In the eighteenth century, and in the early years of the nineteenth, Newcommen, Watt, and others, were busy making steam engines. At that time the academic communities in the universities of Europe held that heat was a fluid. Real progress was not made until that mistaken idea was finally buried by an English brewer: James Prescott Joule.
Joule was a simple man. He lacked the sophistication and manners of the intelligentsia of the day, but he was careful, determined, and persistent to the point of obsessional. He showed, in an improbable, painstaking, difficult set of measurements made by heating water in a barrel with paddles driven by falling weights, that heat was a form of energy. From that starting point, and the brilliant definition and analysis of an ideal heat engine by Sadi Carnot, sprung the branch of physics known as thermodynamics.
Starting points:
1 A piston is free to slide in a cylinder which contains a 'Hooke's law' spring. The spring is fully extended at left when no force is applied to the piston. The red line on the graph shows how the force must increase to push the piston down reducing the distance from the bottom to x.
The piston is now compressed (pushed down), reducing the displacement from Lo to x.
2 The spring is replaced with air (ie. the piston is sealed in the cylinder and the spring is removed). Air behaves like an ideal gas. The red line in the graph shows how the force must increase to push the piston down, reducing the distance from the bottom to x. The graph is not a straight line. The piston is now pushed down, reducing the displacement from Lo to x.
3 The force-compression graph for the piston filled with air can be redrawn as a pressure-volume graph. Pressure is force over the cross sectional area Ax and volume is the distance x times Ax. The shape of the graph is the same, since the cross sectional area Ax is constant. Pushing the piston down slowly so that any heat generated escapes to the surroundings reduces the volume to V which equals x.Ax and increases the pressure. The curve on the PV diagram is part of a hyperbola. The area under the graph has the units of Newton meters since....
The work done is the area under the pressure-volume graph.
| Note: a line on a PV graph at constant temperature is a hyperbola (Boyle's law). The area can be found numerically, (or as a log function), by finding the integral P.dV between P1 and P2. |
That compressing a confined gas suddenly, heats the gas, and eventually the surroundings, is demonstrated with a fire piston.
Look again at the cylinder with the spring. Pushing the piston down does work DW. The work done is stored in the spring as elastic potential energy DU. Releasing the piston returns the work with no losses (in the ideal case). The quantity DU is called the internal energy of the system.
Compressing air is more complicated and interesting. Pushing the piston down does not store elastic potential energy in the gas. In fact, if the piston is pushed down very slowly so any heat generated in the gas [because molecules bounce of the descending piston with increased speeds] escapes to the surroundings, then the temperature of the gas is unchanged and the internal energy of the gas remains the same. DU is zero, and the work done on the gas is equal to the heat expelled to the surroundings.
We come now to the conservation of energy expressed as the first law of thermodynamics. A gas is confined in a cylinder under pressure. The gas in the cylinder is called the working substance. Adding heat (DQ) to the gas both increases the internal energy (DU) [average KE of the molecules] and may do work DW.
+ DQ is heat input.
+ DU is an increase in internal energy (temperature rise).
+ DW is work done by the gas.
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Sign convention In this treatment, the first law describes the workings of a heat engine. The sign convention employed by engineers is used. It is natural to think of the work done by the engine as positive. The opposite convention is used by chemists who are interested in the system itself, rather than the work done by the system. Chemists define work done on the system, which increases internal energy, as positive. It is very important to be clear about, and to state, the convention being used. |
If DQ were negative, heat would be given out by the gas. If DU were negative, internal energy would be reduced (temperature would fall) and if DW were negative, work would be done on the gas.
Example
Air is confined in a cylinder with a frictionless piston. Heat is removed from the gas and the temperature falls. The piston is forced down by external air pressure and work is done in a brake pressing on a turning wheel.
1 The heat removed from the gas DQ is negative.
2 The change in internal energy DU is negative.
3 Work is done on the system by external air pressure. The work done during the contraction DW is negative.
When the piston rises work done is positive. When the piston falls work done is negative.
Note: additional work is done by external air pressure to generate heat in the brake that escapes to the surroundings. Work is not done by the system on the brake.
1 Two cylinders containing the same mass of air are heated. No heat escapes to the surroundings. Temperature is color-coded red. The more red, the higher the temperature.
Explain why the gas on the right is at a higher final temperature than the gas on the left.
2 Two identical cylinders of air at room temperature are fitted with sealed pistons. The pistons are pushed down at different speeds.
a Explain the two processes.
b Explain why an isothermal process cannot be realized exactly in practice.
c The diagram for question 2 is redrawn. In what ways does the second diagram more nearly conform to a real case.
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A clever trick Air and water are enclosed in a container with a small hole below the water line. 1 Heat (DQ) is added to the air in the head. 2 The internal energy (DU) of the air increases. (ie. The temperature rises). 3 The hot air expands, doing work (DW) as water is forced out the hole |
3 A Hooke's law spring is extended to its natural length at left. The spring constant is 2000 N/m and the scale is in cm.
a Estimate the length of the spring in the three positions.
b Sketch the force compression graph.
c Find the work done in moving the piston from the left hand to the middle diagram.
d Find the work done in moving the piston from the middle to the right hand diagram.
4 The spring is removed and the piston is sealed in the cylinder so that it slides freely and confines air in the piston. The piston is in equilibrium in the left hand diagram and the compressions are done slowly.
a Sketch the force compression graph.
b Mark the area on your graph that represents the work done in moving the piston from the left hand to the middle diagram.
c Mark the area on your graph that represents the work done in moving the piston from the middle to the right hand diagram.
For a more detailed discussion of the First Law see The Carnot cycle.