Q.Extensive properties depend on the quantity of matter but intensive properties do not. Explain whether the following properties are extensive or intensive.
Mass, internal energy, pressure, heat capacity, molar heat capacity, density, mole fraction, specific heat, temperature and molarity.

The key idea is the distinction between extensive and intensive properties based on their dependence on the quantity of matter.

Extensive properties depend on the amount of matter in a system, meaning they are additive. Intensive properties do not depend on the amount of matter; they are characteristic of the substance itself, regardless of quantity.

• Mass: Extensive, as it increases with the quantity of matter.
• Internal energy: Extensive, as it increases with the quantity of matter.
• Pressure: Intensive, as it does not change with the quantity of matter at a given state.
• Heat capacity: Extensive, as more matter requires more heat to change its temperature by a certain amount.
• Molar heat capacity: Intensive, as it is heat capacity per mole (an extensive property divided by another extensive property).
• Density: Intensive, as it is mass per unit volume (an extensive property divided by another extensive property).
• Mole fraction: Intensive, as it is a ratio of moles of a component to total moles (an extensive property divided by another extensive property). …

In thermodynamics, properties of a system are broadly categorized into two types: extensive and intensive. Understanding this distinction is fundamental because it helps us characterize how a system behaves when its size or amount of substance changes.

Concept and Intuition

1. Extensive Properties: These are properties that depend on the amount of matter present in the system. If you increase the amount of substance, the value of an extensive property will generally increase.

   • Intuitive Test: Imagine you have a system (e.g., a block of iron or a container of gas). If you divide this system into two equal halves, an extensive property will also be halved in each part.

2. Intensive Properties: These are properties that are independent of the amount of matter present in the system. Their values remain the same regardless of how much substance is present.

   • Intuitive Test: If you divide a system into two equal halves, an intensive property will have the same value in each part as it did in the original whole system.

A useful general rule is that the ratio of two extensive properties often results in an intensive property. For example, density is mass (extensive) divided by volume (extensive), and density itself is intensive.

Let's classify each property from the given list:

1. Mass ($m$)

   • Reasoning: If you take a system and divide it into two equal parts, the mass of each part will be half of the original system's mass. For instance, a 10 kg block of iron, when cut in half, yields two 5 kg blocks.
   • Classification: Extensive

2. Internal Energy ($U$)

   • Reasoning: Internal energy represents the total energy contained within a system due to the motion and interaction of its molecules. If you have more molecules (more matter), you have more total internal energy. Dividing a system in half would also halve its total internal energy.
   • Classification: Extensive

3. Pressure ($P$)

   • Reasoning: Pressure is force per unit area. If you have a gas in a container at a certain pressure and you insert a partition to divide the container into two halves, the pressure in each half will remain the same as the original pressure (assuming the temperature and volume per mole remain constant). Pressure does not depend on the total amount of gas, but rather on the force exerted by molecules per unit area.
   • Classification: Intensive

4. Heat Capacity ($C$)

   • Reasoning: Heat capacity is the amount of heat required to raise the temperature of a substance by one degree Celsius (or Kelvin). A larger amount of substance requires more heat to achieve the same temperature rise. For example, it takes more heat to raise the temperature of 2 kg of water by $1^\circ\text{C}$ than to raise the temperature of 1 kg of water by $1^\circ\text{C}$.
   • Classification: Extensive

5. Molar Heat Capacity ($C_m$)

   • Reasoning: Molar heat capacity is the heat capacity per mole of substance. It is defined as $C_m = \frac{C}{n}$, where $C$ is heat capacity (extensive) and $n$ is the number of moles (extensive). Since it's a ratio of two extensive properties, it becomes intensive. It tells you how much heat is needed to raise the temperature of one mole by $1^\circ\text{C}$, regardless of the total number of moles.
   • Classification: Intensive

6. Density ($\rho$)

   • Reasoning: Density is defined as mass per unit volume ($\rho = \frac{m}{V}$). Both mass ($m$) and volume ($V$) are extensive properties. If you divide a system in half, both its mass and volume are halved, but their ratio (density) remains constant. For example, a 1 kg block of iron has the same density as a 10 kg block of iron.
   • Classification: Intensive

7. Mole Fraction ($x_i$)

   • Reasoning: Mole fraction of a component $i$ in a mixture is defined as the number of moles of component $i$ ($n_i$) divided by the total number of moles in the mixture ($n_{total}$). Both $n_i$ and $n_{total}$ are extensive properties. If you take a sample from a homogeneous mixture, the ratio of moles of a component to the total moles will remain the same.
   • Classification: Intensive

8. Specific Heat ($c$) …