Thermodynamic Processes

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					Thermodynamic States
The ideal gas law relates the pressure (P), volume (V), number of moles (n) and temperature
(T) of a gas.

                                              PV = nRT

In SI units, R = 8.31 J/(mol*K). This law is approximate. It only applies to cases where the
temperature is well above the condensation point and the volume of the molecules is much
less than the volume of the container.

Thermodynamic Processes
ΔU = Q – W

The first law of thermodynamics is a statement of conservation of energy. A system can
exchange energy with its environment with heat (Q) or work (W, also denoted as Ws). This
results in changes in its internal or thermal energy (U, also denoted as Eth). By convention,
heat coming into the system is considered positive and heat going out of the system is
considered negative. Work done by the system is considered positive, while work done on
the system is considered negative (note that this is the opposite of the classical mechanics
sign convention for work).

W = ∫PdV (± the area under the curve in the PV diagram) always. For an isochoric process,
W = 0. For an isobaric process, W = P∫dV = PΔV. For an isothermal process, the ideal gas
law can be used to find P as a function of V and substituted into the equation for work. W =
∫(nRT/V)dV = nRT*ln(Vf/Vi).

The definitions of Cp and Cv can be used to obtain a formula for heat during isobaric and
isochoric processes. For an isobaric process, dQ = nCpdT. If Cp and n are constant (usually
the case, at least approximately), then Q = nCpΔT. Likewise, for an isochoric process, dQ =
nCvdT. If Cv and n are constant (usually the case, at least approximately), then Q = nCvΔT.

This formula for heat during an isochoric process can be used with the first law and the
formula for work to obtain expressions for the change in internal energy for any process. For
an isochoric process, W = 0, so ΔU = Q = nCvΔT. This formula for ΔU can be generalized
to all processes since internal energy is path independent.

Physical Situation       Name          State       P      V       T      ΔU      Q             W
                                       Variables
Insulated    Add         Adiabatic     PVγ =       Up     Down    Up     nCvΔT   0             -nCvΔT < 0
Sleeve       weight to   compression   constant;                         >0
             piston                    TVγ-1 =
                                       constant
Insulated    Remove      Adiabatic     PVγ =       Down   Up      Down   nCvΔT   0             -nCvΔT > 0
Sleeve       weight      expansion     constant;                         <0
             from                      TVγ-1 =
             piston                    constant
Heat gas     Lock        Isochoric     V fixed;    Up     Fixed   Up     nCvΔT   nCvΔT > 0     0
             piston                    PαT                               >0
Cool gas     Lock        Isochoric       V fixed;   Down    Fixed   Down    nCvΔT   nCvΔT < 0       0
             piston                      PαT                                <0
Heat gas     Piston      Isobaric        P fixed;   Fixed   Up      Up      nCvΔT   nCpΔT > 0       PΔV > 0
             free to     expansion       VαT                                >0
             move
Cool gas     Piston      Isobaric        P fixed;   Fixed   Down    Down    nCvΔT   nCpΔT < 0       PΔV < 0
             free to     compression     VαT                                <0
             move
Immerse      Add         Isothermal      T fixed,   Up      Down    Fixed   nCvΔT   nRT*ln(Vf/Vi)   nRT*ln(Vf/Vi)
gas in       weight to   compression     PV =                               =0      <0              <0
large bath   piston                      constant
Immerse      Remove      Isothermal      T fixed,   Down    Up      Fixed   nCvΔT   nRT*ln(Vf/Vi)   nRT*ln(Vf/Vi)
gas in       weight      expansion       PV =                               =0      >0              >0
large bath   from                        constant
             piston
Unknown      Unknown     No Name         PV =       ?       ?       ?       nCvΔT   ΔU + W          ∫PdV = ± area
                                         nRT                                                        under curve in
                                                                                                    PV diagram
Tips: Know which formulas are specific to a particular process and which are true for any
process. See the next section for notes on Cp, Cv and γ.


Heat Capacities
Cv, the molar heat capacity at constant volume (zero work), can be estimated using the
number of degrees of freedom multiplied by ½R. For a monatomic gas, there are three
degrees of freedom from the translation of the particles in three dimensions, so C v = 3/2*R.
For a diatomic gas, there are five degrees of freedom from the three directions of translation
and two axes of rotation. The third possible axis does not have a significant rotational kinetic
energy and is therefore insignificant. Therefore, Cv = 5/2*R. For solids, there are three
degrees of freedom from translation and three from vibration, so Cv = 3R (Dulong-Petit).
All three formulas are theoretical and classical, and generally give reasonable agreement with
empirical evaluations. Deviations from these formulas can be explained with quantum
mechanics which is beyond the scope of this course.

Cp, the molar heat capacity at constant pressure, can be calculated for an ideal gas. For an
isobaric process where n and Cp are constant, Q = nCpΔT. The change in internal energy can
be calculated with the general formula ΔU = nCvΔT. W = P∫dV = PΔV by the definition of
work. Using the ideal gas law, PΔV = nRΔT. ΔU = Q – W by the first law of
thermodynamics. Combine the above formulas to obtain an expression for Cp.

                                      ΔU = Q – W
                                      nCvΔT = nCpΔT – nRΔT
                                      Cp = Cv + R

The ratio of heat capacities is denoted by the letter gamma (γ), and is defined by the
following formula:

                                      γ = Cp/Cv
For a monatomic ideal gas, γ = (3/2*R + R)/(3/2*R) = 5/3. For a diatomic ideal gas, γ =
(5/2*R + R)/(5/2*R) = 7/5.

