Note: This is an ADVANCED HIGHER LEVEL topic. Although all may learn from it, only HL students will be tested on its content.
Thermodynamics is the study of the relationships between heat, energy, work and temperature. In this higher level only topic we will look in more detail at how we can analyse energy transfer and energy storage in a closed system. We will look at how we can use this information to predict the future state of a system and finally we will look at the concept of entropy and its implications - including the implications for the future of our whole universe!
Conservation of energy states that the total amount of energy in an isolated system must remain the same. No energy or mass can move in or out of an isolated system. In a closed system no mass can enter or leave but energy can be transferred in both directions as heat, Q, or work, W. In other words, if an object, representing the closed system, has a total internal energy, U, then the possible changes to the internal energy are heat energy transfers from or to the surroundings, Q, or the object doing work on it's surroundings or having work done on it, W. All of these quantities represent energies so they all have the same unit, joules.
This relationship can be described using the equation:
Q = ∆U + W
Not that sign convention, as shown in the following example and called the Clausius convention, is important.
The classic example to illustrate this system is a sample of ideal gas in a frictionless piston. The gas has its internal energy (U) which, as we saw in the previous topic, is wholly comprised of its total kinetic energy, so is directly proportional to the temperature of the gas. If the gas is heated (+Q) and the gas is not changing volume (so W is zero) then the internal energy must increase. If the gas is cooled (-Q) and the gas is not changing volume (so W is zero) then the internal energy must decrease. W represents the work done by the piston. This is mechanical work, which if you recall from the first topic can be described as force multiplied by displacement. In other words, if our piston expands, which requires a force being applied over a distance then this is positive work being done by the piston (+W). Similarly the opposite, the piston being compressed, would be work done on the piston (-W).
Again, by conservation of energy, any change to Q, U or W will result in a corresponding to change to one, or both, of the other variables.
Above we gave the example of piston expanding as work being done by a gas. Expanding a piston will result in an increase in the volume of the gas (since gases expand to fill the volume of the container). A consequence of this is that the pressure, P, of the gas will decrease. From the argument developed in the pressure - velocity relationship last unit increasing the volume will increase the time between collisions for the gas particles and the walls of the container, which will reduce the rate of change of momentum and therefore the force and therefore the pressure. This relationship can be described by the equation:
W = P ∆V
where W is the work done in joules, P is the gas pressure in pascals and ∆V is the change in volume in m³.
At the end of the last topic we introduced the equations describing the internal energy of an ideal gas:
U = 3/2 × N kB T and U = 3/2 × R n T
both of these indicate that internal energy, U, and temperature, T, are directly proportional. This corresponds with our existing knowledge. The only component of internal energy for an ideal gas is the total kinetic energy of the molecules and the average kinetic energy of the molecules is what determines the temperature. Therefore any change in the internal energy of gas, U, by heating, Q, and/or doing work, W, will result in a change in the temperature of the gas by the equation:
∆U = 3/2 × N kB T = 3/2 × R n T
Note that is possible for heating a gas not to increase the temperature if the gas does work at the same rate as the heating, i.e. Q and W balance leaving U unchanged. This situation is called isothermal expansion.
The entropy of a system, represented by the symbol S, is a thermodynamic property that is related to the amount of disorder of the particles of a system. Disorder (and entropy) can be hard to define, but in the case of systems such as the ones we have been examining (arrangements of particles) a highly disordered system (one with high entropy) is one where many different arrangements of microscopic particles will give identical macroscopic properties. For example, gases have more disorder than liquids, and liquids than solids because there is a far greater range of particle positions and velocities that describe a given gas than describe a given liquid or solids. To put it another way, describing a gas in terms of volume, temperature and pressure will give us less information about the arrangement of the atoms than a similar description of a liquid and, particularly a solid. A hot gas has more entropy than a cold gas for similar reasons, a hot gas has a greater range of possible particle velocities. There are many connections between entropy and information.
Mathematically this relationship can be described by the equation:
S = kB ln Ω
where S is entropy, kB is again Boltzmann's constant and the upper-case Greek letter omega Ω represents the number of microstates of the system, i.e. the number of possible arrangements that can lead to an identical macroscopic state. An example of difference microstates may be found in pile of coins of different value. By shaking and making a pile by using the same set of coins then you can have a large number of equally probably but different arrangements of the coins (the microstates), while the sum remains the same (the macrostate).
An application of entropy to an ideal gas system relates the change of entropy to the heat energy added to a system via the equation:
∆S = ∆Q / T
where ∆S is the change in entropy, ∆Q is the heat energy added in joules and T is the temperature of the system in kelvin. From this relationship we can see that the units of entropy are J/T. Note that this equation assumes a constant T, which is often not the case when heat is added or subtracted from a system. One such system is isothermal expansion as described above. In this case you can see that by this formula adding energy will increase the entropy of the system. This also matches the first equation, because in isothermal expansion the volume of the system must increase (hence expansion) resulting a greater number of microstates as there are more positions the particles can occupy.
