Systems, surroundings and the bookkeeping of energy

A thermodynamic system is the part of the physical world we choose to analyse, focusing on its energy transfers and changes of state. Everything outside that choice is the surroundings. Together, the system and surroundings form the thermodynamic “universe” for the problem. Pick the system first. Many sign errors in this topic come from leaving that choice vague.

A closed system is a thermodynamic system that cannot exchange mass with its surroundings, though it can still exchange energy by heating or by doing work. The standard IB example is a fixed mass of gas under a piston: gas particles do not enter or leave, but energy can cross the boundary.

The first law of thermodynamics states conservation of energy for a closed system. It links the energy transferred into the system to the change in its internal energy and to the work done by the system:

Q = ΔU + W, where Q is the resultant energy transferred to the system by heating (J), ΔU is the change in internal energy of the system (J), and W is the resultant work done by the system on the surroundings (J).

IB uses the Clausius sign convention. Positive Q means energy is supplied to the system by heating. Positive ΔU means the system’s internal energy store increases. Positive W means the system does work. So work done on a system is negative, while work done by a system is positive. Keep that convention throughout a calculation; if you mix conventions, the answer can look algebraically neat but be physically meaningless.

What the first law does and does not say

The first law tells us that work and heating transfer energy across the system boundary. It does not tell us which processes can happen naturally. Conservation of energy alone, for example, would not forbid energy flowing spontaneously from a cold object to a hot object; the second law deals with that direction-of-change question later in this topic.

Joule’s paddle-wheel experiments mattered historically because they showed that mechanical work done on water can raise its temperature. In modern terms, doing work on a system can increase its internal energy, just as transferring energy by heating can.

Boundary work

When a gas pushes a movable piston, the system boundary shifts. The gas applies a force through a distance, so it does work. For a small change at constant pressure,

W = PΔV, where P is the gas pressure (Pa) and ΔV is the change in volume of the gas (m³).

During expansion, ΔV is positive, so the gas does positive work on the surroundings. During compression, ΔV is negative, so the work done by the gas is negative; equivalently, work is being done on the gas.

The same result comes from thinking about a piston of cross-sectional area A, where A is the piston area (m²), moving a distance Δx, where Δx is the piston displacement (m). The gas force is P A, and the volume change is ΔV = AΔx. So the work is force × distance = P AΔx = PΔV.

P–V diagrams and area

A P–V diagram is a graph of pressure against volume used to represent a gas process. On a pressure–volume graph, the work done by the gas is the area under the process curve. That still applies when the pressure changes; in quantitative IB problems, you may have to estimate the area by counting squares or by splitting the curve into thin vertical strips.

For a curved path, you can approximate the area by adding rectangular strip areas: W ≈ ΣPᵢΔVᵢ, where Pᵢ is the pressure for strip i (Pa), ΔVᵢ is the width of strip i in volume (m³), and Σ means “sum over all strips”. Thinner strips give a better estimate, but use practical judgement too: in an exam, the test is usually whether your area estimate is sensible.

Watch out for false origins. If a P–V graph does not start from zero pressure and zero volume, the visible rectangle may not be the whole area under the graph. Work is the full area down to the volume axis, not just the neat-looking part inside the plotted grid.

Internal energy of a monatomic ideal gas

Internal energy is the total microscopic kinetic and potential energy associated with the particles of a system. In the ideal gas model, there is no intermolecular potential energy, so the internal energy depends only on temperature. For a monatomic ideal gas,

ΔU = (\frac{3}{2})N kB ΔT = (\frac{3}{2})n RΔT, where N is the number of particles, kB is the Boltzmann constant (J K⁻¹), ΔT is the change in absolute temperature (K), n is the amount of substance (mol), and R is the molar gas constant (J mol⁻¹ K⁻¹).

The useful thing about this equation is that the route taken does not matter. Stir the gas, compress it, heat it slowly, or heat it quickly: for a monatomic ideal gas, ΔT alone fixes the change in internal energy.

Combining with the first law

The first law tells you where that temperature change comes from. If energy is supplied to a gas while the gas expands, part of the supplied energy increases internal energy, and part leaves as work done on the surroundings. If a gas is compressed adiabatically, Q = 0, and the negative value of W means ΔU is positive: work done on the gas raises its temperature.

Quantitative problems in this topic are limited to monatomic ideal gases, including cases where pressure is not constant. When pressure varies, don’t force W = PΔV using one pressure value unless the process is actually constant pressure. Use the area under the P–V curve for work, and use the temperature change equation for ΔU.

Entropy and disorder

Entropy is a thermodynamic state quantity: it measures how many microscopic arrangements can give the observed macroscopic state of a system. Its symbol is S, where S is entropy (J K⁻¹).

The phrase “degree of disorder” can help, but only if used carefully. A gas spread through a whole container is more disordered than the same gas confined to one corner, because its particles can occupy far more possible positions. A crystal with a missing atom has more disorder than a perfect crystal, since the vacancy could be in many possible places.

