Problem 4 - A Complete Resource Book in Chemistry for JEE Main

Question

The dimensions of pressure are same as that of (a) energy (b) energy per unit volume (c) force per unit area (d) force per unit volume

Step-by-Step Solution

Answer

The dimensions of pressure are the same as (b) energy per unit volume and (c) force per unit area.

Step 1: Understand Pressure Dimensions

Pressure is defined as force per unit area. The dimensional formula for force is \([M][L][T]^{-2}\) and for area is \([L]^2\). Therefore, the dimension of pressure is \(\frac{[M][L][T]^{-2}}{[L]^2} = [M][L]^{-1}[T]^{-2}\).

Step 2: Analyze Energy Dimensions

The dimensional formula for energy is \([M][L]^2[T]^{-2}\). Since this does not match the dimensions of pressure, energy is not the correct choice.

Step 3: Analyze Energy Per Unit Volume Dimensions

Energy per unit volume has dimensions \(\frac{[M][L]^{2}[T]^{-2}}{[L]^{3}} = [M][L]^{-1}[T]^{-2}\). These dimensions match those of pressure \([M][L]^{-1}[T]^{-2}\).

Step 4: Analyze Force Per Unit Area Dimensions

Force per unit area has dimensions \(\frac{[M][L][T]^{-2}}{[L]^{2}} = [M][L]^{-1}[T]^{-2}\), which also matches the dimensions of pressure.

Step 5: Analyze Force Per Unit Volume Dimensions

Force per unit volume is \(\frac{[M][L][T]^{-2}}{[L]^{3}} = [M][L]^{-2}[T]^{-2}\), which does not match the dimensions of pressure.

Key Concepts

Pressure Dimension

Pressure is an essential concept in physics. It represents how much force is applied over a specific area. The pressure dimension formula comes from the relationship between force and area. If you think about applying force with your hand onto a table, the pressure is the force you exert spread out over the area of your hand touching the table. The dimensional formula for pressure is derived as follows:

• Force has the formula \([M][L][T]^{-2}\).
• Area is expressed as \([L]^2\).

By dividing the dimensions of force by area, we simplify to the formula for pressure: \([M][L]^{-1}[T]^{-2}\). This dimension tells us that pressure involves mass and time, with a dependency on how these quantities are spread over length.

Energy Dimension

Energy is the ability to do work or cause change. It can come in various forms such as kinetic, potential, thermal, or chemical energy. Dimensional analysis helps us understand the fundamental compositions of these forms.

• The dimensional formula for energy is \([M][L]^2[T]^{-2}\), which is derived from its relation to work done (force times distance).
• This shows that energy is dependent on mass and velocity (length/time) components squared.

This complex dimension suggests that energy involves interactions over time and space. Comparatively, the dimensions of energy differ from the pressure dimensions, indicating different fundamental relationships.

Force Per Unit Area

Force per unit area is the essence of understanding pressure. It's simply the amount of force distributed over a specific area. Imagine pressing down with a pen on a sheet of paper. The smaller the tip of the pen, the greater the pressure because the force is concentrated over a tiny area. The dimensional formula here, like that of pressure, is:

• Force: \([M][L][T]^{-2}\)
• Area: \([L]^2\)

Thus, the formula becomes \([M][L]^{-1}[T]^{-2}\). These dimensions perfectly match the pressure dimensions, clearly explaining why force per unit area is fundamentally the same as pressure itself.

Energy Per Unit Volume

Energy per unit volume is a relationship that connects how much energy is held within a given space. Visualize this idea like a balloon filled with energetic gas molecules. The energy per unit volume is related to how tightly packed and energetic those molecules are. This concept is expressed dimensionally by:

• Energy: \([M][L]^2[T]^{-2}\)
• Volume, which is the cube of length: \([L]^3\)

So, the formula for energy per unit volume becomes: \([M][L]^{-1}[T]^{-2}\). These dimensions match those of pressure, showing a profound relation between energy density and pressure. This equivalency explains why certain physical phenomena, like sound waves or fluids in a container, can be analyzed in terms of either energy density or pressure.