Thermodynamics


Chapter 12 The Laws of Thermodynamics


Mechanical Work in a Thermodynamic Process

The work done by a system on its environment is given by
 
W = PDV

when DV is small.

If DV is not small, the amount of work done by a system in a thermodynamic process can be found using the areas in a P-V diagram.

Figure 12.4

Note that in addition to its dependence on the initial and the final state of the system, the work done during a thermodynamic process depends also on the path, i.e. it depends on how the system gets there. In contrast, the change of internal energy is independent of the path.

Some specific thermodynamical processes:


Isobaric process:

A process that occurs at constant pressure.

W = PDV

even when DV is large.


Isovolumetric process: 

A process that occurs at constant volume.

W = 0

regardless of pressure.


A cycle: 

A process that returns to its initial state.

The magnitude of the work done by a gas in a cycle process is equal to the area enclosed by the cycle on the P-V diagram. It is positive if the cycle goes clockwise. It is negative if the cycle goes counter-clockwise.


The First Law of Thermodynamics

First Law of Thermodynamics: Energy is conserved. In mathematical terms,

DU = Q - W

which says that the amount of heat transferred into a system minus the amount of work done by the system (on its environment) must be equal to the correponding change in the internal energy of the system.

It is a formulation of the conservation of energy that takes into account the fact that heat is also a form of energy.

Q depends on the path since W depends on it and DU does not.


Isobaric process:

A process that occurs at constant pressure.

W = PDV

DU = Q - PDV

even when DV is large.


Isovolumetric process: 

W = 0

DU = Q


A cycle: 

A process that returns to its initial state.

DU = 0

Q = W

The work done by the system is equal to the net amount of heat absorbed.


Adiabatic process:

A process in which there is no heat transfer.

Q = 0

DU = - W


Heat Engines and the Second Law of Thermodynamics 

A heat engine is a device that converts thermal energy into other forms of useful energy such as mechanical or electrical energy.

A heat engine operates in cycles. The net effect of each cycle is to absorb an amount of heat Qh from the hot reservoir, do an amount of work W, and release an amount of heat Qc to the cold reservoir.

The efficiency of a heat engine is defined as

e W/Qh = (Qh - Qc)/Qh = 1 - Qc/Qh

It describes the efficiency of an engine in transforming heat into work.

One way of stating the second law of thermodynamics: It is impossible to construct a heat engine that, operating in a cycle, produces no other effect than the absorption of heat from a reservoir and the performs of an equal amount of work.

In other words, heat cannot be transformed into mechanical energy with 100% efficiency. In more mathmatical terms, the second law of thermodynamic asserts that it is not possible to build a heat engine which has

Qc = 0 and e = 1.

For example, an automobile based on internal combustion engine must have an exhaust (where the leftover heat goes).

Heat cannot be transformed with 100% efficiency into work because it is the energy that is associated with the random motion of atoms and molecules. This randomness (disorder) can be described by a concept called entropy. A more pedagogic statement of the second law of thermodynamics says that the randomness of an isolated system never goes away, it can only increase. In terms of entropy, it says that the entropy of an isolated system can only increase. It is easy to see that a heat engine with 100% efficiency would violate the second law stated in this fashion.


The Carnot Engine

A Carnot engine is an idealized heat engine that operates through Carnot cycles.

A Carnot cycle is a process that consists of the following four steps:

  1. isothermal expansion in contact with heat reservior at Th
  2. adiabatic expansion in which the temperature drops to Tc
  3. isothermal compression in contact with heat reservior at Tc
  4. adiabatic compression which brings the gas back to its initial state.

By computing Qh, Qc, and W for this cycle, which is not difficult once you know how to do a few simple integrals, one can show that a Carnot engine has a efficiency given by

ec = (Th - Tc)/Th = 1 - Tc/Th

This result is significant because of the Carnot theorem:

No real engine operating between two heat reserviors (one at Th and one and Tc) can be more efficient than a Carnot engine operating between the same two reserviors.

The efficiency of the Carnot engine thus gives us the maximum efficiency that can be achieved with reserviors at Th and Tc.


 Contents | Assignments | Thermodynamics | Vibrations and Waves
Copyright 1997 by Bo Gao. All rights reserved.