Questions: Intensive and Extensive Properties

Question 1

You have a 2-liter sample of water at 50°C. You divide it into two equal 1-liter portions. Which statement correctly describes what happens?

• A. Both temperature and volume remain the same in each portion
• B. Temperature stays at 50°C in each portion; volume becomes 1 liter in each
• C. Both temperature and volume are halved in each portion
• D. Temperature doubles in each portion to compensate for the reduced volume

Answer: (B)

This is the bisection test in action. Temperature is intensive — it characterizes the state of matter at a point and does not depend on how much material is present, so each portion remains at 50°C. Volume is extensive — it scales with the amount of substance — so each portion has 1 liter. The most common mistake is thinking intensive properties also change when the system is divided.

Question 2

Which of the following is an intensive property?

• A. Mass
• B. Internal energy
• C. Entropy
• D. Pressure

Answer: (D)

Pressure is intensive — it has the same value throughout a uniform system regardless of system size, and remains unchanged when you apply the bisection test. Mass, internal energy, and entropy are all extensive: they scale proportionally with the amount of substance and are halved when the system is halved. A common confusion is entropy — while entropy is extensive (total entropy scales with size), entropy density (entropy per unit volume) or molar entropy (per mole) are intensive.

Question 3

If you combine two identical systems, the temperature of the combined system is twice that of each individual system.

• T. True
• F. False

Answer: False

Temperature is intensive — it is independent of system size and does not add when systems are combined. If two identical 50°C water samples are combined, the result is still 50°C. Contrast this with volume, mass, or internal energy (all extensive): combining two identical systems does double those quantities. The bisection test makes this clear: temperature in each half equals the original temperature, so doubling the system can't double it either.

Question 4

Entropy is an extensive property of a thermodynamic system.

• T. True
• F. False

Answer: True

Entropy S is extensive: it is additive for subsystems and scales proportionally with the amount of substance. If you double the size of a system (holding intensive variables constant), the entropy doubles. This is directly expressed in the Euler relation U = TS − PV + μN, where S appears as an extensive variable multiplied by its intensive conjugate T. The common confusion is with entropy density or molar entropy (S/n), which are intensive — but those are derived quantities, not entropy itself.

Question 5

Using the bisection test, explain why temperature is an intensive property while internal energy is an extensive property.

Model answer: When a system is mentally divided in half, intensive properties remain the same in each half while extensive properties are halved. Temperature characterizes the average kinetic energy per particle — this does not change when you split the system, because each half has the same molecular motion. Internal energy is the total energy of all particles; with half the particles, each half contains half the total energy. Temperature is a property of the state of matter at a point; internal energy is a property of the entire collection.

The bisection test is the clearest operational definition of the intensive/extensive distinction. Temperature, pressure, density, and chemical potential are intensive — they describe local state. Volume, mass, entropy, and internal energy are extensive — they describe how much. Understanding this distinction is essential for using thermodynamic equations correctly: the Euler relation and equations of state like PV = nRT can be written in intensive (per-mole) form precisely because intensive variables are independent of system size.