Entropy is not a substance. Nor is it “mess” in the everyday sense. It is a state property linked to probability: macroscopic states with many microscopic arrangements are overwhelmingly more likely than macroscopic states with very few microscopic arrangements. That is why thermodynamics can predict the future evolution of large systems statistically, even though individual particle motion is random.

Macroscopic definition

For a reversible energy transfer at constant temperature, calculate the entropy change using

ΔS = ΔQ / T, where ΔS is the change in entropy (J K⁻¹), ΔQ is the energy transferred reversibly into the system by heating (J), and T is the absolute temperature at which the transfer occurs (K).

Watch the sign. When energy enters the system, ΔQ is positive, so the system’s entropy increases. When energy leaves the system, ΔQ is negative, so the system’s entropy decreases. Use this equation when the temperature is constant or changes only negligibly; calculus versions are not required for IB Physics.

Microscopic definition

A microstate is one specific microscopic arrangement of the particles that gives the same observed macroscopic state. For example, if ten particles are divided between the left and right halves of a box, “five on the left” is a macrostate; the particular choice of which five particles are on the left is a microstate.

Entropy can also be written as

S = kB ln Ω, where Ω is the number of possible microstates of the system and ln is the natural logarithm.

In the simple models used here, all microstates are assumed to be equally probable. That is why counting helps: a macrostate with more microstates is more likely to be observed.

A coin or counter model shows the idea well. If each counter can be in either of two boxes, every extra counter doubles the total number of arrangements. The evenly split macrostate has many more microstates than the “all counters in one box” macrostate, so it is much more likely. For a mole of gas particles, the difference in probabilities is so extreme that the unlikely macroscopic states are never seen in practice.

Linking the two views

The macroscopic equation ΔS = ΔQ / T and the microscopic equation S = kB ln Ω describe the same physical idea in different ways. When a gas expands into a larger volume, Ω increases because the particles have more possible positions. For an isothermal expansion, energy supplied to the gas is converted into work while the temperature remains fixed, and the same entropy increase can be described either by the energy transfer or by the increased number of microstates.

Three equivalent ways to say the second law

An isolated system is a thermodynamic system that cannot exchange either mass or energy with its surroundings. So for an isolated system, there is no “outside” energy transfer that can step in and make an impossible process happen.

The second law of thermodynamics restricts the direction of real thermodynamic processes. Energy conservation is not enough on its own; the second law tells us which energy transfers can occur spontaneously.

In the Clausius form: energy cannot be transferred spontaneously from a colder body to a hotter body. To make this happen, work must be done on a device or system. That’s why a fridge needs electrical work to cool its interior.

In the Kelvin form: no heat engine operating in a cycle can take energy from a single hot reservoir and convert it entirely into work. Some energy has to be rejected to a colder reservoir. This is the deeper reason no cyclic engine is 100% efficient.

In entropy form: the entropy of an isolated system cannot decrease. For a reversible process in an isolated system, the entropy change is zero. For an irreversible process in an isolated system, the entropy increases.

Consequences for the universe

If we treat the universe as the largest isolated system, the second law says that its total entropy tends to increase. Temperature differences act as the “fuel” for useful processes, such as engines, weather systems, and stars radiating to space. In the very long term, if all temperature differences were smoothed out and entropy reached a maximum, no heat engine-like process could extract useful work. This idea is often called the heat death hypothesis. It is a physical extrapolation from the second law, not an experiment we can actually run on the universe.

Why real processes do not simply run backwards

A reversible process is an idealised process that can be reversed by an infinitesimal change, with no net change left in the system or the surroundings. For a process to be reversible, it must be quasi-static: the system is effectively in thermodynamic equilibrium at every stage.

An irreversible process is a real process that cannot be undone without causing a net change somewhere else. Friction, turbulence, unrestrained expansion, mixing and heat transfer across a finite temperature difference all make processes irreversible.

In real isolated systems, processes are almost always irreversible because energy spreads out among microscopic degrees of freedom. A piston moving with friction warms its surroundings; a gas expanding freely into a vacuum fills the available volume; two gases mix. Conservation of energy is not being broken in any of these examples. The reverse process would need a fantastically unlikely microscopic rearrangement.

For an isolated system, irreversible change increases entropy. “Almost always” is included because the laws of mechanics do not forbid all particles from accidentally moving into a more ordered arrangement, but with macroscopic numbers of particles the probability is so tiny that it is physically irrelevant.

Local order is allowed

A non-isolated system is a thermodynamic system that can exchange energy, and possibly mass, with its surroundings. Its own entropy can go down, as long as the surroundings gain at least as much entropy.

Freezing water gives a simple example. As the water freezes, it becomes more ordered, so the entropy of the water decreases. At the same time, the water releases energy to colder surroundings. Those surroundings gain more entropy than the water loses, so the total entropy change for water plus surroundings is positive.

