Which Of These Is An Extensive Property Of A Substance

Extensive properties are fundamental characteristics of matter that reveal a substance's overall magnitude or size. These properties change when the amount of the substance changes, making them essential for understanding various scientific and engineering applications.

Defining Extensive Properties

An extensive property is any property of a system that depends on the amount of matter present. This means if you double the amount of a substance, the value of its extensive property will also double. Understanding extensive properties is crucial in fields such as thermodynamics, chemistry, and material science because they help predict how systems will behave under different conditions and scales.

Examples of Extensive Properties

• Mass: The most straightforward example of an extensive property is mass. If you combine two identical objects, the total mass is the sum of their individual masses.
• Volume: Similar to mass, volume is an extensive property. The volume of a substance increases linearly with the amount of substance.
• Energy: The total energy of a system, including kinetic energy, potential energy, and internal energy, is extensive because it depends on the number of particles in the system.
• Enthalpy: A thermodynamic property of a system, enthalpy is the sum of the internal energy and the product of pressure and volume. It's extensive because it accounts for the total heat content of the system.
• Entropy: A measure of the disorder or randomness of a system, entropy increases with the amount of substance.
• Heat Capacity: The amount of heat required to raise the temperature of a substance by a certain amount is extensive. It depends on the quantity of the substance.
• Electric Charge: The total electric charge in a system is extensive as it is directly proportional to the number of charged particles.
• Length: For a uniform object, length is extensive. If you join two identical rods end to end, the total length is the sum of their individual lengths.
• Number of Moles: In chemistry, the number of moles of a substance is an extensive property. It represents the amount of substance in terms of the number of particles (atoms, molecules, ions).

Differentiating Extensive from Intensive Properties

To fully grasp the concept of extensive properties, it is essential to distinguish them from intensive properties. Intensive properties do not depend on the amount of substance. Examples of intensive properties include:

• Temperature: The temperature of a system remains the same regardless of how much material you have.
• Pressure: Pressure is intensive; the pressure in a container of gas does not change if you remove half the gas (assuming the volume and temperature remain constant).
• Density: Density is the mass per unit volume and is intensive because it is a ratio of two extensive properties.
• Melting Point: The temperature at which a substance melts is an intensive property as it is characteristic of the substance itself, not the amount.
• Boiling Point: Similar to melting point, the boiling point is an intensive property.
• Color: The color of a substance is intensive; a small sample will have the same color as a large sample.
• Hardness: The resistance of a material to deformation is intensive.
• Concentration: The amount of solute in a solution is intensive.
• Refractive Index: This property, which describes how light bends when passing through a substance, is intensive.

Mathematical Representation

Mathematically, an extensive property ( E ) can be represented as:

[ E = n \cdot e ]

Where:

• ( n ) is the amount of substance (e.g., mass, number of moles).
• ( e ) is the specific property (the property per unit amount of substance, which is intensive).

For example, if ( E ) is volume ( V ), then:

[ V = n \cdot v ]

Where ( v ) is the specific volume (volume per unit mass or mole).

Examples and Illustrations

1. Mass and Volume:
   • If you have 1 kg of water, its volume is approximately 1 liter. If you add another 1 kg of water, you now have 2 kg of water with a volume of approximately 2 liters. Both mass and volume have doubled, illustrating they are extensive.
2. Energy:
   • Consider a battery. A larger battery (with more material) will have more energy stored in it than a smaller battery. The total energy the battery can supply is directly related to the amount of active material inside it.
3. Heat Capacity:
   • Heating a small cup of water requires less heat than heating a large pot of water to the same temperature. The heat capacity (the amount of heat needed to raise the temperature by 1 degree) is greater for the larger volume of water.
4. Number of Moles:
   • In a chemical reaction, the amount of reactants and products is measured in moles. If you double the number of moles of a reactant, you will double the amount of product formed (assuming sufficient amounts of other reactants are available).

Extensive Properties in Thermodynamics

In thermodynamics, extensive properties play a vital role in defining the state of a system. The state of a thermodynamic system is described by its properties, such as pressure, temperature, volume, and internal energy. While intensive properties like temperature and pressure are essential for understanding equilibrium conditions, extensive properties like volume, internal energy, enthalpy, and entropy are critical for quantifying the energy and matter content of the system.

