The philosophy of biology

by James Johnstone

THE CARNOT POSITIVE CYCLE

We have therefore a substance which can be heated by contact with a hot body, and which can then expand, doing mechanical work by raising a piston, and perhaps turning a flywheel, and on which work is then done so that it returns to its original condition. This is a cycle of operations. If we consider only the changes which occur in the working substance we can represent these changes by a diagram.

First operation, (1→2). We suppose that the valve is turned so that the non-conducting plug closes the cylinder. The piston is in the position II (Fig. [31]). Heat cannot then enter or leave the gas. But the latter already contains heat: it is at a temperature of T₂°, so that it can expand doing work. Let it expand, forcing up the piston. During this operation the pressure of the gas will fall from a point on the vertical axis opposite 1 to a point opposite 2, and its volume will increase from a point on the horizontal axis beneath 1 to a point beneath 2. It will cool because it has expanded, and no heat is allowed to enter it during this act of expansion. The expansion is therefore adiabatic; the temperature falls from T₂° to T₁°; and work is done by the gas.

Fig. 32.

Second operation, (2→3). The piston is now at the position I, that is, at the upper end of its stroke, and we must bring it back again to the lower end of the cylinder. The valve is turned so that the bottom of the cylinder is placed in thermal communication with the refrigerator (−), and the piston is pushed in to the position II. The gas is therefore compressed until its volume decreases from a point beneath 2 to a point beneath 3. As it is being compressed, heat is generated and its temperature would rise, but as this heat is generated it flows into the refrigerator, so that the temperature of the gas remains the same during the operation. The contraction is therefore an isothermal one; the temperature remains at T₁°; and work is done on the gas from outside.

Third operation, (3→4). But the piston is not at the lower end of its stroke yet. We turn the valve so that the bottom of the cylinder is closed by the non-conducting plug O, and then push in the piston until it reaches the position III. The gas is still further compressed, and this compression generates heat. But the heat cannot escape, so that the temperature of the gas rises until it reaches T₂°. The contraction is therefore an adiabatic one. Work is done on the gas.

Fourth operation, (4→1). The piston is now at the lower end of its stroke. We turn the valve so that the bottom of the cylinder is placed in communication with the source of heat (+). The gas expands from the point beneath 4 to the point beneath 1, raising the piston to the position II. This expansion of the gas would lower its temperature, but it is in communication with the source of heat, and so it does not cool, but draws heat from the source and remains at a constant temperature, T₂°. The expansion is therefore an isothermal one. Work is done by the gas.

This completes the cycle. But the gas is heated, and when the piston is at position II, the valve is turned so as to close the cylinder by the non-conducting plug O. The heat already contained in the gas continues to expand, the latter doing more work, but this expansion causes the temperature to fall from T₂° to T₁°. This is the operation with which the cycle commenced.

Summarising the positive Carnot cycle, we see that the engine takes heat from a source (+) and gives up part of this to a refrigerator (−), (in an actual steam-engine heat is taken from the boiler and given up to the condenser water). If we measure the quantity of heat taken from the boiler in the steam which enters the cylinders we shall find that this quantity of heat is greater than the quantity which is given up to the condenser water. What becomes of the balance? It is converted into the mechanical work of the engine. The Carnot engine therefore takes a quantity of heat, Q₂, from the source and gives up another quantity of heat, Q₁, to the refrigerator. We find that Q₂ is greater than Q₁ and the balance, Q₂ − Q₁, is represented by the work done by the engine. Heat-energy falls from a state of high, to a state of low potential, and is partly transformed into mechanical work.