Latent Heat: Formula & Dimensions Explained

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Hello there! This is your friendly guide to understanding latent heat and its dimensional formula. You've asked a great question, and I'm here to provide you with a clear, detailed, and correct answer. Let's dive right in!

Correct Answer

The dimensional formula for latent heat is [M¹L²T⁻²], where M represents mass, L represents length, and T represents time.

Detailed Explanation

Alright, let's break down latent heat, its formula, and how we arrive at its dimensional formula. We'll go step-by-step, making sure everything is crystal clear.

What is Latent Heat?

Latent heat is the heat energy absorbed or released during a phase transition or a change in the physical state of a substance, such as a change from solid to liquid (melting), liquid to gas (boiling), or vice versa, without any change in temperature. Unlike sensible heat, which causes a temperature change, latent heat causes a change in state.

Think of it like this: Imagine you're boiling water. As you add heat, the temperature of the water rises until it reaches 100°C (at standard atmospheric pressure). At this point, the water starts to boil, and you continue to add heat. However, the temperature of the water doesn't change! Instead, all the added heat is used to convert the liquid water into steam. This heat is the latent heat of vaporization. Similarly, when ice melts, the heat absorbed is the latent heat of fusion.

Key Concepts

Before we jump into the formula, let's define some key terms:

  • Heat (Q): Energy transferred due to a temperature difference. The SI unit for heat is the Joule (J).
  • Mass (m): The amount of matter in a substance. The SI unit for mass is the kilogram (kg).
  • Latent Heat (L): The heat energy required to change the state of a unit mass of a substance without a change in temperature. The SI unit for latent heat is J/kg.

The Formula for Latent Heat

The formula that connects these concepts is:

Q = mL

Where:

  • Q is the heat absorbed or released (in Joules).
  • m is the mass of the substance (in kilograms).
  • L is the latent heat (in Joules per kilogram).

To find the latent heat L, we can rearrange the formula:

L = Q / m

Finding the Dimensional Formula

Now, let's find the dimensional formula for latent heat L. We'll do this by analyzing the dimensions of the quantities in the formula L = Q / m.

  1. Dimensions of Heat (Q): Heat is a form of energy. The dimensional formula for energy (and thus heat) is [M¹L²T⁻²]. You can derive this from the formula for kinetic energy (KE = 1/2 * mv²): Mass (M) * (Length/Time)² = M¹L²T⁻².
  2. Dimensions of Mass (m): The dimensional formula for mass is [M¹].

Now, substitute the dimensional formulas into the formula for latent heat:

L = Q / m

[L] = [M¹L²T⁻²] / [M¹]

Divide the dimensions:

[L] = [M¹⁻¹L²T⁻²]

[L] = [M⁰L²T⁻²]

Therefore, the dimensional formula for latent heat L is [M¹L²T⁻²]. Notice that the mass dimension (M⁰) cancels out and you are left with just mass per se.

Examples of Latent Heat

Here are a few common examples to illustrate the concept:

  • Melting Ice: When ice melts, it absorbs latent heat of fusion to change from a solid to a liquid state. The temperature remains at 0°C during the phase change.
  • Boiling Water: When water boils, it absorbs latent heat of vaporization to change from a liquid to a gaseous state (steam). The temperature remains at 100°C (at standard atmospheric pressure) during the phase change.
  • Condensation of Steam: When steam condenses, it releases latent heat of vaporization to change from a gaseous state to a liquid state. The temperature remains at 100°C.
  • Freezing Water: When water freezes, it releases latent heat of fusion to change from a liquid to a solid state (ice). The temperature remains at 0°C.

Importance of Dimensional Analysis

Dimensional analysis is a powerful tool in physics. It allows us to:

  • Check the consistency of equations.
  • Derive relationships between physical quantities.
  • Convert units from one system to another.
  • Understand the fundamental nature of physical quantities.

In the case of latent heat, knowing its dimensional formula helps us to understand how it relates to other physical quantities like energy, mass, and time. It also helps us to ensure that the units in our calculations are consistent.

Common Mistakes to Avoid

  • Confusing Latent Heat with Sensible Heat: Remember, latent heat causes a change of state without a temperature change, while sensible heat causes a temperature change.
  • Using Incorrect Units: Always use consistent units (e.g., Joules for energy, kilograms for mass) in your calculations.
  • Forgetting the Mass: Make sure to consider the mass of the substance when calculating the latent heat required or released.

Key Takeaways

  • Latent heat is the heat absorbed or released during a phase change (solid-liquid, liquid-gas, etc.) at a constant temperature.
  • The formula for latent heat is Q = mL, where Q is heat, m is mass, and L is latent heat.
  • The dimensional formula for latent heat is [M¹L²T⁻²].
  • Understanding latent heat is crucial for understanding phase transitions and energy transfer.

That's it! You've successfully learned about the dimensional formula of latent heat. If you have any further questions, feel free to ask! I'm always here to help you learn and understand complex concepts in a simple and easy way.