The study of the molecules of a gas is a good example of a physical situation where statistical methods give precise and dependable results for macroscopic manifestations of microscopic phenomena. For example, the pressure, volume and temperature calculations from the ideal gas law are very precise. The average energy associated with the molecular motion has its foundation in the Boltzmann distribution, a statistical distribution function. Yet the temperature and energy of a gas can be measured precisely.
Monday, February 23, 2009
Temperature and pressure are macroscopic properties of gases. These properties are related to molecular motion, which is a microscopic phenomenon. The kinetic theory of gases correlates between macroscopic properties and microscopic phenomena. Kinetics means the study of motion, and in this case motions of gas molecules.
At the same temperature and volume, the same numbers of moles of all gases exert the same pressure on the walls of their containers. This is known as Avogadros principle. His theory implies that same numbers of moles of gas have the same number of molecules.
Common sense tells us that the pressure is proportional to the average kinetic energy of all the gas molecules. Avogadros principle also implies that the kinetic energies of various gases are the same at the same temperature. The molecular masses are different from gas to gas, and if all gases have the same average kinetic energy, the average speed of a gas is unique.
Based on the above assumption or theory, Boltzmann (1844-1906) and Maxwell (1831-1879) extended the theory to imply that the average kinetic energy of a gas depends on its temperature.
They let u be the average or root-mean-square speed of a gas whose molar mass is M. Since N is the Avogadro's number, the average kinetic energy is (1/2) (M/N) u2 or
M 3R T 3
K.E. = --- u2 = ---- = --- k T
2 N 2 N 2
= (3 R T / M)1/2.
= (3/2) 8.3145 J/(mol K) * 300 K
= 3742 J / mol (or 3.74 kJ/mol)
Sunday, February 22, 2009
The Clausius statement implies the Kelvin-Planck statement: -
So if we had a magic heat engine, we could have a magic fridge. Therefore, the statement that you can't have a magic fridge implies that you can't have a magic heat engine.
The Kelvin-Planck statement implies the Clausius inequality: -
We now wish to prove that the statement that you can't have a magic heat engine implies the Clausius inequality - that for any closed, stationary system operating in a cycle
where Q is as usual the heat transferred to the system and T is the temperature at the boundary of the system.
We shall prove this with the assistance of the piece of imaginary equipment shown in the diagram below. We choose any closed, stationary system (hereafter referred to as "the chosen system"). We imagine that the chosen system is attached to a reversible system which is connected to a thermal reservoir, as shown on the diagram.
By convention, the diagram shows heat flowing in to the chosen system and work coming out of it. We emphasize, however, that this is not a restriction on what the chosen system is allowed to do: it merely means that work input or heat output will have negative sign.
It is important to note that the inequality we're going to derive relates only to the boundary temperature and heat flow of the chosen system, and is therefore true whether or not the chosen system is connected to a reversible system which is connected to a thermal reservoir. The reversible system and the thermal reservoir are, so to speak, intellectual scaffolding, which will be discarded after we have produced our result.
So, consider the diagram. The combined system of the chosen system and the reversible system to which it's attached is, like everything else, subject to the First Law of Thermodynamics, which we give for the combined system in its differential form: δQr - δW = dE.
Now, as the reversible system is reversible, then if we are using a Thermodynamic Temperature Scale (eg degrees kelvin) then by the definition of a thermodynamic temperature scale we have δQr / δQ = Tr / T.
For convenience, we shall assume that the reversible system performs an whole number of cycles in the same time that the chosen system takes to execute one. Now, let's integrate the equation above over one cycle of the chosen system.
Of course, is simply W. To evaluate , recall that over a cycle, the combined system can neither gain nor lose energy - or it wouldn't be a cycle - and so . This gives us
which we shall rearrange to give
Now, here's the clever part. There is no such thing as a magic heat engine. That's the Kelvin-Planck statement of the Second Law, and the whole thing we're basing our argument on.
Now, look at the diagram given above, and observe that the combined system is connected to only one thermal reservoir. If the net work output W was greater than zero, the combined system would be a magic heat engine. Therefore, the net work output W of the combined system is less than or equal to zero, and so
Finally, we note that Tr is a constant (being the temperature of a thermal reservoir) and positive, since we are using a thermodynamic temperature scale. So we can divide both sides of the inequality through by Tr, giving us
The statement made by Kelvin-Planck for third law of thermodynamics says, “It is impossible for a heat engine to produce net work in a complete cycle if it exchanges heat only with bodies at a single fixed temperature.” Thus to produce the work the cycle should exchange heat with two reservoirs which are a different temperatures. The high temperature reservoir is called as source and low temperature reservoir is called as sink.
W = QH – QC
As per the above statement the net work will be produced in the cycle as long as there is difference in temperature between the source and sink. In due course of time if source loses too much heat and sink gains too much heat and their temperatures become equal, the net work produced in the cycle will be zero.
The Kelvin-Planck statement tells the condition for producing work within the cycle, the Carnot’s statement tells maximum work or efficiency that can be obtained within the cycle.
The ratio of the maximum mechanical work obtain to the total heat supplied to the engine is known as maximum thermal efficiency (ηmax) of the engine. Mathematically
ηmax = Maximum work obtain / Total heat supplied
= QH – QC/QH = 1 – QC/QH = 1 – TC/TH
For a reversible engine, QH/TH = QC/TC
- When a closed system undergoes a thermodynamic cycle, the net heat transfer is equal to the net work transfer. This statement does not specify the direction of flow of heat and work (i.e. whether the heat flows from a hot body to a cold body or from a cold body to a hot body). It also does not give any condition under which these transfers take place.
