Basic concepts and definitions
Thermodynamics begins with a decision rather than an equation: what, exactly, are you talking about. Draw that line in the wrong place and a two-minute problem becomes unsolvable, while the algebra you eventually write looks perfectly correct.
The definitions in this topic look like the sort of thing you skim on the way to the interesting material. They are not. Almost every thermodynamics problem that refuses to come out has gone wrong in the first thirty seconds, in the choice of what to call the system, or in a quiet assumption that some property is fixed when it is not. The equations of the first and second laws are short. What makes them hard to apply is that they refer to a system, a state and a process, and each of those three words carries more weight than it appears to.
System and boundary
A system is whatever quantity of matter or region of space you have chosen to study. Everything else is the surroundings, and the surface between them is the boundary. The point to hold on to is that none of these is discovered. You choose them, the same way you choose a free body in statics, and the choice is the first and most consequential step in the solution.
A boundary may be real or imaginary, and fixed or moving. The steel wall of a tank is a real fixed boundary. The face of a piston is a real moving one. A plane drawn across a pipe where the flow enters a turbine is entirely imaginary, and it is no less legitimate for that. What the boundary has to be is closed in the geometric sense — it must have an inside and an outside — and it must have zero thickness, so that it contains no mass and stores no energy of its own.
Three cases matter. A closed system, also called a control mass, is one whose boundary no mass crosses: \(m\) is fixed, and the same molecules are inside at the end as at the beginning. Energy is free to cross in the form of heat or work, and the boundary may move, as it does when a gas pushes a piston out. An open system, or control volume, is a region that mass flows through — a nozzle, a pump, a steady-flow device of any kind. An isolated system is one that nothing crosses at all, neither mass nor energy.
Isolation sounds like an idealisation with no use, and as a description of real hardware it is. Its real job is as a construction: take any system together with every part of its surroundings it interacts with, and the composite is isolated by definition. Much of the second law is argued that way, by insisting that something can never decrease for the system and its surroundings taken as a whole.
The reason the choice of boundary is worth this much attention is that it decides which quantities cross it, and therefore which terms appear in your energy balance. Consider a rigid insulated tank with a stirring paddle inside, driven by a falling weight. Take the gas as the system and there is shaft work crossing the boundary at the shaft. Take the gas, the paddle, the shaft and the weight as the system and there is no work crossing at all — the weight has simply fallen, inside. Both analyses are correct and they give the same answer, but they do not contain the same terms, and mixing the two accounts is how the paddle work ends up counted twice.
One consequence is worth stating flatly, because it is the root of a persistent confusion. Heat and work are defined only at a boundary, during a process. They are things that cross, not things a system has. A system contains energy; it does not contain heat, and it contains no work whatsoever. Sentences beginning “the heat in the gas” are not merely loose, they are the reason the first law comes out wrong. This is developed properly in heat, work and energy.
Hardware is not a system. Nothing has been decided yet.
Properties and state
A property is any macroscopic characteristic of a system to which a value can be assigned: pressure, temperature, volume, mass, internal energy, and a good many more. Properties come in two kinds, and the test that separates them is the only one you need. Divide the system in half. Whatever halves along with it is extensive; whatever does not is intensive.
Mass, volume, total energy and total entropy are extensive. Pressure, temperature and density are intensive: half a tank of air at 300 K and 200 kPa is still at 300 K and 200 kPa. Divide an extensive property by the mass and you get a specific property, which is intensive by construction, and which is written in lower case:
The case of the letter is doing real work there, and it is worth being fussy about it from the start, because \(V\) in cubic metres and \(v\) in cubic metres per kilogram look almost identical in a hurried line of algebra and differ by a factor of the mass. Note also that specific volume is the reciprocal of density, \(v = 1/\rho\), so any relation quoted in one can be rewritten in the other.
The state of a system is the complete set of values its properties take at one instant. Give the state and you have given everything: no history, no account of how the system arrived, just the numbers. That is what makes a property a point function. Its change over a process depends only on the two end states,
and never on the route between them. Heat and work are not properties and do not behave this way, which is the whole reason they are written with a different differential and the whole reason they cannot be tabulated.
