ ⎠ 11/18/2003 PHY 113 -- Lecture 20 19 Peer instruction questions Match the following types of processes of an ideal gas with their corresponding P-V relationships, assuming the initial pressures and volumes are Pi and Vi, respectively. 1. Isothermal 2. Isovolumetric 3. Isobaric 4. Adiabatic (A) P=Pi (B) V=Vi (C) PV=PiVi (D)PVγ=PiViγ 11/18/2003 PHY 113 -- Lecture 20 20 Examples process by an ideal gas: Pf B C A→B B→C C→D D→A P (1.013 x 105) Pa Q Vi ( Pf − Pi ) γPf (V f − Vi ) − V f ( Pf − Pi ) -γPi (V f − Vi ) γ -1 γ -1 γ -1 γ -1 W 0 Pf(Vf-Vi) 0 -Pi(Vf-Vi) Pi A D ∆Eint Vi ( Pf − Pi ) Pf (V f − Vi ) − V f ( Pf − Pi ) -Pi (V f − Vi ) γ -1 γ -1 γ -1 γ -1 Vi Efficiency as an engine: Vf e = Wnet/ Qinput 11/18/2003 PHY 113 -- Lecture 20 21 11/18/2003 PHY 113 -- Lecture 20 22
Recommended
View more...
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks