Systems and state functions

What thermodynamics is about

Burning fuel warms a room. A battery drives a phone. Ice melts in a drink. In every case energy changes form or moves from one place to another. Thermodynamics is the study of these energy changes.

Its laws deal with bulk matter, a huge number of molecules at once. They tell us whether a change can happen and how much energy it moves. They do not tell us how fast it happens. That is the job of chemical kinetics.

The system and the surroundings

To study a change, we first decide what to look at.

• The system is the part of the universe we study. It might be the gas in a cylinder or the solution in a beaker.
• The surroundings are everything else that can exchange energy or matter with the system.
• The boundary separates the two. It can be real, like the walls of a flask, or imagined.

Together, the system and the surroundings make up the universe.

Open, closed and isolated systems

Systems differ in what can cross the boundary.

• An open system exchanges both energy and matter with its surroundings. Water boiling in a pan without a lid is open: heat goes in and steam escapes.
• A closed system exchanges energy but not matter. Water heated in a sealed steel flask is closed: heat crosses the walls, but no water gets out.
• An isolated system exchanges neither. Hot coffee in a good vacuum flask is nearly isolated. Neither heat nor coffee crosses the walls.

Most chemistry in this chapter uses closed systems, such as a gas trapped in a cylinder under a piston.

The state of a system

A system is in a definite state when its properties have fixed values. For a gas, four quantities describe the state: pressure $p$, volume $V$, temperature $T$ and amount $n$. These are the state variables.

They are not all independent. For an ideal gas, $pV = nRT$ links them. Fix three and the fourth is fixed too. One mole of an ideal gas at 273.15 K and 1 bar always has a volume of about 22.7 L.

We do not need to know where each molecule is. A few bulk quantities are enough. This is why thermodynamics works with large samples.

State functions and path functions

Think of a hike from a car park at 1200 m up to a hilltop at 1600 m. You gain 400 m of height whichever route you take. The height gained depends only on where you start and where you stop. The distance you walk depends on the route.

A state function depends only on the state of the system, not on how that state was reached.

Pressure, volume, temperature and internal energy are state functions. A change in a state function depends only on the start and the end:

Heat and work are different. They are energy in transit, and their amounts depend on the path. They are path functions. A system does not "contain" heat or work, so we never write $\Delta q$ or $\Delta w$.

Bend the route over a ridge, then through a valley. The distance changes each time. Does the height gained ever change?

Extensive and intensive properties

Some properties depend on how much matter there is. Others do not.

• An extensive property grows with the amount of matter. Mass, volume, amount and internal energy are extensive.
• An intensive property does not depend on the amount. Temperature, pressure, density and molar volume are intensive.

Pour two beakers of water, each 100 mL at 25 °C, into one jug. The volume is 200 mL, but the temperature is still 25 °C. A quick test: cut the system in half in your mind. An extensive property halves. An intensive one stays the same.

The ratio of two extensive properties is intensive. Mass divided by volume gives density.

Practise with new numbers

Try it in Python

The program finds the volume of an ideal gas in four states. Then it heats the gas and cools it back, and checks the change in volume over the round trip.

Common mistakes

• Calling a thermos of tea open. Matter does not cross, and very little heat does. It is nearly isolated.
• Writing $\Delta q$ or $\Delta w$. Heat and work are not state functions. A system has no heat to change.
• Thinking every property is extensive. Temperature and pressure do not add when you join two samples.
• Expecting thermodynamics to give the rate. It says whether a change can happen, not how fast.

Remember these

• System, surroundings and boundary make up the universe.
• Open: energy and matter cross. Closed: only energy. Isolated: neither.
• State variables: $p$, $V$, $T$ and $n$. A state function depends only on the state.
• Heat and work are path functions.
• Extensive properties depend on the amount. Intensive ones do not.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Ice cubes melt in an open glass of juice. Is the glass of juice open, closed or isolated?

2. Which of these are state functions: temperature, heat, internal energy, work, volume?

3. A gas is heated from 300 K to 400 K and then cooled back to 300 K. What is $\Delta T$ for the whole trip?

4. Is molar volume extensive or intensive?