Lecture he First Law o hermodynamics (h.) Lecture - we introduced macroscopic parameters that describe the state o a thermodynamic system (including temperature), the equation o state (,,) 0, and linked the internal energy o the ideal gas to its temperature. Outline:. Internal Energy, Work, Heating. Energy onservation the First Law 3. Enthalpy 4. Heat apacity
Internal Energy he internal energy o a system o particles,, is the sum o the kinetic energy in the reerence rame in which the center o mass is at rest and the potential energy arising rom the orces o the particles on each other. system boundary environment system Dierence between the total energy and the internal energy? kinetic + potential A he internal energy is a state unction it depends only on the values o macroparameters (the state o a system), not on the method o preparation o this state (the path in the macroparameter space is irrelevant). In equilibrium [ (,,)0 ] : (, ) depends on the kinetic energy o particles in a system and an average inter-particle distance (~ -/3 ) interactions. For an ideal gas (no interactions) : () - pure kinetic
Internal Energy o an Ideal Gas he internal energy o an ideal gas with degrees o reedom: 3 (monatomic), 5 (diatomic), 6 (polyatomic) (here we consider only trans.+rotat. degrees o reedom, and neglect the vibrational ones that can be excited at very high temperatures) How does the internal energy o air in this (not-air-tight) room change with i the external const? N in room k N in room k - does not change at all, an increase o the kinetic energy o individual molecules with is compensated by a decrease o their number.
Work and Heating ( Heat ) WORK We are oten interested in Δ, not. Δ is due to: Q - energy low between a system and its environment due to Δ across a boundary and a inite thermal conductivity o the boundary HEAING heating (Q > 0) /cooling (Q < 0) (there is no such physical quantity as heat ; to emphasize this act, it is better to use the term heating rather than heat ) W - any other kind o energy transer across boundary - work Work and Heating are both deined to describe energy transer across a system boundary. Heating/cooling processes: conduction: the energy transer by molecular contact ast-moving molecules transer energy to slow-moving molecules by collisions; convection: by macroscopic motion o gas or liquid radiation: by emission/absorption o electromagnetic radiation.
he First Law he irst law o thermodynamics: the internal energy o a system can be changed by doing work on it or by heating/cooling it. Δ Q + W From the microscopic point o view, this statement is equivalent to a statement o conservation o energy. Sign convention: we consider Q and W to be positive i energy lows into the system. For a cyclic process ( i ) Q - W. I, in addition, Q 0 then W 0 An equivalent ormulation: erpetual motion machines o the irst type do not exist. erpetual motion machines come in two types: type violates the st Law (energy would be created rom nothing), type violates the nd Law (the energy is extracted rom a reservoir in a way that causes the net entropy o the machine+reservoir to decrease).
Quasi-Static rocesses Quasi-static (quasi-equilibrium) processes suiciently slow processes, any intermediate state can be considered as an equilibrium state (the macroparamers are welldeined or all intermediate states). Advantage: the state o a system that participates in a quasi-equilibrium process can be described with the same (small) number o macro parameters as or a system in equilibrium (e.g., or an ideal gas in quasiequilibrium processes, this could be and ). y contrast, or nonequilibrium processes (e.g. turbulent low o gas), we need a huge number o macro parameters. Examples o quasiequilibrium processes: isochoric: const isobaric: const isothermal: const adiabatic: Q 0 For quasi-equilibrium processes,,, are well-deined the path between two states is a continuous lines in the,, space.
Work A the piston area he work done by an external orce on a gas enclosed within a cylinder itted with a piston: W (A) dx (Adx) - d orce Δx he sign: i the volume is decreased, W is positive (by compressing gas, we increase its internal energy); i the volume is increased, W is negative (the gas decreases its internal energy by doing some work on the environment). W - d - applies to any shape o system boundary W (, ) d d Q d he work is not necessarily associated with the volume changes e.g., in the Joule s experiments on determining the mechanical equivalent o heat, the system (water) was heated by stirring.
W D A W and Q are not State Functions (, ) d diagram Δ Q + W - we can bring the system rom state to state along ininite # o paths, and or each path (,) will be dierent. Since the work done on a system depends not only on the initial and inal states, but also on the intermediate states, it is not a state unction. W is a state unction, W - is not thus, Q is not a state unction either. net W A + WD ( ) ( ) ( )( ) < 0 - the work is negative or the clockwise cycle; i the cyclic process were carried out in the reverse order (counterclockwise), the net work done on the gas would be positive.