Using the above relationship we can approximate the entropy change during a process. We will find that although we can reduce the entropy of part of a system, make that part less disordered, then in other parts of the system the entropy will be increased, and always by the same of more than the first part was reduced. In fact, in all real world situations:
The total entropy of any system (including its environment) will increase as a result of any natural process.
This means that any physical process or procedure that claims to result in an overall reduction in entropy is impossible. Unlike most physical laws which are a statement of a conservation law, where some physical quantity is conserved, this is an opposite situation. Entropy always increases. A consequence of this law is that natural processes always move from order to disorder.
Intuitively this makes sense. It is often easy to create more disorder, by burning a log, dropping a plate or mixing two powders. It usually takes a lot more energy to return these processes to their starting point, if it is possible at all. Most of them can be considered irreversible.
In the previous section on gas laws, it was mentioned that changes in the equation of state of an ideal gas can be plotted in a pressure - volume diagram. Here we will describe certain categories of change and how they appear on a P-V diagram.
Isovolumetric: This is equivalent to Gay-Lussac's law. Volume is held constant. Example is heating a rigid container of gas that cannot expand or contract. A vertical line on a P-V diagram.
Isobaric: This is equivalent to Charles's law. Pressure is held constant. Example is heating a balloon that can easily expand or contract such that the internal pressure always matches the external pressure. A horizontal line is a P=V diagram.
Isothermal: This is equivalent to Boyle's law. Temperature is held constant. Example is expanding or contracting a piston very slowly such that the piston always has time to be in equilibrium withe the surroundings. A relatively shallow curved line on a P-V diagram that follows an "isotherm"
Adiabatic: In this process no heat transfer is allowed, Q is zero is the first law equation, Q = ∆U + W. In real-world situations this is achieved by rapid expansion or compression that does not allow time for heat transfer to occur. Shown by a relatively steep curved line on a P-V diagram that connects two different isotherms. Approximate equation for an adiabatic process in a monoatomic ideal gas is PV^5/3 = constant. Note that all quantitative problems in thermodynamics will be restricted to monoatomic ideal gases.
A cyclic gas process is one in which all of the variables of the equation of state return to their original values and can be repeated. On a P-V diagram this is represented by a closed loop. A heat engine is a process for continuously converting heat energy to mechanical work. It was the invention of development of heat engines that drove the industrial revolution and, consequently, it was the quest to build a better heat engine that drove the development of much of the science discussed in this section.
In all heat engines heat, Q, be transferred from a hot temperature, TH, reservoir to a cold temperature, TC, reservoir. Some (but never all) of that heat energy can be diverted to doing work, W. Steam engines and internal combustion engines are classic examples of heat engines.
Different heat engine designs use different cyclic processes and plot different P-V diagrams. Operationally their main distinction is efficiency. The symbol for efficiency is the Greek letter eta, η.
η = useful work / input energy
As a ratio, η has no units.
One cyclic process is called the Carnot cycle (or Carnot engine). In an ideal Carnot cycle (no real world application gets quite this good), the efficiency of the heat engine is determined by the operating temperatures (TH and TC) of the reservoirs used by the engine.
ηCARNOT = 1 − (TH / TC)
The fact that no heat engine can be 100% efficient is a direct consequence of the Second Law of Thermodynamics and was the origin of its formulation. In the Kelvin form, as written by Lord Kelvin (As William Thompson, forty years before he was created 1st Baron Kelvin) in 1851:
It is impossible, by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.
In other words, for mechanical work to be done the heat engine must always be warmer than the surroundings, TH > TC.
Alternatively, as Rudolf Clausius put in (1854 in German, 1856 in English, known as the Clausius form):
Heat can never pass from a colder to a warmer body without some other change, connected therewith, occurring at the same time.
Or to put this in the reverse, heat always flows from hot to cold (got things will get colder and cold things will get hotter) unless some mechanical action takes place to reverse the flow. We can do this - it is the basis of refrigeration and cooling by air conditioning, but in all cases the local decrease in entropy is more than exceed elsewhere by an overall increase in entropy.
A final statement of the second law is known as the Kelvin-Planck statement. It says that because no engine can be 100% efficient:
no device is possible whose sole effect is to transform a given amount of heat completely into work.
This statement itself is sometimes called the third law of thermodynamics.
Cosmology is the science of the development and the future of the universe, and the second law of thermodynamics has significant consequences in cosmology. The current best and most widely-accepted theory of cosmology states that our universe began in a hot dense state about 13.75 billion years ago. This was a very low entropy system with all matter soon being a sea of mostly hydrogen at high temperatures. As the universe has evolved it has expanded and much of the hydrogen collapsed to form stars and galaxies. These stars are, through nuclear fusion reactions, transforming much of that matter into high entropy pure energy as time passes. Indeed some physicists and philosophers believe that it is this consequence of the second law of thermodynamics that gives time direction, that entropy provides the arrow of time. The second law predicts that eventually the universe will reach an end state in which all matter is converted into diffuse energy. This end state is known as the heat death of the universe.
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