BIT 1002 Thermodynamics

Rubber Band heat engine (Haverford college)

Thermodynamics: this will introduce you to First Law Heat engines Second Law Entropy

What is heat?

The First Law of Thermodynamics

• When a gas expands, its energy can change, and it can change the energy of its surroundings

  Change in internal energy of a system =Heat input - work done
  \color{red}{ \Delta U = Q - W}
• Internal Energy U can be calculated for a gas
• \color{red}{ U = \frac{3}{2}nRT}
• If the piston is allowed to move,then the gas will do work
• This is pretty obvious: it's essentially conservation of energy

Thermodynamic Processes

Force is \color{red}{F = PA} Work done by an ideal gas:if the piston is allowed to move,then the gas will do work: \color{red}{W.D. = F\Delta L} Difficulty is that pressure can change as we go along

In general, whenever we move on the PV diagram, we will do work

Isovolumetric (easiest). No change in volume so no work done. (would need to cool gas, hold piston steady)

Isobaric: For a displacement \color{red}{\Delta L} , \color{red}{W.D. = F\Delta L,F = PA,\Delta V = A\Delta L} then \color{red}{W = P\Delta V}

Isothermal process: one which occurs at a constant temp \color{red}{ P_1 V_1 = nRT = P_2 V_2 } so \color{red}{ \Delta U = 0 = Q - W} i.e. work done by the gas = energy added to the gas

• To evaluate the W.D needs calculus (since the pressure changes): one gets

  W.D. = F dL= P A dV =  nRT dV 
                    A     V
  =  nRT ln (V1/V0)

• or

  W = nRT ln (P0/P1)
  

• Pressure drops as the gas expands so P₀> P₁, so positive work is done
• e.g. 2 moles of a gas expand from 3 litres to 5 litres in an isothermal process at room temp.
• What is the initial and final pressures?
• How much work is done?

no heat is lost or gained. Simulation is actually an adiabatic one

First Law of Thermodynamics:

Change in internal energy of a system =Heat input - work done

• For an ideal gas,

  U = 3/2 NkT
  

• This result is almost obvious if we think in terms of the kinetic theory
• e.g. Allow gas to expand, doing work, without allowing temp. to change: this is an isothermal expansion
• \color{red}{ \Delta U = 0 \Rightarrow Q = W}
• is simply saying internal energy of gas does not change, so Heat added = work done by gas
• e.g. Allow gas to expand, doing work, without adding heat: this is an adiabatic expansion
• \color{red}{ Q = 0 \Rightarrow \Delta U = - W}
  is simply saying, KE of particles is changing
• change in K.E. = work done by external forces

Summary

When you pump up your bike tires, the base of the pump gets hot. THis is because

1. It is an adiabatic expansion of the air
2. It is an adiabatic compression of the air
3. It is an isothermal expansion of the air
4. It is an isothermal compression of the air

• P vs V curves are steeper for an adiabatic process, because the gas will cool down (T0 < T1)

• Any other way to get from an initial pressure and volume to a final one

• Change in internal energy of a gas in going from P0,V0 to P1,V1 is indep. of how we do it e.g.

• 1. "expand at constant pressure" (which means adding heat) followed by "reduce pressure at constant volume" which means cooling.
  2. "reduce pressure at constant volume" which means cooling followed by "expand at constant pressure" (which means adding heat)
  3. "expand at constant temperature" (which means adding heat)

• This result looks a lot like the result that we needed to introduce P.E., where the W.D. is independent of the path (if the force is conservative).
• However, W.D. by the gas is not the same by going along different paths, and by adding and subtracting heat, we can get an ideal gas to do work.

  hence convert heat into mechanical work. e.g. in the previous diag, W.D. = Δ P dV = P0(V1-V0) + P1(V0-V1) = (P0-P1)(V1-V0)

• e.g.: suppose P₀ = 2 atm, P₁= 1.5 atm, V₁ = 5 l, V₀ =10 l: what is the energy produced?
• Why don't the vertical moves matter?
• Since P₁ < P₀ and V₁ > V₀, this is a device which converts thermal energy into mechanical: a HEAT ENGINE

Second Law of Thermodynamics

Second Law of Thermodynamics

A man who is ignorant of the second law of thermodynamics can no more claim to be educated than a scientist who has never read Shakespeare or Milton (C. P. Snow, paraphrased)

• For example, why can't we have (e.g) a boat that takes in water at 20°C, extracts some heat, turns it into energy and exhausts cold water

• Doesn't violate first law

• In symbolic terms, why can't we have

• Shares for investment in the company will be available after the class.

