Lecture 14

• Interaction of 2 systems at different temperatures

• Irreversible processes: 2nd Law of Thermodynamics

• Chapter 19: Heat Engines and Refrigerators

Thermal interactions

• T’s change via collisions at boundary (not mechanical interaction)

• elastic collision (total energy conserved): energy transfer from faster atom to slower atom

• on average, energy transferred from 1 to 2 ( )

Etot = E1i + E2i constant withE1i = 3

2n1RT1i and E2i = 32n2RT2i

T1i > T2i

Equilibrium

• 2 systems reach common final T due to energy exchange via atomic collisions (in reality via wall, but still no mechanical interaction)

E1f

N1= E1f

N1= Etot

N1+N2⇒

E1f = N1N1+N2

Etot; E2f = N2N1+N2

Etot

W = 0 (barrier does not move) ⇒ 1st law givesQ1 = ∆E1 = E1f − E1i; Q2 = ∆E2 = E2f − E2i

and Q1 = −Q2 (energy conservation)

Example• 2.0 g of helium at an initial temperature of 300 K

interacts thermally with 8.0 g of oxygen at an initial temperature of 600 K. (a) What is the initial thermal energy of each gas? (b) What is the final thermal energy of each gas (c) How much heat is transferred, and in which direction? (d) What is the final temperature?

Irreversible processes

• heat not transferred cold to hot (conserves energy)

• reversible microscopic (molecular) motions irreversible macroscopic phenomena? New law (past vs. future)

Statistics of (Very) Large Numbers

• small probability for 1 to increase, especially if

• net result of many collisions is to transfer energy from 1 to 2

• equilibrium state is most probable

N1 = 1020

2nd Law of Thermodynamics• negligible probability (not impossible) for atoms to spontaneously

order: more random arrangements...

• state variable entropy: probability for macroscopic state to occur (measures disorder)

• 2 systems with different T’s: lower entropy (more order) than...

• formally: entropy of isolated system never decreases (can order by reaching from outside e.g. refrigerator for cold to hot)

• order turns into disorder;information lost; “system runs down” (vastly more random states: laws of probability)

• informally (i): heat transferred from hotter to colder

• informally (ii): irreversible evolution from less-likely to more-likely macroscopic state gives time direction (entropy increases is “future”)

Chapter 19• physical principles for all heat engines (transform heat

energy into work) and refrigerators (uses work to move heat from cold to hot)

• 2nd law: limit on efficiency (Carnot cycle)

• general concepts of turning heat into work; heat engines and refrigerators

Today

Heat Work• thermodynamics: transformation of energy e.g. heat into work

obeys (i)1st law (energy conservation): (ii) 2nd law: heat flows from hotter to colder (spontaneously)

• Work done by system, (vs. work done on system by external force, W: heat and work are 2 ways to transfer energy to system)equilibrium: F̄gas = −F̄ext ⇒ Ws = −W = the area under the pV curveWs > 0 (W < 0) during expansion (energy transferred out of system)1st law: Q = Ws + ∆Eth (heat used to do work or stored as thermal)

∆Eth = W + Q

Ws

Energy Transfer diagrams

• energy reservoir (hot or cold): much larger than system, temperature does not change when heat transferred between it and system due to difference in temperaturesQH, C(> 0) = heat transferred to/from a hot/cold reservoirQ = −QC in 1st law (heat transferred from system...)1st law: Q = Ws + ∆Eth refers to systemQ = QH −QC ; Ws = 0; ∆Eth = 0 (steady state) ⇒QH = QC (system provides route for energy transfer from hot to cold)heat transferred from cold to hot: 1st law not violated if QH = QC ,but 2nd law does not allow spontaneous transfer...

Efficiency of Heat Work

• 100 % efficient: e.g. warm up rocks from ocean by rubbing ( ); back into ocean ( ); continue as long as there is motion

• isothermal expansion: 100% efficient, but one-time process (piston reaches end of cylinder)

• practical device must return to initial state for continued use, but 2nd law does not allow perfect engine (100% efficient): asymmetry of 2 conversions similar to heat transfer

Work into heat

Heat into Work

W → ∆Eth

∆Eth → QC

Heat engines• closed cycle device (e.g. car engine: p, T

inside cylinder repeated) extracts heat (combustion of fuel); does useful work (move pistons...); exhausts heat (radiator...): all state variables return to initial once every cycle

• thermal efficiency

• perfect engine ( ) not possible: must exhaust energy (waste heat:

(∆Eth)net = 0 (over 1 full cycle)1st law: (∆Eth)net = Qnet −Wout

with Qnet = QH −QC ⇒(energy

conservation)

= 1− QC

QH

η = 1

energy extracted from hot reservoir, not transformed into work)

A Heat-Engine Example• useful work of lifting mass during

isobaric expansion...step (e): no net change in gas (start lifting mass again)

• heat engines require source and sink

• reservoirs not explicitly shown: highest system temperature; coldest...

TH >TC <