Question

The compressibility of a substance equals......... (A) \((\Delta \mathrm{V} / \mathrm{PV})\) (B) \(\\{(\mathrm{P} \Delta \mathrm{V}) / \mathrm{V}\\}\) (C) \(\\{\mathrm{V} /(\mathrm{P} \Delta \mathrm{V})\\}\) (D) \((\mathrm{PV} / \Delta \mathrm{V})\)

Short answer

None of the given options are correct. The correct formula for compressibility is: \(compressibility = \frac{-1}{V} \cdot \frac{\Delta V}{\Delta P}\).

Step-by-step solution

Option A: Check if the given formula represents compressibility.

The formula in this option is: (A) \((\Delta \mathrm{V} / \mathrm{PV})\) However, this formula does not represent compressibility, since it represents the change in volume divided by the product of initial pressure and volume. The correct formula for compressibility should have the pressure change in the numerator.

Option B: Check if the given formula represents compressibility.

The formula in this option is: (B) \(\\{(\mathrm{P} \Delta \mathrm{V}) / \mathrm{V}\\}\) This formula also doesn't represent compressibility, as it has the product of the initial pressure and the change in volume divided by the initial volume. The correct formula for compressibility should have the pressure change in the numerator.

Option C: Check if the given formula represents compressibility.

The formula in this option is: (C) \(\\{\mathrm{V} /(\mathrm{P} \Delta \mathrm{V})\\}\) Once again, this formula doesn't represent compressibility, as it has the initial volume divided by the product of the initial pressure and the change in volume. The correct formula for compressibility should have the pressure change in the numerator.

Option D: Check if the given formula represents compressibility.

The formula in this option is: (D) \((\mathrm{PV} / \Delta \mathrm{V})\) Again, this formula doesn't represent compressibility, as it has the product of the initial pressure and the initial volume divided by the change in volume. The correct formula for compressibility should have the pressure change in the numerator. In conclusion, none of the given options represent the correct formula for compressibility. The correct formula for compressibility is: \(compressibility = \frac{-1}{V} \cdot \frac{\Delta V}{\Delta P}\)

Key concepts

Thermodynamic Properties

Thermodynamic properties are essential characteristics of systems that describe their physical and energetic states. These properties help us understand how substances behave under varying conditions.

• Common examples include pressure, volume, and temperature.
• Thermodynamic properties can further be categorized as extensive or intensive.
• Extensive properties depend on the amount of substance present (e.g., volume), while intensive properties do not (e.g., temperature).

Understanding these properties allows us to predict how a substance will react or change when exposed to different environments. For example, knowing the pressure and volume changes, one can assess how compressibility, a key thermodynamic property, affects a system.

Volume Change

Volume change refers to the variation in the size or capacity of a system when subjected to changes in pressure or temperature. It's a vital aspect in fields like physics and engineering.
Volume change occurs when external conditions alter the space a substance occupies. For instance, when you heat a gas, its volume typically increases if the pressure is constant. This principle is described by Charles's law. Observing volume changes helps in understanding thermodynamic behavior and allows for the calculation of compressibility:

• The concept of \[\Delta V\] represents this change in volume.
• Through experimental data, volume change provides insights into how materials and substances react under different conditions.

Pressure Change

Pressure change can significantly impact the physical state of a substance and its interactions. It is crucial for understanding material and system behaviors across many scientific and engineering applications.
Pressure is defined as force per unit area. When the force exerted on a substance changes, the pressure it experiences also alters, leading to different reactions based on the substance's nature.

• In the context of gases, an increase in pressure typically reduces volume (Boyle’s law).
• Pressure change is depicted as \[\Delta P\].
• In fluids, pressure changes can affect density and flow dynamics.

Recognizing how pressure affects system variables, you'll find it's key to analyzing compressibility, where pressure change is part of the fundamental formula.

Formula Derivation

Deriving the correct formula for compressibility involves understanding how volume and pressure changes interplay.
Compressibility is a measure of a substance's ability to withstand changes in volume under pressure shifts. The correct formula for compressibility is: \[compressibility = \frac{-1}{V} \cdot \frac{\Delta V}{\Delta P}\]

• The negative sign indicates volume decreases as pressure increases for most materials.
• \( V \) is the initial volume, reflecting how compressibility is proportional to the inverse of volume.
• \( \Delta V \) and \( \Delta P \) represent the change in volume and pressure, respectively.

Exploring formula derivations like this one helps to enhance your insight into thermodynamic equations and improves your ability to apply them in real-world scenarios.