• In order to get work out of a system, one must have a very asymmetrical system
• e.g. High pressure one side of a piston, low pressure the other side. Can this arise by chance?
• e.g. high temp. one side of a piston Can this arise by chance?
• Given 6 atoms, what is probability of finding them all one side of a room? Can model this via coin tossing

Entropy

• Watch the simulation!

• Hot gas + cold gas ⇒ warm gas
• Gas molecules will randomize themselves very fast
• Microscopic Version of Second Law
• a system will always tend towards the most random arrangement
  

• Low entropy Macintosh!

• High Entropy Macintosh!

• It is very probable that dropping a Mac will rearrange it in a more randomly ordered form!
• Dropping it again (once or one million times) is not likely to get it working again!

Entropy tends to increase in a closed system.

• Of course we can decrease entropy locally:

• How about a fridge? Initially room and fridge at same temp., afterwards T0 < T1

• Fridge is not a closed system: must include power station. How about hydro-power?

Where does the hydro power come from?

• Degradation of energy: high temp. energy in sun ⇒ low temp. energy here
• Note that the nomenclature complicates things unnecessarily: it would be easier if heat was called energy (or maybe heat energy) and entropy was called heat. Then paraphrase of 2nd Law would be
• All forms of energy get converted into heat energy. Once all the heat is at the same temperature, can get no further work.

Murphy's versions of the laws of thermodynamics

• 1st: You can't win
• 2nd: You can't break even
• 3rd: You can't quit the game
• No, don't put them on an exam!

Cannot build a heat engine working in a closed cycle which removes heat from one place and converts it entirely into useful work

• How well can we do?

• Carnot engine: special kind of heat engine whose properties are easily calculated. Carnot engine is series of "moves" on PV plane. The efficiency of such an engine can be found to be \color{red}{ \eta = 1 - \frac{{T_1 }}{{T_0 }}}

• What is maximum efficiency of engine? Why can't we get there?
• To have an engine with 100% efficiency would need to have sink temp. of 0⁰K
• Note that the nomenclature complicates things unnecessarily: it would be easier if heat was called energy (or maybe heat energy) and entropy was called heat. Then paraphrase of 2nd Law would be
• All forms of energy get converted into heat energy. Once all the heat is at the same temperature, can get no further work.

Refrigerators/Heat pumps/Air conditioners

• Can run Carnot cycle backwards, which means we have to supply power, but we extract heat: Efficiency for engine ⇔ Coef. of performance for Heat pump.

• \color{red}{ CP = \frac{{T_0 }}{{T_0 - T_1 }}}
• e.g. if T₀ = 20 °C, T₁ = -10 °C what is η

2nd law and evolution

• 2nd law has profound philosophical consequences: e.g:

• Clearly complexity of animals has increased over history of earth. We are more ordered than amoebas (no moral judgents here!)

• Therefore evolution contradicts 2nd law?
• Not a closed system!

• Or better, how do you know if a movie film is being run backwards?

• The "arrow of time" is defined via an increase in entropy.

• What happens in the end? i.e how does the universe evolve, assuming that it is expands for ever?
• All processes increase entropy, hence end of universe will come when entropy becomes a maximum
• When temperature of everything is the same, then can do no work, hence .....nothing! Heat Death of the Universe

• "This is the way Worlds end, not with a Bang, but a Whimper" T.S. Eliot

Heat in summary:

First law

  δU = Q-W

Second law:

Note what we have done in all this discussion: we have taken Newton's laws of motion Conservation of Momentum Conservation of Energy and a model for what a gas is.