CHEM 331 Physical Chemistry Fall 2014 Application of the First Law To The Construction of an Engine So, now we are in the thick of it. We have established a method for measuring the Internal Energy change U, which is a consequence of a system's transit from one equilibrium state to another. We have also considered how to measure the Heat flow Q across the wall separating the system from its surroundings during this process. And we have codified the relationship between U, Q and the work done by or on the system W in the form of a fundamental law of nature. This Law, the First Law of Thermodynamics states that for a given process: U = Q + W Now we wish to leverage the First Law to construct an engine, the system, capable of producing work in the system's surroundings. Why we want to do this will only become fully clear later. However, in short, an analysis of the efficiency of engines will lead us to another law of thermodynamics and another state function, the Entropy (S). This will allow us to define yet another state function, the Gibbs Free Energy (G), which will conclude our exploration of the thermodynamic framework. The Gibbs function will allow us to return to analyses of chemical problems using the thermodynamic framework embedded in the Gibbs function. to chemical systems. But this is getting ahead of ourselves; we still need to fully develop the framework. What type of engine to build? Our goal is to analyze the efficiency of an engine capable of converting heat into work. So, a piston based engine, one which uses heat to drive the expansion of gases within a cylinder, doing work as a result of this expansion, will serve our purposes. "Sectioned Petrol Engine" http://en.wikipedia.org/wiki/piston
First, we are familiar a number of these engines. Hot Steam injected into a cylinder fitted with a piston drives a Steam Engine. A gasoline or diesel engine uses hot gases, the products of combustion, in the engine's cylinder to push a piston outward and to produce work, propelling a car forward. The heating of a gas within the cylinder of Sterling engine also drives a piston which produces work. Also, it was the analysis of Steam engines that led to an enunciation of the Second Law of Thermodynamics. Now, importantly, will this choice of a piston based engine affect the generality of the conclusions we draw from its analysis? No! It turns out that our results will be completely general. But, again, we are getting ahead of ourselves. We must also choose a working fluid for our piston; an Ideal Gas. This choice is made practically because the expansion of gases will give us large amounts of work. Gases are capable of significant expansion, so the movement of the piston will also be considerable. We could use a hot liquid or solid, but the expansion of these substances will be minimal, so the resulting work will be likewise minimal. And, again, will the use of a gas as the working fluid lead to results which are completely general. Yes! But, again, we are getting ahead of ourselves. Finally, will choosing the gas to be an Ideal Gas affect the generality of our conclusions. No! This is simply a choice which simplifies our example calculations. As with our previous choices, this will in no way affect the generality of our results. But, again.. So, imagine a cylinder fitted with a piston and containing a compressed Ideal Gas; the massless piston being held in place by means of stops. Upon the top of the piston rests a mass m. The stops are removed, allowing the gas to expand and push the piston outward, lifting the mass. The system, the compressed gas, is doing work in the surroundings; lifting the mass. This work can be calculated according to: W = - mgh Since the mass is providing a force downward on the piston, we also have:
F downward = mg = P op A where P op is the pressure exerted on the piston which has a surface area of A. This opposing pressure is measurable because it is a part of the surroundings and how the surrounding interacts with the system. Now, W = - P op Ah Ah is simply the volume swept out by the piston as the gas expands. So, Ah equals the volume change of the cylinder V as a result of the gas's expansion. This gives us our final result: W = - P op V If the expansion is incremental, then: W = - P op dv Allowing for the fact that P op could change from one incremental expansion to the next, the Work done during an expansion process can be calculated according to: W = - As a specific example, consider an Ideal Gas at 2 atm confined in a cylinder to 10 L. A constant opposing pressure of 0.5 atm is applied to the piston. The stops are removed and the gas is allowed to expand to 20 L, at which point its pressure is 1 atm. Initially the gas is in an equilibrium state, denoted as State A, and will end in another equilibrium state, denoted State B. However, during the process which takes the gas from State A to State B, the gas's state is not specified. The piston will expand outward rapidly, causing the gas within the cylinder to become turbulent; hence its pressure is no longer well defined and it is not at equilibrium. This process can be represented by a P-V indicator diagram.