- The heat energy and mechanical work are mutually convertible. Though the mechanical work can be fully converted into heat energy, but only a part of heat energy can be converted into mechanical work. This means that the heat energy and mechanical work are not fully mutually convertible. In other words, this is a limitation on the conversion of one form of energy into another form.
- Path function: Their magnitudes depend on the path followed during a process as well as the end states. Work (W), heat (Q) are path functions.
Process A: WA = 10 kJ
Process b: WB = 7 kJ
- Point Function: They depend on the state only, and not on how a system reaches that state. All properties are point functions.
Process A: V2 - V1 = 3 m3
Process B: V2 - V1 = 3 m3
Heat is energy transferred from one system to another solely by reason of a temperature difference between the systems. Heat exists only as it crosses the boundary of a system and the direction of heat transfer is from higher temperature to lower temperature.
For thermodynamics sign convention, heat transferred to a system is positive; Heat transferred from a system is negative.
The heat needed to raise a object's temperature from T1 to T2 is:
Q = cp m (T2 - T1)
cp = specific heat of the object (will be introduced
in the following section)
m = mass of the object
Unit of heat is the amount of heat required to cause a unit rise in temperature of a unit mass of water at atmospheric pressure.
- Btu: Raise the temperature of 1 lb of water 1 oF
- Cal: Raise the temperature of 1 gram of water 1 oC
J is the unit for heat in the S.I. unit system. The relation between Cal and J is
1 Cal = 4.184 J
Notation used in this book for heat transfer:
- Q : total heat transfer
- : the rate of heat transfer (the amount of heat transferred per unit time)
- δQ: the differential amounts of heat
- q: heat transfer per unit mass
Modes of Heat Transfer: -
Conduction: Heat transferred between two bodies in direct contact.
If a bar of length L was put between a hot object TH and a cold object TL , the heat transfer rate is:
kt = Thermal conductivity of the bar
A = The area normal to the direction of heat
Convection: Heat transfer between a solid surface and an adjacent gas or liquid. It is the combination of conduction and flow motion. Heat transferred from a solid surface to a liquid adjacent is conduction. And then heat is brought away by the flow motion.
Newton's law of cooling:
h = Convection heat transfer coefficient
Ts = Temperature of the solid surface
Tf = Temperature of the fluid
The atmospheric air motion is a case of convection. In winter, heat conducted from deep ground to the surface by conduction. The motion of air brings the heat from the ground surface to the high air.
Stefan - Boltzmann law:
σ = Stefan - Boltzmann constant
ε = emissivity
Ts = Surface temperature of the object
The three modes of heat transfer always exist simultaneously. For example, the heat transfer associated with double pane windows are:
- Conduction: Hotter (cooler) air outside each pane causes conduction through solid glass.
- Convection: Air between the panes carries heat from hotter pane to cooler pane.
- Radiation: Sunlight radiation passes through glass to be absorbed on other side.
Work is the energy transfer associated with a force acting through a distance.
Like heat, Work is an energy interaction between a system and its surroundings and associated with a process.
In thermodynamics sign convection, work transferred out of a system is positive with respect to that system. Work transferred in is negative.
Units of work is the same as the units of heat.
- W : total work
- δW: differential amount of work
- w: work per unit mass
- : Power, the work per unit time
A system without electrical, magnetic, gravitational motion and surface tension effects is called a simple compressible system. Only two properties are needed to determine a state of a simple compressible system.
Considering the gas enclosed in a piston-cylinder device with a cross-sectional area of the piston A.
- Pressure P1
- Volume V1
- Pressure P2
- Volume V2
δW = F ds = P A ds = P dV
Heat and Work - A Path Function: -
There are many similarities between heat and work. These are
- The heat and work are both transient phenomena. The systems do not possess heat or work. When a system undergoes aa change, heat transfer or work done may occur.
- The heat and work are boundary phenomena. They are observed at the boundary of the system
- The heat and work represent the energy crossing the boundary of the system.
- The heat and work are path functions and hence they are inexact differentials. They are written as δQ and δW.
Saturday, February 14, 2009
- Zeroth Law of Thermodynamics
- First Law of Thermodynamics and
- Second Law of Thermodynamics
Zeroth Law of Thermodynamics: -
"If A and B are each in thermal equilibrium with C, A is also in thermal equilibrium with B."
- For example, if two systems of ideal gas are in equilibrium, then P1V1/N1 = P2V2/N2 where Pi is the pressure in the ith system, Vi is the volume, and Ni is the "amount" (in moles, or simply the number of atoms) of gas.
a) "The heat and mechanical work are mutually convertible". According to this law, when a closed system undergoes a thermodynamic cycle, the net heat transfer is equal to the net work transfer.
b) "The First Law states that energy cannot be created or destroyed". The amount of energy lost in a steady state process cannot be greater than the amount of energy gained. This is the statement of conservation of energy for a thermodynamic system.
The only difference here is that δW is the work done on the system. So, when the system (e.g. gas) expands the work done on the system is − PdV whereas in the previous formulation of the first law, the work done by the gas while expanding is PdV. In any case, both give the same result when written explicitly as:
And also the first law can be expressed as the Fundamental Thermodynamic Relation:
Second Law of Thermodynamics: -
The entropy of a thermally isolated macroscopic system never decreases. However, a microscopic system may exhibit fluctuations of entropy opposite to that dictated by the Second Law.