Thermodynamics deals with processes which involve energy changes as a result of heat flow to or from a system and/or work done on or by a system.
Often the ‘system’ considered is a fixed mass of gas separated from its surroundings by a cylinder and a piston:
The energy in a system is called its internal energy.
The state of a gas at a given moment is defined in terms of the particular values of mass, volume, pressure, temperature and internal energy it has at that moment.
Heat and work
Heat and work are terms used to describe energy in the process of transfer:
The term ‘internal energy’ of a system is preferred to the term ‘heat content’, since, for example, in the above system, if the piston is pushed in a little, work is done on the gas, and its temperature and internal energy will rise, but no heat has entered the gas.
The zeroth law of thermodynamics (this law was formally stated after the first one)
If a hot and cold body are brought into contact, after a while all heat transfer between them will stop. The bodies are then said to be in thermal equilibrium - they share the common property which we call temperature.
The first law of thermodynamics
The internal energy of a system can be:
The signs assigned to the various quantities is a matter of convention. Here we will specify:
Work done by an expanding gas
Gas pressure, P = force/area, so force = pressure * area, i.e. F = PA.
The piston is held in position by the force PA exerted by the gas and the external force F.
Suppose that the volume changes by a finite amount. A graph such as the following of P against V during such a change is called an indicator diagram:
The area between the graph and V1 to V2 on the axis can be divided into strips like the one shown. The sum of the areas of all such strips equals the total work done. Hence, the total work done in the finite expansion from V1 to V2 equals the area below the graph.
If pressure P is constant, the change is called an isobaric change, in which case the indicator diagram looks like:
Thus, for an isobaric change, the work done is given by:
The equation applies equally if the gas is compressed at constant pressure, in which case W is the work done on the gas.
To calculate work done using W = P(V2- V1) we assume that P is the same at every stage of the change. This implies that the system is in equilibrium (i.e. all its parts are at the same temperature and pressure) at every instant of the change. Thus, during a change we regard a system as passing through an infinite series of states of equilibrium. Since it would be possible to take the system back through the same set of equilibrium states, such a change is said to be reversible.
PRINCIPAL HEAT CAPACITIES OF A GAS (return to start of page)
Recall that the heat capacity of something is the amount of heat required to raise its temperature by 10C (or 1K).
Consider heat being supplied to a gas trapped in a cylinder by a piston, as in the earlier diagram. The heat required to warm the gas by 10C will depend upon how much the gas expands, since heat is needed to:
The principle molar heat capacities of a gas are defined as:
Derivation - relationship between Cp and Cv.
We consider a fixed mass of gas trapped in a cylinder by a frictionless piston. We specify ‘frictionless’ because if the piston moves, no energy is ‘lost’ due to friction.
The gas is heated till its temperature rises by 10C (= 1K), with:
Recalling the equation for the first law of thermodynamics:
Since R is a positive constant, CP > CV.
ISOTHERMAL AND ADIABATIC CHANGES (return to start of page)
So, in an isothermal expansion, the heat supplied to the gas equals the work done by the gas (and conversely for an isothermal compression).
Also, since the temperature is constant, the expansion curve follows a Boyle’s law P-V curve:
The area enclosed by the graph and the axes, between volumes V1 and V2, equals the work done during the volume change.
Conversely, in an adiabatic compression, work is done on the gas by an external agent, increasing the internal energy and so the temperature of the gas rises.
Notice that the slope (gradient) of the adiabatic curve through any point is greater than the slope of the isothermal curve through the same point.
Boyle’s law (PV = a constant) assumes that the temperature is constant, and so does not apply to an adiabatic change.
HEAT ENGINES (return to start of page)
A heat engine converts heat into mechanical work.
Heat engines operate by taking some ‘working substance’ (e.g. the gas in the cylinder of an internal combustion engine) around a repeating cycle, in which:
Efficiency of a heat engine
It can be shown that for an ideal heat engine:
The efficiency given by this expression is the maximum theoretically achievable for the temperatures of the source and sink. Real heat engines are much less efficient.
For example, suppose a steam turbine is driven by steam at 6000C which is exhausted at 1000C. Adding 273 to each temperature to get kelvins, then:
The actual efficiency is about 30%. The efficiency of a petrol engine is only about 20%.
Carnot’s ideal heat engine (1824)
Carnot considered an ideal heat engine, in which a working substance is taken reversibly through a cycle represented by:
The petrol engine cycle (or Otto cycle)
This is a four-stroke cycle: down - up - down - up
The diesel engine cycle
In both the above cases, the fuel is burnt inside the cylinder, and so both are internal combustion engines.
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A Level Physics - Copyright © A
C Haynes 1999 & 2004