omment on State Functions,,, and are the state unctions, Q and W are not. Speciying an initial and inal states o a system does not ix the values o Q and W, we need to know the whole process (the intermediate states). Analogy: in classical mechanics, i a orce is not conservative (e.g., riction), the initial and inal positions do not determine the work, the entire path must be speciied. In math terms, Q and W are not exact dierentials o some unctions o macroparameters. o emphasize that W and Q are NO the state unctions, we will use sometimes the curled symbols δ (instead o d) or their increments (δq and δw). d d S d - an exact dierential S ( ) ( )dy y z(x,y ) d a Ax x, y dx + Ay x, y - it is an exact dierential i it is the dierence between the values o some (state) unction z(x,y ) z(x,y) at these points: d a z( x + dx, y + dy) z( x, y) x A ( x y) Ay ( x, y) x, A necessary and suicient condition or this: y x I this condition z z z ( ) ( x, y) z ( ) ( x, y) Ax x, y Ay x, y d z dx + dy holds: x y x y y x e.g., or an ideal gas: δq d + d d + d - cross derivatives are not equal
roblem Imagine that an ideal monatomic gas is taken rom its initial state A to state by an isothermal process, rom to by an isobaric process, and rom back to its initial state A by an isochoric process. Fill in the signs o Q, W, and Δ or each step., 0 5 a A const Step Q W Δ A + -- 0, m 3 A -- + -- + 0 +
Another roblem During the ascent o a meteorological helium-gas illed balloon, its volume increases rom i m 3 to.8 m 3, and the pressure inside the balloon decreases rom bar (0 5 N/m ) to 0.5 bar. Assume that the pressure changes linearly with volume between i and. (a) I the initial is 300K, what is the inal? (b) How much work is done by the gas in the balloon? (c) How much heat does the gas absorb, i any? (a) (b) (c) δw ( ) d δw ON Δ δ ON W Y i δw Y δ Q + δw ON i i i 0.5bar.8m 300K 3 bar m 3 i 70K i ( ) 0.65 bar/m 3 +.65bar - work done on a system δw Y ( ) d 3 3 3 4 ( 0.5 0.8bar m + 0.5 0.4bar m ) 0.6bar m 6 0 J ( ) d i 3 δq Δ δwon i ON i i Y i i - work done by a system 3 5 4 4 ( ) W + δw.5 0 J ( 0.) + 6 0 J 4.5 0 J
Quasistatic rocesses in an Ideal Gas, isochoric ( const ) W 0 3 Q ( ) > ( Δ ) 0 d Q (see the last slide) W isobaric ( const ) ( ) 0 (, ) d < 5 Q 0 d W Q ( ) > ( Δ ) +
Isothermal rocess in an Ideal Gas isothermal ( const ) : W i W i ln W d (, ) d ln Q d 0 W W i- > 0 i i > (compression) W i- < 0 i i < (expansion)
Adiabatic rocess in an Ideal Gas adiabatic (thermally isolated system) he amount o work needed to change the state o a thermally isolated system depends only on the initial and inal states and not on the intermediate states. W Q 0 d W (, ) d to calculate W -, we need to know (,) or an adiabatic process d d d ( the # o unrozen degrees o reedom ) d + d d d + d d d d 0, + + + ln ln d + const d 0
Adiabatic rocess in an Ideal Gas (cont.) const An adiabata is steeper than an isotherma: in an adiabatic process, the work lowing out o the gas comes at the expense o its thermal energy its temperature will decrease. W (, ) d +/3.67 (monatomic), +/5.4 (diatomic), +/6.33 (polyatomic) (again, neglecting the vibrational degrees o reedom) d
roblem Imagine that we rapidly compress a sample o air whose initial pressure is 0 5 a and temperature is 0 ( 95 K) to a volume that is a quarter o its original volume (e.g., pumping bike s tire). What is its inal temperature? Rapid compression approx. adiabatic, no time or the energy exchange with the environment due to thermal conductivity For adiabatic processes: also / const 0.4 95 K 4 95 K.74 54 K - poor approx. or a bike pump, works better or diesel engines const
Non-equilibrium Adiabatic rocesses (Joule s Free-Expansion Experiment) const - applies only to quasiequilibrium processes!!!. const increases decreases (cooling). On the other hand, Δ Q + W 0 ~ unchanged (agrees with experimental inding) ontradiction because approach # cannot be justiied violent expansion o gas is not a quasistatic process. must remain the same.
Isobaric processes ( const): he Enthalpy d Q - Δ Q -Δ() Q Δ + Δ() H + - the enthalpy he enthalpy is a state unction, because,, and are state unctions. In isobaric processes, the energy received by a system by heating equals to the change in enthalpy. isochoric: isobaric: Q Δ Q Δ H in both cases, Q does not depend on the path rom to. onsequence: the energy released (absorbed) in chemical reactions at constant volume (pressure) depends only on the initial and inal states o a system. he enthalpy o an ideal gas: (depends on only) H + + +
Heat apacity he heat capacity o a system - the amount o energy transer due to heating required to produce a unit temperature rise in that system is NO a state unction (since Q is not a state unction) it depends on the path between two states o a system ( isothermic, adiabatic 0 ) +d Q δ Δ 3 i he speciic heat capacity c m
and d d d d Q + δ const const the heat capacity at constant volume the heat capacity at constant pressure H o ind and, we need (,,) 0 and (,) d d d d d d d d + + + + d d +
and or an Ideal Gas nr d/d 3/ 5/ 7/ 0 00 000, K ranslation Rotation ibration o one mole o H R d d + ( or one mole ) 0 nr + + + For an ideal gas # o moles For one mole o a monatomic ideal gas: R R 5 3