Chapter Nine - Heat Engines and The Second Law

where does the work come from in a reversible isothermal expansion for an ideal gas

all of the heat transfer from the temperature reservoir is converted into mechanical work

how do we know W= Q for reversible isothermal expansion of an ideal gas

change in internal energy is 0 in isothermal expansion as U= U(T) for an ideal gas so Q= W

why is the reversible isothermal expansion of an ideal gas not ideal

this process stops when P2 equals ambient pressure but we want a continuously working engine

what is a cyclic heat engine

working substance is cycled back from some state through a series of different states and back to original state, during the cycle the working substance alternately absorbs heat from the hot reservoir and rejects heat to the cold reservoir

what is the structure of a cyclic heat engine

QH extracted reversible from a hot temperature bath, performs work W and rejects heat QC reversible to a cold temperature bath

what does the reversibility in the cyclic heat engine mean

the system has to be at a temperature that is only differentially different from that of the respective temperature baths

what is the equation for the efficiency of reversible cyclic heat engine

efficiency = 1 - Qc/QH

how is the equation for efficiency derived

in one cycle change in U = (QH - QC) - W = 0 for cyclic so W = QH - QC, efficiency is what you get out divided by what you get in so efficiency = W/QH = QH - QC/QH

what is carnots theorem

any reversible cyclic heat engine working between TH and TC operates at the maximum possible efficiency of any engine working between these temperatures

what is proof of carnots theorem

if there is a super engine with a higher efficiency that extracts Wsuper, Wsuper>nQH, feed this into ordinary engine which needs Wordinary= nQH, so there is a net work creation of W = Wsuper - Wordinary >0, in a cycle there is no net Q or change in U, therefore not possible to have work

what is efficiency in terms of temperature

for T2<T1, n = 1 - T2/T1

how is efficiency in terms of T found

carnots theorm implies n can only depend on TH and Tc so n = 1 - f(T1,T2) where H is one and C is 2, for an engine transferring Q1 from T1 and dumping Q2 at T2 via an intermediate source T’ then f(T2,T1) = Q2/Q1 = Q2/Q’/Q1/Q’ = f(T2,T’)/f(T1,T’), f(T2,T1) cant depend on T’ as its arbitrary so f(T,T’) = g(T) , define g(T) as aT where a is constant

what are the four steps of the carnot cycle

AB - Isothermal expansion absorbing heat Q1 from hot source doing work W1 = R@1ln(Vb/VA) and Q1= W1, BC - adiabatic expansion (Vc/Vb)^(gamma-1) = @1/@2, Isothermal compression rejecting heat Q2 to cold source absorbing W2 = R@2ln(Vc/Vd) and Q2 = W2, Da adiabatic compression (Va/Vd)^(gamma-1) = @2/@1

how is the ideal gas scale @ idential to the absolute temperature T

efficiency = W1 - W2/Q1 = W1- W2/W1 = 1- @2ln(vc/Vd)/@1ln(Vb/Va) but we know from the equations at Bs and DA that Vc/Vd= Vb/Va so n = 1 - @2/@1

what is the net work done by the reversible carnot cycle

integral d(dash) W = integral PdV

what is kelvin plancks statement of the second law

it is not possible to have a cyclic heat engine whose sole effect is to absorb heat from a single thermal reservoir and convert it entirely to work

what is Clausius form of the second law

It is not possible for heat to pass from a cold to a hot body without the expenditure of work

what is a heat pump

a carnot cycle in reverse to deliver heat from a cold to a hot source by taking in external work

what is a heat pumps coefficient of performance

COP = QH/W

what is COP in terms of T

COP = TH/ TH -TC

how is COP in terms of T found

QH = nQH + QC in the most efficient operation and using our equation for n we get COP = TH/TH-TC

when are heat pumps most energy efficient

is there is a source of heat at a temperature only a little below the target inside temperature