The “Burning” Issue: In the US
and elsewhere in the world we seem to use energy
inefficiently. Why is this so? Are we really not clever enough to build heat
engines that are highly efficient at converting heat energy (random
thermal
kinetic energy) into mechanical or electrical energy (ordered energy)? While the First Law of Thermodynamics
(Conservation of Energy) does not limit our efficient design, the
Second Law of
Thermodynamics (Entropy or disorder must increase in all actual
processes)
does. It would be really useful to be
able to know how to calculate the efficiency of heat engines since they
are
central to our lives at this moment. We
will then know what to change to increase their efficiency (and thereby
reduce
our energy use).
Enter Ideal Gases: The working fluids of many heat engines are gases such as air, steam, or a combusting fuel/air mixture. Over a wide range of temperatures and pressures gases have similar behavior to each other. In the laboratory we measure the pressure on the walls, the volume of the container, the mass (from weight) of the gas inside and the temperature of the gas using a thermometer. The experimental relationship among these measurable variables, for all gases over a wide range of each variable, is simply:
pV = nRT, sometimes called the Ideal Gas Law where
p = pressure [N/m2], V = volume [m3], T = absolute temperature [K], n = mass/(molecular weight) = number of moles, and R is the called the universal gas constant:
R = 8.315 J∙K-1mole-1 = 1.99 cal K-1 mole-1.
It says lots of nice things. For example, if the temperature of a gas is held constant, the pressure must be inversely proportional to the volume.
Avogadro’s Hypothesis: Amedeo Avogadro (1776-1856) and Italian scientist hypothesized that equal volumes of gas at the same pressure and temperature contain equal numbers of molecules. Since the number of molecules per mole is defined as Avogadro’s constant, NA = 6.02 x 1023 molecules/mole, the number of molecules N = nNA, and we may write the Ideal Gas Law in terms of the number of molecules.
pV = NkT, the Ideal Gas Law where k, Boltzmann’s constant equals k = R/NA = 1.38 x 10-23 J/K. The universal gas constant R is related to Boltzmann’s constant by
nR = Nk.
Temperature in the Microscopic Picture: Interestingly, an ideal gas may be described as a collection of non-interacting objects in a box. These independent molecules are not sticky. That is, there is no long range force and consequent potential energy between them. They do have kinetic energy, they bounce elastically off the walls of the box, and when allowed to collide elastically, they will share their kinetic energy with each other. Also, their collisions with the walls exert pressure on the walls of the box. When you add energy to a gas by doing work on the gas or by adding heat energy, the temperature increases. Not surprisingly, then, temperature is simply related to the internal energy of the gas, that is to the sum of the kinetic energies of the molecules in the gas. From the microscopic view temperature is proportional to the average kinetic energy of a single molecule of the gas. That is:
3/2 kT = <1/2 mv2> = <KE per molecule> (for a monatomic gas).
Again, k is Boltzmann’s constant, k = 1.38 x 10-23 JK-1(molecule)-1. The total internal energy of the gas of N molecules is then
U = N<1/2 mv2> = 3/2 NkT = 3/2 nRT
and for a small change
in internal
energy, dU =
3/2 Nk dT =
3/2 nR dT. Notice that both U depends only on the
temperature of the gas, not on the pressure or volume.
The First Law of Thermodynamics:
dQ
= dU
+ dW
dQ = 3/2 nR dT + p dV [monatomic ideal gas]
The Definitions of
Specific Heats
cv
= 1/n dQ/dT|v
(partial derivative)
But since volume is constant, no work is done. And from first
law: dQ = dU
cv
=
1/n dU/dT|v
(partial derivative)
However, U is only a function of T for ideal gases (and by experiment
for real
gases) so
cv
= 1/n dU/dT (regular
derivative), hence dU = cv
n dT always for ideal gases.
At the same time:
cp = 1/n dQ/dT|p
(partial derivative)
The Relationship of the Specific Heats:
Starting with
the
first law: dQ =
dU + dW
At constant volume, dW = p dV = 0
Therefore, dQ
= dU
and the first law becomes
cv n dT
= 3/2 nR dT
[monatomic ideal gas]
Hence, cv = 3/2 R = 3 cal/mol/0C.
[monatomic ideal gas]
At
constant pressure, Vdp
= 0, so dW = pdV
= nR dT
because
the derivative of the Ideal Gas Law gives: pdV
+ Vdp = nRdT.
So
the first law becomes:
cp n dT = cv
n dT + nR dT [all
ideal gases]
And, in general: cp = cv + R [all ideal gases]
Hence, cp = 5/2 R = 5 cal/mol/0C. [monatomic ideal gas]
Names for the Processes:
dU = 0 for isothermal processes.
dQ = 0 for adiabatic processes.
dQ = cpn dT for isobaric or constant pressure processes.
dQ = cvn dT for isochoric or constant volume processes.
dW = p dV and is zero for isochoric processes.
Adiabatic processes the fall in pressure with increase in volume is faster than for isothermal processes because some of the internal energy is being used to do the work of expanding the gas. For adiabatic processes: pVg = constant, where g = cp/cv = 5/3 = 1.667 for a monatomic gas, and g = 7/5 = 1.4 for a diatomic gas… You might show that, with the ideal gas law, the adiabatic relationship can also be written as: TVg-1 = constant.
Otto Cycles: The Otto Cycle is an approximation of the real ICE engine. We will calculate its efficiency and show that it is given by: h = 1 – (V1/V2)1-g where V1/V2 = the compression ratio, ~9 for an ICE engine.