Question

Dimensional formula of latent heat is ........ (a) \(\mathrm{M}^{0} \mathrm{~L}^{2} \mathrm{~T}^{-2}\) (b) \(\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-2}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-1}\) (d) \(\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-1}\)

Short answer

The dimensional formula of latent heat is \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\).

Step-by-step solution

Identify known dimensional formulas

The given equation for Latent heat (L) has three variables: 1. mass (m) 2. specific heat (c) 3. temperature change (ΔT) The dimensional formula of mass is M. The dimensional formula of temperature change is K. We need to find the dimensional formula for specific heat.

Find dimensional formula of specific heat

The specific heat can be determined using the following formula: specific heat (c) = energy / (mass × temperature change) We know that: Energy (E) has the dimensions of mass (M), length (L), and time (T): \([E] = M^1 L^2 T^{-2}\) Temperature change (ΔT) has the dimensions of Kelvin (K): \([ΔT] = K^1\) Now, we can solve for the dimensional formula of specific heat (c): \([c] = \frac{[E]}{[m][ΔT]}\)

Calculate the dimensional formula of specific heat

To calculate the dimensional formula of specific heat (c), substitute the known dimensional formulas for energy, mass, and temperature change: \([c] = \frac{M^1 L^2 T^{-2}}{M^1 K^1}\) Now, simplify the expression: \([c] = L^2 T^{-2} K^{-1}\)

Determine the dimensional formula of latent heat

Now that we have the dimensional formula for specific heat, we will find the dimensional formula of latent heat (L) by using the equation: Latent heat (L) = (mass × specific heat × temperature change) In terms of dimensional formulas, we have: \([L] = [m][c][ΔT]\) Substitute the known dimensional formulas for mass, specific heat, and temperature change: \([L] = M^1 \cdot (L^2 T^{-2} K^{-1}) \cdot K^1\) Now, simplify the expression: \([L] = M^1 L^2 T^{-2}\) Therefore, the dimensional formula of latent heat is \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-2}\), which matches option (c).

Key concepts

Specific Heat

Specific heat is a key concept in thermodynamics that describes how much heat energy is required to raise the temperature of a given mass of a substance by a certain amount, often by 1 degree Celsius. It is a unique property of each material, influencing how it reacts to heat.
To calculate specific heat (\(c\)), we use the formula:

• \(c = \frac{Q}{m \Delta T}\)

Where \(Q\) is the heat energy, \(m\) is the mass, and \(\Delta T\) is the change in temperature.
Understanding this concept is essential for solving problems involving heat transfer in various substances. By knowing the specific heat, we can calculate how much energy is needed for heating or cooling a substance under specific conditions.

Latent Heat

Latent heat refers to the amount of heat that is absorbed or released by a substance during a phase change without changing its temperature. Common phase changes include melting, freezing, boiling, and condensation.
The two primary types of latent heat are:

• Latent heat of fusion - for transitions between solid and liquid (melting or freezing).
• Latent heat of vaporization - for transitions between liquid and gas (boiling or condensation).

The dimensional formula of latent heat is intriguing because it relates to how the substance's mass and its phase change interact with the absorbed or released energy.
Understanding latent heat is crucial when studying processes that involve heat transfer and phase transitions, which are common in natural and industrial processes.

Dimensional Formula

In physics, the dimensional formula is a way to express a physical quantity in terms of its basic dimensions, like mass (\(M\)), length (\(L\)), and time (\(T\)), among others. It’s a concise way to summarize a concept’s physical nature and helps us understand different phenomena by comparing their dimensions.
For example, the dimensional formula for energy is \(M^1 L^2 T^{-2}\). You can use these formulas to check the consistency of physical equations and derive relations between different physical quantities.
In the context of the exercise, dimensional analysis of latent heat involves breaking it down into its fundamental units, which helps ensure that all terms in an equation are compatible.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, temperature, and energy transformations. It plays a vital role in understanding how energy moves and changes form. The field is governed by four main laws which describe how heat and energy are conserved and transferred across systems.
Key concepts in thermodynamics include:

• The First Law (Law of Energy Conservation) - energy can neither be created nor destroyed, only transformed.
• The Second Law - heat flows naturally from hot to cold and systems become more disordered over time (entropy increases).
• The Third Law - as temperature approaches absolute zero, the entropy of a system approaches a constant minimum.
• The Zeroth Law - if two systems are each in thermal equilibrium with a third system, they are in thermal equilibrium with each other.

Thermodynamics explains why specific heat and latent heat are essential for understanding how energy is distributed and transformed in physical systems.