Find
here examples of how 1st and 2nd Law intution tells
us what is likely NOT (and what may be) possible, without delving into details
about how to pull it off. The focus is on energy and uncertainty flows into,
and out of, ``steady-state engines'' of virtually any type. The equations treat
the world outside the engine as a contained universe. This is possible since
steady-state engines (by definition) process energy and information (often cyclicly) while their own state of being is maintained
(i.e. at the end of each cycle, the state of the engine itself is to first
order unchanged). This page helps set up the problems, writing out color-coded
terms for 1st and 2nd law equations, and it provides the numeric answer for one
example problem in each case as a reality check for your own deductions. **However,
there is a catch!**

We
ask that you work out for yourself the final, generally useful, formula for solving
each problem type.

*Can you do it?*

The
templates provided may, of course, help solve possibility problems we haven't
thought of as well. Suggestions for problem types to add to this list are
invited. E-mail your suggestions to pfraundorf[AT]umsl[DOT]edu.

**1st Law:** Heat_OUT *plus* Work_OUT *equals* Heat_IN *plus* Work_IN**
2nd Law:** Uncertainty_INCREASE

Net_Surprisal_Irreversibly_Lost, *which is greater than or equal to ***Zero**.

Note that Uncertainty_INCREASE can often be expressed
as...

Heat_OUT/Temperature_OUT *minus* Heat_IN/Temperature_IN,

and Correlation is information on *the relation
between* *subsystems* in [J/K] or
[bits].

[work available] [work required] [heating costs] [irreversibility losses] [information engines]

**1). Work available from a heat engine
(e.g. operating on fossil fuels or via photosynthesis)...**

Q_{exhaust} + W_{out} = Q_{hot},
and Q_{exhaust}/T_{exhaust}
- Q_{hot}/T_{hot}_{ } >=
0,

implies that *Carnot** Efficiency* W_{out}/Q_{hot}
<= (1-T_{exhaust}/T_{hot}).*
Case Study: *An automobile
engine, with T

and T

**2). Work needed to keep the frost on a
6-pack: Refrigerator coefficient of performance...**

Q_{room} = Q_{cold}
+ W_{in}, and Q_{room}/T_{room} - Q_{cold}/T_{cold}
>= 0,

implies a *Refrigerator C.O.P.*
of Q_{cold}/W_{in} <= T_{cold}/(T_{room}-T_{cold}).*
Case Study:* For each joule
of electricity, a freezer in a 295K room

may remove up to 9.9 joules of heat from its 268K air.

**3). Work needed to pump winter heat from
the outside in, or heat pump C.O.P....**

Q_{room} = Q_{cold}_{
}+ W_{in} ,
and Q_{room}/T_{room}
- Q_{cold}/T_{cold}
>= 0,

implies a *Heat Pump C.O.P.* of Q_{room}/W_{in} <= T_{room}/(T_{room}-T_{cold}).*
Case Study:* For each joule
of electricity, heat pumps might

bring inside up to 7.4 joules of heat from a 0 F backyard.

**4). Reversible home heating with a
flame: Getting lots more BTU's for the
buck...**

Q_{room} = Q_{flame}_{ }+ Q_{cold} , and Q_{room}/T_{room}
– (Q_{flame}/T_{flame} + Q_{cold}/T_{cold})
>= 0,

implies a *Reversibility Gain* of
Q_{room}/Q_{flame} <= (T_{room}/T_{flame})(T_{flame}-T_{cold})/(T_{room}-T_{cold})*
Case Study:* When T

a room to T

**5). Zero energy ovens for eskimos, or how to cook food in cold weather for free...**

Q_{oven} = Q_{room}_{ }+ Q_{cold} , and Q_{oven}/T_{oven}
- ( Q_{room}/T_{room} + Q_{cold}/T_{cold})
>= 0,

implies a *Heat Transfer Ratio *of
Q_{oven}/Q_{room} <= (T_{oven}/T_{room})(T_{room}-T_{cold})/(T_{oven}-T_{cold}).*
Case Study:* After "the
turkey is done", this transfer may be

spontaneously reversed (see previous example) for no net heat loss!

**6). Reversibility losses from an oven
leaking heat...**

Q_{room} = Q_{oven},
and S_{irr}
>= (Q_{room}/T_{room}
- Q_{oven}/T_{oven}),

implies minimal net_surprisal
losses of S_{irr} = Q_{oven}(1/T_{room}-1/T_{oven})*
Case Study:* A 473K oven
irreversibly raises state uncertainty

by nearly 10^20 nats, per joule of heat leaked to a 295K room.

**7). Reversibility losses from ice melting
in the ****North Sea****
(or a glass of iced tea)...**

Q_{ice} = Q_{liquid},
and S_{irr}
>= (Q_{ice}/T_{melt}
- Q_{liquid}/T_{melt}),

implies minimal net_surprisal
losses of S_{irr} = Q_{ice}(1/T_{melt}
– 1/T_{liquid}).*
Case Study:* Chipped ice in
an ice-cold slurry melts reversibly, resulting in S

**8). Reversibility losses as your coffee
gets cold...**

dQ_{room} = dQ_{coffee},
and dS_{irr}
>= dQ_{room}/T_{room}
- dQ_{coffee}/T_{coffee},
can be integrated

from T_{coffee} down to T_{room}
using dQ_{coffee} = HeatCapacity
dT_{coffee},

to get Net_Surprisal_Loss
>= HeatCapacity × ξ[T_{coffee}/T_{room}],
where ξ[x] ≡ x - 1 - ln[x].*
Case Study:* For water
cooled from a boil, with dimensionless

HeatCapacity = 9/molecule, S

**9). Ice water
invention: Convert hot water to cold reversibly, with ambient exhaust,
work-free...**

dQ_{room} = dQ_{hot}_{ } + dQ_{cold}, and
dQ_{room}/T_{room}
– (dQ_{hot}/T_{hot}+dQ_{cold}/T_{cold})
>= 0 can be integrated

to T_{room} for dQ_{hot}
from T_{hot}_{,} and for dQ_{cold} from T_{cold}_{,
}using dQ_{water} = HeatCapacity
dT_{water},

to get HeatCapacity (ξ[T_{hot}/T_{room}]
- ξ[T_{cold}/T_{room}])
>= 0, so that T_{room} <= (T_{hot}-T_{cold})/ln[T_{hot}/T_{cold}].*
Case Study:* For water
cooled from a boil,

this invention will make ice water as long as T_{room} <= 100/Ln[373/273] = 320.4[K] or
47.4[C].

**10). Energy required to clear one's mind
(or a quantum computer's memory)...**

Q_{ambient} = W_{in},
and Q_{ambient}/T_{ambient} - I_{open} >= 0,

implies work to erase old data (and clear space for new) of
W_{in} >= I_{open}
× T_{ambient}.*
Case Study:* At room
temperature, nature thus requires W

**11). Maximum astrophysical (or other)
observation rates, per observer per meal...**

Q_{ambient} = W_{food},
and Q_{ambient}/T_{ambient} - I_{recorded}
>= 0,

limits mutual information created to I_{recorded}
<= W_{food}/T_{ambient}.*
Case Study:* Human observers
with typical caloric intake must therefore themselves

create less than 10^21 Gigabytes of correlation information
per day.

(Aside: Some of us, yours truly included, produce MUCH LESS!)

See also this page on the ice water invention, our page on correlation based complexity and heat capacity in bits, and (oldest of all) our information physics page.

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