2005

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Course Name: 2005 Thermodynamics
Lecturer: Dr. Graham Cross
Course Page: http://www.tcd.ie/Physics/People/Graham.Cross/
Textbook: Thermal Physics by C.B.P Finn


Contents

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The basics

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Temperature

When studying thermodynamics we confine our attention to a particular part of the universe called the system. Everything else is called the surroundings. The system and surroundings are seperated by a boundary and in general exchange energy and matter. In a closed system there is no matter exchange.

The S.I unit of temperature is the kelvin K

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Equilibrium state

An equilibrium state is one in which all the bulk physical properties are uniform throughout the system and do not change with time.

We require two variables to specify the equilibrium state of a simple system and these variables are called state variables. E.g for a stretched wire we could use the force F and the length or extension l or for an ideal gas we can pick any two of P, V and T.

Functions of state variables are called state functions. They are functions which take unique values at each equilibrium state. It does not matter how a particular state was reached.

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Zeroth law

If each of two systems is in thermal equilibrium with a third then they are in thermal equilibrium with one another. i.e If we have two systems say A and B and in A objects 1 and 3 are in thermal equilibrium with each other and in B objects 2 and 3 are in thermal equilibrium then 1 and 2 are in thermal equilibrium.

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Reversible processes

A process is the mechanism of bringing about a change from one equilibrium state to another.

Quasistatic is used for a process which is a sucession of equilibrium states. As an example consider a pendulum being displaced from one equilibrium position to another. If we pull on the bob with a force which is only infinitesimally greater than the restoring force at every stage of the displacement (gravity in this case) then if we stop the process at any stage then the pendulum would stay where it is (equilibrium state for a pendulum) . We have to perform work in effecting this displacement. The pendulum then goes through a series of equilibrium states. If we were to reduce the force by a very small amount the reverse would happen. The work done against us would be exactly the same as in the initial displacement.

Reversible processes then are quasistatic processes where no dissipative forces such as friction are present.

For an ideal gas, the equation of state PV = nRT, holds for each point in a reversible process but it does not hold for intermediate stages in an irreversible process.

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Work

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The sign convention for work

When the surroundings do work on the system that work is positive conversely when the system does work on the surroundings that work is negative. The system is the important thing to the physicist.

  • When work leaves/exits the system we define the work to be negative. W<0. "When work is done by the system W is negative".
  • When work enters the system we define the work to be positive. W>0. "When work is done on the system W is positive".
  • When heat enters/is added to the system Q>0, Q is positive.
  • When heat leaves/is removed Q<0, Q is negative.
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Gas in a cyclinder

Suppose we have a gas in the familiar cyclinder situation: a cyclinder with one end closed containing a gas and a piston with a force applied to it. The surface area of the face of the piston is A and the force is F.

Suppose we have a gas in the initial equilibrium state and we allow it to expand to a new equilibrium state by decreasing the external balancing force on the piston and allowing it to slide out. If no friction is present all the work the gas does in pushing the piston out goes into performing work on the surroundings provided the expansion is performed quasistatically so that the pressure is well defined and uniform throughout the gas. In other words we have to expand the gas reversibly.

Suppose at one of the intermediate states the pressure of the gas is P. Then F = PA. Thus dW = -PAdx = -PdV. The total work performed is

W = - \int_{v_1}^{v_2}PdV

Work in general is path dependent and it cannot be expressed simply as the difference between two end point values of some state function.

  1. It is important to be clear as to what is meant by "the system".
  2. \bar{d}W = PdV is applicable only to reversible processes.
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The first law of thermodynamics

The first law tells us that, in any process, energy is conserved. It may be converted from one form to another but the total amount remains unchanged.

Work done under adiabatic conditions is called adiabatic work.

If a thermally isolated system is brought from one equilibrium state to another, the work necessary to achieve this change is independent of the process used.

That is saying that the adiabatic work W_{ad} expended in a process is path independent, depending only on the end equilibrium points and this is true whether or not the process is reversible. So there must exist a state function whose difference between the two end points 2 and 1 is equal to the adiabatic work. We call this state function the internal energy U with

W_{ab} = U_2 - U_1

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Heat

If the system is not thermally isolated, it is found that the work W done in taking the system between a pair of equilibrium points depends on the path.

Now for a given change \Delta U = U_2 - U_1 is fixed but W is not now equal to \Delta U.

There is a difference between the adiabatic work required to bring about a change between two equilibrium states and the non-adiabatic work required to effect the same change. We call this difference the heat Q. The generalization of the equation is then

\Delta U = W + Q

which is the mathematical statement of the first law.

It tells us that the internal energy can be increased by either doing work on or by supplying heat to the system. It is true for all processes whether reversible or irreversible.

Heat is the non-mechanical exhange of energy between the system and the surroundings because of their temperature difference.

For an infinitesimal process

dU = \bar{d}W + \bar{d}Q

The Heat capacity of a system is defined as the limiting ratio of the heat introduced reversibly into the system divided by the temperature rise

C = \lim_{\Delta t \rightarrow 0}\frac{Q}{\Delta T} = \frac{\bar{d}Q}{dt}

The bars indicate that and W and hence Q are in general path dependent. We shall consistantly write \bar{d}W since W is not a state function.

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The second law of thermodynamics

The Clausius Statement of the Second Law:

  • It is impossible to construct a refrigerator which, operating in a cycle, will produce no other effect than the transfer of heat from a cooler body to a hotter one.

The Kelvin-Planck statement of the second law:

  • It is impossible to construct an engine which, operating in a cycle, will produce no other effect than the extraction of heat from a reservoir and the performance of an equivalent amount of work.

A more modern version of the law would be:

  • The entropy of any totally isolated system not at thermal equilibrium will tend to increase over time, approaching a maximum value.
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Carnot cycles

Work could be obtained from an engine if there were heat sources at different temperatures (\Delta U = W + Q).

It is possible for heat to flow from a hot body to a cold body with no work being performed, the flow continuing until thermal equilibrium is attained. Thus since any return to thermal equilibrium could be used to produce work any return to equilibrium without the production of work must be considered a loss. So any temperature difference may be utilized in the production of work or it may be wastefully dissipated in a spontaneous flow of heat.

In an efficient engine all transfers of heat should be between bodies of nearly equal temperature.

[Picture of isotherm]

Take an ideal gas around a cycle.

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