Living organisms show the same idea: “entropy always increases” is too loose a statement. A plant can grow ordered structures because it is not an isolated system. Energy from the Sun and energy transfers to the environment make the total entropy change of the larger Sun–Earth–surroundings system positive. The second law forbids a total entropy decrease for an isolated system; it does not forbid local organisation.

Four standard gas processes

A gas process is a change from one thermodynamic state of a gas to another. On a P–V diagram, each process has its own shape, and that shape shows how much work is done.

An isovolumetric process is a gas process in which volume is constant. Since ΔV = 0, no boundary work is done, so the first law becomes Q = ΔU. For an ideal gas, P / T = constant during a constant-volume process.

An isobaric process is a gas process in which pressure is constant. Its P–V graph is horizontal, and the work done is simply W = PΔV. For an ideal gas, V / T = constant, where V is the gas volume (m³).

An isothermal process is a gas process in which absolute temperature is constant. For an ideal gas, internal energy stays constant, so ΔU = 0 and the first law gives Q = W. The ideal gas equation is P V = n R T. So, for fixed n and constant T, P V = constant. On a P–V graph, an isothermal expansion follows a downward-curving isotherm.

An adiabatic process is a gas process in which no energy is transferred by heating across the system boundary, so Q = 0. In practice, this is approximated by good insulation or by a process so rapid that there’s no time for significant thermal energy transfer.

Slow, well-conducted changes give the best approximation to isothermal behaviour, since the gas has to remain at the same temperature while energy crosses the boundary. Rapid, insulated changes give the best approximation to adiabatic behaviour, because energy transfer by heating must be negligible.

The adiabatic model

For a monatomic ideal gas undergoing an adiabatic process,

P V^(5/3) = constant.

The exponent 5/3 applies specifically to a monatomic ideal gas. IB quantitative problems in this topic use monatomic ideal gases, so use this adiabatic model unless the question says otherwise.

In adiabatic expansion, the gas does positive work on the surroundings. Since Q = 0, the first law gives ΔU = −W, so the internal energy falls and the temperature drops. In adiabatic compression, W is negative because work is done on the gas; ΔU is therefore positive, and the temperature rises.

On a P–V diagram, an adiabatic curve is steeper than an isothermal curve through the same state. For expansion, pressure drops more quickly in an adiabatic process because the gas cools as it expands. For compression, pressure rises more quickly because work done on the gas heats it.

Cycles and net work

A cyclic process is a sequence of thermodynamic changes that brings the system back to its initial state. Internal energy is a state property, so when a gas completes a full cycle, the total ΔU over the cycle is 0.

On a P–V diagram, the net work done by the gas in one cycle equals the area enclosed by the loop. A clockwise loop shows positive net work done by the gas; an anticlockwise loop shows net work done on the gas.

A heat engine is a cyclic device that takes in energy from a high-temperature reservoir, produces useful work, and rejects some energy to a lower-temperature reservoir. For a heat engine, Qₕ is the energy transferred from the hot reservoir to the engine per cycle (J), Qc is the energy rejected by the engine to the cold reservoir per cycle (J), Tₕ is the absolute temperature of the hot reservoir (K), and Tc is the absolute temperature of the cold reservoir (K).

After one complete cycle, the engine has returned to its starting internal energy. The useful work output is therefore the energy taken in minus the energy rejected: Wnet = Qₕ − Qc, where Wnet is the net useful work output per cycle (J).

Steam engines, internal combustion engines and gas-turbine systems all use cyclic ideas, although real engineering cycles are more complicated than tidy textbook curves. The central idea stays the same: return the working substance to its initial state so the device can keep operating.

Efficiency as useful output divided by input

Efficiency is a dimensionless ratio: useful energy output compared with total energy input. For a heat engine,

η = useful work / input energy, where η is the efficiency of the engine.

For a cyclic heat engine, it is usually written as

η = Wnet / Qₕ = (Qₕ − Qc) / Qₕ.

Engine cycles do not all have the same efficiency. They follow different paths on a P–V diagram, do different amounts of work, and reject different amounts of energy. Engineers often compare cycles using the area inside the P–V loop, but they also have to ask how much input energy is needed to produce that loop.

Why efficiency matters beyond gas diagrams

Efficiency matters in motors and generators because any energy transfer that is not useful shows up as heating, sound, turbulence, frictional losses or some other wasted output. An electric motor is not usually treated as a heat engine, but its useful mechanical output still has to compete with unwanted heating. In a power station, a generator is often driven by a turbine, and that turbine may belong to a thermal cycle. So thermodynamic efficiency limits how much electrical energy can be obtained from the fuel or nuclear energy source.

Steam power gives a clear example of physics changing society. Once people understood enough thermodynamics to improve engines, steam engines could drive factories, pumps, locomotives and ships. This was more than a better machine. It helped move many societies away from local agricultural production and toward industrial production, urbanisation and large-scale transport. The same physics brought costs too: pollution, fossil-fuel dependence and new social pressures.