Listing every property to specify a state would be hopeless, and the state postulate says you do not have to. The state of a simple compressible system — one with no significant electrical, magnetic, gravitational, motion or surface-tension effects — is completely fixed by two independent intensive properties. Two numbers, and every other property in the tables follows. The count is not arbitrary: it is one for each quasi-equilibrium work mode the system has, plus one, and a simple compressible substance has exactly one work mode, the moving boundary.
Everything then rests on the word independent, and this is where the postulate is misapplied. Two properties are independent if one can be varied while the other is held constant. Pressure and temperature are independent for a single-phase substance, and they are emphatically not independent during a phase change: water boiling at 101.3 kPa sits at 100 °C whatever you do, so quoting both tells you one thing rather than two, and cannot distinguish nearly-liquid from nearly-vapour. Under the saturation dome you need a pair that still varies, such as pressure and specific volume, or pressure and quality. That case is the subject of properties of pure substances.
One vessel of gas. Its state is the whole list at once.
Equilibrium and processes
Every property value quoted so far assumed something that is almost never true: that the system has one pressure and one temperature rather than a distribution of them. A system in equilibrium has no unbalanced potentials inside it, so a single value describes it throughout, and only then can it be marked as a point on a chart. Away from equilibrium the question “what is the pressure of the gas?” has no answer, because different parts of it have different pressures.
Equilibrium is not one condition but four, and a system is in thermodynamic equilibrium only when it satisfies all of them. Thermal equilibrium means the temperature is the same everywhere, so nothing drives heat from one part to another. Mechanical equilibrium means the pressure does not change with time, allowing for the hydrostatic variation with depth that gravity insists on. Phase equilibrium means the mass of each phase has stopped changing. Chemical equilibrium means the composition has stopped changing — no reaction is still running.
A process is any change from one equilibrium state to another, and the series of states passed through on the way is the path. A process is specified by its end states, its path, and the interactions with the surroundings along it. The names follow the property held constant: isothermal at fixed \(T\), isobaric at fixed \(P\), isochoric at fixed \(v\), adiabatic with no heat crossing the boundary at all. Adiabatic is the odd one out — it names a boundary condition, not a constant property, and an adiabatic process changes the temperature quite happily.
Here is the difficulty. If a process must run through equilibrium states, and equilibrium takes time to reach, then no real process qualifies, and no real process could be drawn as a line on a chart. The repair is the quasi-equilibrium process: one carried out slowly enough that the system stays infinitesimally close to equilibrium throughout, so that at every instant it has a definite pressure and temperature. It is an idealisation, and it is worth keeping for two reasons. Many real devices are close enough to it — a piston in an engine moves far more slowly than the molecules redistributing themselves behind it — and, more importantly, quasi-equilibrium work is the best that can be done, delivering the most work from an expansion and demanding the least for a compression. It is the benchmark real machinery gets measured against.
The practical consequence is one that gets ignored every year. If a gas expands suddenly into a vacuum, or a piston is released and slams outward, there is no single pressure during the process, so the path does not exist and the integral \(\int P\,dV\) cannot be evaluated. Drawing a neat curve between the two states and integrating under it is not an approximation in that case, it is an answer to a different question. The end states are still perfectly well-defined and can be marked on the chart; only the line joining them is illegitimate, and it is conventionally drawn dashed to say so.
When a process returns a system to the state it started from, it is a cycle. Because properties depend only on state, every property change around a cycle is exactly zero,
which is why engines and refrigerators are analysed as cycles: whatever the working fluid did in between, it has finished where it began, and the net heat and net work are all that is left to account for. Steady flow is a related but distinct idea, and the two are often confused. A steady-flow device is not in equilibrium — its inlet and its exit are at quite different states — but nothing at any fixed point in it changes with time.
Two states. Each is a point: one value for every property.
Temperature and the zeroth law
Temperature is the property everyone has an intuition for and nobody can define without going in a circle. Hot and cold are sensations, and poor ones: a steel rail and a wooden sleeper at the same temperature feel nothing like each other. What thermodynamics needs is not a sensation but a rule guaranteeing that a number measured on one body can be compared with a number measured on another, and that rule has to be assumed rather than derived.
The zeroth law states that if two bodies are each in thermal equilibrium with a third body, then they are in thermal equilibrium with each other. It reads like an obvious remark about transitivity, which is presumably why it was written down decades after the first and second laws and then had to be numbered backwards. It is also exactly what a thermometer relies on. Touching a thermometer to a block tells you about the thermometer; the zeroth law is the permission to conclude something about two blocks that were never touched to each other. Every temperature measurement ever made is an application of it.