Note the use of a "dashed" curve to represent a process for which the internal state of the system is not determinable. We can now calculate the work done by the system as: W = - = - (0.5 atm) (20 L - 10 L) = - 5 Latm Work is done on the surroundings by the system, hence W is negative. Also, the unit Latm is a unit of energy. This could be easily converted to Joules, but why bother. Finally, the work is represented in the indicator diagram as the area under the P op vs. V curve. This is typical. Now imagine this process occuring as a result of a large number of infintesimal expansion stages in which the opposing pressure is always incrementally dp less than the pressure of the gas within the cylinder. Then, P op = P - dp
If the process is carried out extremely slowly, then the gas in the cylinder will be able to respond rapidly enough so that its equilibrium state can be quickly restored. Thus, we can assume the gas is always in an equilibrium state throughout the process. If at the completion of any incremental expansion, an opposing pressure of P + dp is applied, the gas will be compressed an return to its previous state. Hence, we refer to this path as Reversible. A reversible path is an idealization that can only be approximated for real systems. An engine operating reversibly will suffer no frictional or other extranseous energy losses during its operation. It will run cool to the touch, with no rocking and emitting no sound. During each incremental expansion of the gas, the work done will be represented as in our modified indicator diagram; an incremental slice of the P vs V curve. This work can now be calculated according to: W = - We have taken: as dp dv ~ 0 when the incremental change is small enough. For our Ideal Gas example, if 1 mole of the gas is maintained at a temperature of T = 243.7 K and the process is carried out reversibly, then: W = -
= - (1 mol) (0.08206 ) (243.7 K) = - 13.9 Latm The work produced by proceeding along this reversible and isothermal path is considerably greater than the work produced by proceeding along the previous irreversible path. In fact, by consulting the indicator diagram we note that the maximum work produced is produced by the reversible path. Reversing the process requires that we do work on the piston to compress the gas. If, for our example system, we carry-out the compression reversibly, then +13.9 Latm of work must be done to carry the system from State B back to State A. Suppose instead that we carry-out the compression in a sinlge stage with an opposing pressure of 5 atm. Then the work required for the compression is: W = - P op V = - (5 atm) (10 L - 20 L) = + 50 Latm More work is required to carry-out the compression irreversibly than is required for a reversible compression. Again, by consulting the indicator diagram, we note that the minimum amount of work required for the compression is that required for the reversible compression. This leads to a fundamental problem for our engine. After the expansion of the gas, we must recompress the gas in order to carry out another expansion. This cyclic operation of the engine is required if the engine is to operate is continuously. However, the maximum work produced is produced during a reversible expansion and the minimum work required for the compression is the same; - 13.9 Latm and + 13.9 Latm in our example: W cycle = W expansion + W compression = 0
Hence, no net work is produced as a result of the cyclic operation of the piston. In fact, it could be worse. If the piston is operated irreversibly, it may require that work be done on the piston to carryout the cycle. We will resolve this problem at a later date. It is really not all that difficult to resolve. But you should begin thinking about how to resolve the problem. We now turn to some general comments concerning our situation. We desire to drive our engine with heat input and obtain some work as a result. ( U = 0 for the system because the engine that is our system is operating cyclically.) In the famous experiments of Count Rumford on the boring of brass cannon barrels with a dull bit, Rumford showed that work can be converted into Heat. He drove the turning of the bit with the work of horses. He immersed the cannon into a barrel of Water and the resulting heat output raised the temperature of the Water to boiling. He argued that the heat output was a result of the motive force impressed on the cannon barrel. In fact, 100% of the work done on the system can be converted into heat.
Count Rumford Previously, heat output was thought to be the result of a caloric fluid leaving the system and entering the surroundings. For instance, a caloric fluid leaving a burning candle might enter a beaker full of water, thereby raising the Water's temperature. In fact, the calorie was defined as the heat needed to raise the temperature of one gram of Water from 14.5 o C to 15.5 o C. The results of Rumford's experiments demonstrate that there is a mechanical (work) equivalent to heat. Work, measured in Joules, can be used to generate heat, measured in calories. A early quantification of this equivalence was provided by the paddle-wheel experiment of James Prescott Joule.
James Prescott Joule http://jacques.boudier.pagespersoorange.fr/c_online/anglais/scientist/joule.htm http://www.juliantrubin.com/bigten/mechanical_equivalent_of_heat.html
In Joule's experiment, a weight is connected to the shaft of a paddle wheel immersed in a bucket of Water. The weight is allowed to fall a measured distance, doing a quantifiable amount of work on the Water via the turning of the paddle-wheel. The resulting temperature change of the Water is then measured, allowing for a calculation of the heating of the Water. The mechanical work is then equal to the heating of the Water. Joule found this equivalence to be: 1 calorie = 4.154 Joules Today, by agreement, the calorie is defined as exactly: 1 calorie = 4.184 Joules Can a cyclic engine be designed that will allow us to convert 100% of the heat input into work? An analysis by Carnot demonstrated that the answer to this question is no. He further demonstrated that there is an upper limit to the efficiency with which we can convert heat input into work. But, again, more about this later.