                       Gas       Cv    Cp    γ   Examples
                       monatomic 3/2*R 5/2*R 5/3 He, Ne, Ar
                       diatomic  5/2*R 7/2*R 7/5 H2, N2, O2

Thermodynamic Cycles
A cycle must have a total ΔU = 0. Therefore ΣQ = ΣW by the first law. Normally, it is
useful to separate the values of Q that are positive from those that are negative. Positive
values represent heat input into the system and negative values represent heat output from
the system.

The total work for a cycle can be calculated graphically with the area enclosed in the PV
diagram. If the cycle is clockwise, then the device is a heat engine and W > 0. If the cycle is
counter-clockwise, then the device is a refrigerator/air conditioner or heat pump and W < 0.

Heat Engines
A heat engine has a characteristic called efficiency, e (also denoted η).

                                         e = ΣW/ΣQH

The summation of work includes all processes, regardless of sign. The summation for heat
includes only heat input from the “hot reservoir” which in this case includes only the
processes with positive heat.

There is a theoretical upper limit on the efficiency of an engine operating between two
temperature extremes. This is the Carnot efficiency.

                                       ecarnot = 1 – Tc/Th

Heat Pumps, Refrigerators, and Air Conditioners
If a cycle has a total work less than zero, then the device might be a refrigerator/air
conditioner or a heat pump. These devices have heat input from a lower temperature system
and have heat output to a higher temperature system. The physical construction for these
two devices can be exactly the same, but the use and desired outcomes are different. With a
refrigerator/air conditioner, the goal is to transfer heat from the colder system. With a heat
pump, the goal is to transfer heat to the hotter system. With any of these devices, the energy
input is in the form of work. This work is typically done by a compressor (you pay for the
energy to run this).

A refrigerator/air conditioner has a characteristic called the coefficient of performance,
C.O.P. or K.

                                        K = ΣQc/|ΣW|
The summation for heat includes only the heat input from the “cold reservoir” which in this
case includes only the processes with positive heat. The summation of work includes all
processes, regardless of sign.

There is a theoretical upper limit on the coefficient of performance for a refrigerator/air
conditioner operating between two temperature extremes. This is the Carnot coefficient of
performance and is based on the second law of thermodynamics.

                                    Kcarnot = Tc/(Th – Tc)

A heat pump also has a characteristic called the coefficient of performance, C.O.P. or K.

                                       K = ΣQh/ΣW

The summation for heat includes only the heat output to the “hot reservoir” which in this
case includes only the processes with negative heat. The summation of work includes all
processes, regardless of sign.

There is a theoretical upper limit on the coefficient of performance for a heat pump
operating between two temperature extremes. This is the Carnot coefficient of performance
and is based on the second law of thermodynamics.

                                    Kcarnot = Th/(Th – Tc)

Entropy
Entropy, S, is often characterized as a measure of disorder. The second law of
thermodynamics states that for an isolated system (no energy or matter exchanged with
external agents), the total entropy of the system cannot decrease:

                                           ΔS ≥ 0

There are two general methods for calculating entropy. The first is useful when there are
exchanges of energy in the form of heat. The second is useful when there is mixing of
particles:

                               ΔS = ∫(dQ/T) ≈ Q/T
                               S = k*ln(W)

                               S = entropy
                               Q = heat
                               T = temperature
                               k = Boltzmann’s constant = 1.38E-23 J/K
                               W = number of possible microscopic states consistent with
                               the macroscopic state
Misuses of the Laws of Thermodynamics
There are commonly held views among the general public that the laws of thermodynamics
falsify both cosmic and biological evolution. These views are not supported by data.

One view unsupported by data is that the Big Bang couldn’t possibly be correct because it
violates the first law of thermodynamics. The claim is that there is obviously a lot of energy
now and there couldn’t be any energy before the Big Bang. The total energy apparently
increased thus violating the first law of thermodynamics.

For this alleged violation to be true, we must know both the total energy of the universe
both before and after the Big Bang and show that they are different (putting aside objections
that there is no such thing as “before the Big Bang”). No one has calculated the energy of
the universe “before the Big Bang.” But even if one assumes that the energy is zero “before
the Big Bang”, a calculation of the total energy of the universe based on classical physics also
yields a total energy of zero! This was shown in a paper written by E. Tryon in a 1973 article
in the journal Nature (and not refuted to date). How can that be so with all the stuff moving
around (kinetic energy), light energy, etc.? It turns out that the negative gravitational
potential energy balances out the positive energy and the net sum is zero. Even if we assume
the total energy of the universe before the Big Bang is zero, there is no proven violation of
the first law of thermodynamics.

Another view unsupported by data is that the second law of thermodynamics prohibits the
evolution of chemicals to simple organisms or of simple organisms to more complex
organisms, an apparent decrease in entropy.

There are two important objections to the "life violates the second law" argument. First,
there is no calculation that shows that complex life forms are lower entropy than less
complex forms or non-living things. Hand waving metaphors are no substitute for a proper
calculation. Second, life forms are not closed systems. They continuously exchange energy
and particles with their environments, so the second law has nothing to say about them
unless you include their environments in the calculation. Evolution has not been proven to
violate the second law of thermodynamics.

				
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