Given that, a scale is a matter of convention. Celsius and Fahrenheit were fixed by picking two reproducible states and dividing the interval between them. The thermodynamic scales, Kelvin in SI and Rankine in English units, are absolute: their zero is not a chosen reference point but the temperature at which the pressure of an ideal gas held at constant volume would extrapolate to nothing. The relations are
and that extrapolated zero, \(-273.15\ ^\circ\)C, cannot be reached, only approached. It is worth noticing that a constant-volume gas thermometer defines the scale by a property relation rather than by a substance, which is what makes it absolute in any useful sense; the ideal-gas behaviour it depends on is the subject of ideal and real gases.
Because the two scales differ only by an offset and not by a scale factor, a temperature difference is the same number in either:
A 20-degree rise is 20 K and 20 °C, and adding 273.15 to it is simply wrong. The mirror-image error is more expensive. Wherever an absolute temperature appears on its own rather than as a difference — in \(Pv = RT\), in the efficiency of a reversible engine, in any entropy calculation — the value must be in kelvin. Feeding 25 into the ideal gas law because the air was at 25 °C understates the temperature by a factor of twelve, and the answer that comes out is not slightly wrong but nonsense. The habit worth building is to convert every temperature to kelvin the moment it is written down, and to convert back only at the end if somebody asked for Celsius.
Two blocks that have never met, and one small third thing.
Pressure
Pressure is the normal force exerted by a fluid per unit area, measured in pascals, where one pascal is one newton per square metre. The unit is inconveniently small, so real quantities come in kilopascals and megapascals, and in bars and atmospheres in the literature: one bar is exactly 100 kPa and one standard atmosphere is 101.325 kPa, which is close enough to a bar to be dangerous when precision matters. In a fluid at rest the pressure at a point is the same in every direction, which is why a single number describes it at all.
The distinction that causes the most damage is between absolute and gauge pressure. Absolute pressure is measured from a perfect vacuum. Almost every instrument, though, is open to the atmosphere on its other side, and so reads the difference between the pressure in the vessel and the pressure of the air outside:
The second of those is just the first written so that a reading below atmospheric comes out positive, which is a convenience for the person reading the dial and a trap for everybody else. Both are differences, and neither belongs in a property table or an equation of state. Every steam table, every ideal-gas calculation and every saturation temperature is quoted in absolute pressure, so a gauge reading of 200 kPa on a tank at sea level enters the algebra as roughly 301 kPa. Doing otherwise is the most common single arithmetic error in a first thermodynamics course, and it is invisible afterwards, because the number that results is perfectly plausible.
Within a static fluid the pressure varies with depth only, and the governing relation comes from balancing the weight of a column against the pressures on its ends. For a fluid of constant density it integrates to
with \(h\) measured downwards from wherever \(P_0\) is known. Two things follow. Horizontal distance does not appear, so any two points at the same level in the same connected body of the same fluid are at the same pressure, however tortuous the path between them. And \(\rho\) for a gas is so small that the variation across a tank of air is negligible: gas columns in a manometer are almost always ignored, and doing so is justified rather than lazy.
A manometer is that relation used as an instrument. A U-tube of liquid connects the vessel to the atmosphere, and the difference in the two levels reports the pressure difference directly. Worked properly it is a walk from one open end to the other: start at the known pressure, add \(\rho g h\) for every descent, subtract it for every rise, and ignore the gas legs. The errors are consistent and worth naming. Getting the sign backwards, because going down raises pressure and going up lowers it, and the mind wants to associate “up” with “more”. Using the length of the tube rather than the vertical height when the manometer is inclined, which throws away the very sensitivity the inclination was there to provide. And forgetting that the fluid must be continuous and the same throughout for the equal-level argument to hold, so that a mercury and water manometer needs the interfaces tracked one at a time.
The barometer is the same equation with the vessel replaced by a sealed evacuated tube, so that \(P_0 = 0\) and the height of the column measures atmospheric pressure outright. That is where the old habit of quoting pressure in millimetres of mercury comes from, and it explains why the standard atmosphere is 760 of them. It is also the cleanest demonstration of what a manometer actually does: the column never knows the pressure of anything, it only ever balances a difference.