Applications of Extensive Properties

1. Material Science:
   • In material science, understanding extensive properties is crucial for designing structures and machines. For instance, the amount of material needed to build a bridge is determined by its required mass and volume.
2. Chemical Engineering:
   • Chemical engineers use extensive properties to design chemical reactors and separation processes. The amount of reactants needed for a chemical reaction and the volume of the reactor are determined by the desired output.
3. Environmental Science:
   • In environmental science, extensive properties are used to measure pollution levels and assess the impact of human activities on the environment. For example, the total mass of pollutants in a river or the total volume of greenhouse gases emitted by a factory are important extensive measures.
4. Food Science:
   • In food science, understanding extensive properties is essential for scaling recipes and ensuring consistency in food production. The mass and volume of ingredients determine the final product's taste, texture, and nutritional content.

How to Determine if a Property is Extensive

To determine whether a property is extensive, you can perform a simple thought experiment:

1. Divide the System: Imagine dividing a system into two or more identical parts.
2. Measure the Property: Measure the property of interest in each part.
3. Compare the Results:
   • If the value of the property for the whole system is the sum of the values for the individual parts, then the property is extensive.
   • If the value of the property remains the same for each part as it is for the whole, then the property is intensive.

For example, consider a container of water:

• Mass: If you divide the water into two equal parts, each part will have half the original mass. Therefore, mass is extensive.
• Temperature: If you divide the water into two equal parts, each part will have the same temperature as the original water. Therefore, temperature is intensive.

Combining Extensive Properties

Extensive properties can be combined through addition. For example, if you have two systems, A and B, with volumes ( V_A ) and ( V_B ), respectively, the total volume of the combined system is:

[ V_{total} = V_A + V_B ]

Similarly, for mass:

[ m_{total} = m_A + m_B ]

This additivity is a key characteristic of extensive properties.

Ratio of Extensive Properties

When you take the ratio of two extensive properties, you often obtain an intensive property. A classic example is density (( \rho )), which is the ratio of mass (( m )) to volume (( V )):

[ \rho = \frac{m}{V} ]

Density does not depend on the amount of substance; it is a property of the material itself. Similarly, specific heat capacity is an intensive property derived from the ratio of heat capacity (extensive) to mass (extensive).

Mathematical Formalism: Homogeneity

In mathematical terms, an extensive property exhibits homogeneity of degree one. This means that if you scale the amount of substance by a factor ( \lambda ), the value of the extensive property will also scale by the same factor. Mathematically:

[ E(\lambda N) = \lambda E(N) ]

Where:

• ( E ) is the extensive property.
• ( N ) is the amount of substance.
• ( \lambda ) is a scaling factor.

For example, if you double the amount of substance (( \lambda = 2 )), the extensive property will also double.

Common Misconceptions

1. Size and Extensivity:
   • It's important to note that not all properties related to "size" are extensive. For example, while length is an extensive property for a uniform object, its diameter might not be if the object is not uniform.
2. Confusion with Intensive Properties:
   • Sometimes, the distinction between extensive and intensive properties can be confusing. A common mistake is thinking that properties like "heat" are extensive. Heat, by itself, is a process of energy transfer, not a state function. However, the heat capacity, which quantifies the amount of heat required to change the temperature, is extensive.

Examples in Everyday Life

1. Cooking:
   • When cooking, the amount of ingredients (mass, volume) directly affects the final product. A larger cake requires more flour, sugar, and eggs.
2. Construction:
   • The amount of concrete needed for a building's foundation is an extensive property determined by the building's size and design.
3. Gardening:
   • The amount of soil needed for a garden is an extensive property that depends on the size of the garden bed.
4. Fuel:
   • The amount of fuel needed to drive a certain distance is an extensive property that depends on the distance and the vehicle's fuel efficiency.

Conclusion

Extensive properties are fundamental in science and engineering, providing insights into how the size or amount of a substance affects its overall characteristics. They differ significantly from intensive properties, which are independent of the amount of substance. Understanding the distinction between extensive and intensive properties is critical for predicting and analyzing the behavior of systems in various fields, including thermodynamics, material science, chemical engineering, and environmental science. By recognizing the additivity and scaling behavior of extensive properties, scientists and engineers can effectively design, analyze, and optimize processes and systems in a wide range of applications. From the mass and volume of ingredients in a recipe to the energy stored in a battery, extensive properties play a central role in shaping our understanding and manipulation of the physical world.