Specific heat capacity and specific latent heat

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Hello future Physicist! This chapter is all about understanding how energy affects matter, specifically how substances heat up and how they change state. It might seem tricky because sometimes you add heat and the temperature goes up, and sometimes you add heat and the temperature doesn't go up!

Don't worry, we will break down these concepts—Specific Heat Capacity (SHC) and Specific Latent Heat (SLH)—to show exactly where the energy is going. Mastering these concepts is fundamental to understanding thermodynamics and is crucial for exam success.

Section 1: Specific Heat Capacity (SHC)

1.1 What is Specific Heat Capacity?

Imagine you have a metal coin and a small glass of water. If you put them both in the sun, which one gets hot faster? The coin, right?

Specific Heat Capacity (c) is a measure of how much thermal energy (heat) a substance needs to absorb (or release) to change its temperature by a certain amount.

Substances with high SHC (like water) need a lot of energy to get hot. They resist temperature change.
Substances with low SHC (like metals) need very little energy to get hot. They heat up quickly.

The Formal Definition

The specific heat capacity ($c$) of a substance is the energy required per unit mass to raise the temperature of the substance by one kelvin (1 K) or one degree Celsius ($1^\circ\text{C}$).

Key Unit: The SI unit for specific heat capacity is Joules per kilogram per kelvin, or $J\ kg^{-1}\ K^{-1}$. (Since a change of $1^\circ\text{C}$ is the same as a change of 1 K, $J\ kg^{-1}\ ^\circ\text{C}^{-1}$ is also used.)

1.2 The Specific Heat Capacity Formula

The formula connects the energy transferred ($Q$) to the mass ($m$), the specific heat capacity ($c$), and the resulting temperature change ($\Delta\theta$).

\[Q = mc\Delta\theta\]

Breaking Down the Terms:

$Q$: Energy transferred (or heat absorbed/released) in Joules (J).
$m$: Mass of the substance in kilograms (kg).
$c$: Specific heat capacity of the substance in $J\ kg^{-1}\ K^{-1}$.
$\Delta\theta$: Change in temperature (final temperature minus initial temperature) in Kelvin (K) or $^\circ\text{C}$.

Analogy for SHC: Thermal Inertia
Think of SHC like the inertia of a substance. A high SHC material (like water, $c \approx 4200\ J\ kg^{-1}\ K^{-1}$) has high thermal inertia; it's hard to get it moving (change its temperature). A low SHC material (like copper, $c \approx 390\ J\ kg^{-1}\ K^{-1}$) has low thermal inertia; it reacts quickly to heat.

1.3 Experimental Determination of SHC (Quick Review)

To determine the SHC of a material, you need to measure the input energy ($Q$), the mass ($m$), and the temperature rise ($\Delta\theta$).

We often use electrical heaters where the energy supplied can be calculated using $Q = P t = V I t$.

Common Pitfall: In practical experiments, energy is always lost to the surroundings. You must account for these losses (e.g., using insulation or calibration) to get an accurate value for $c$.

Quick Review: Specific Heat Capacity

When the temperature of a substance changes, use: $Q = mc\Delta\theta$.

Section 2: Specific Latent Heat (SLH)

Now, let's look at what happens when you heat ice. Once the ice reaches $0^\circ\text{C}$, you keep adding heat, but the temperature stays at $0^\circ\text{C}$ until all the ice has melted into water. Where did the energy go?

The energy is used to break the intermolecular bonds holding the solid structure together, allowing it to become a liquid. This energy required to change the state without changing the temperature is called Latent Heat.

2.1 The Concept of Latent Heat

When a substance is changing state (e.g., solid $\to$ liquid or liquid $\to$ gas):

The added thermal energy ($Q$) increases the potential energy between the molecules (breaking bonds).
The average kinetic energy of the molecules (which determines temperature) remains constant.

2.2 Specific Latent Heat Defined

The specific latent heat (L) of a substance is the energy required per unit mass to change the state of the substance at a constant temperature.

\[Q = mL\]

Breaking Down the Terms:

$Q$: Energy transferred (or latent heat absorbed/released) in Joules (J).
$m$: Mass of the substance changing state in kilograms (kg).
$L$: Specific latent heat of the process in $J\ kg^{-1}$.

Key Unit: The SI unit for specific latent heat is Joules per kilogram, or $J\ kg^{-1}$.

2.3 Fusion vs. Vaporisation: The Distinction

We must distinguish between the two main types of state change involving latent heat, as required by the syllabus:

1. Specific Latent Heat of Fusion ($L_f$)

This is the energy required to change unit mass of a substance from solid to liquid (melting) or from liquid to solid (freezing), at its melting point.

Process: Melting (Endothermic - absorbs heat) or Freezing (Exothermic - releases heat).
Energy Use: Breaks the rigid crystalline structure of the solid.
Formula: $Q = mL_f$

2. Specific Latent Heat of Vaporisation ($L_v$)

This is the energy required to change unit mass of a substance from liquid to gas (boiling/evaporation) or from gas to liquid (condensation), at its boiling point.

Process: Boiling (Endothermic - absorbs heat) or Condensation (Exothermic - releases heat).
Energy Use: Separates the molecules completely, overcoming all attractive forces.
Formula: $Q = mL_v$

Did You Know?
For water, $L_v$ (vaporisation) is significantly larger than $L_f$ (fusion). This is because much more energy is required to completely separate water molecules into a gas than is needed just to loosen them up into a liquid. This is why steam burns are so severe—a large amount of latent heat is released upon condensation onto your skin.

2.4 Calculating Energy for a Full Process

In many exam questions, you need to calculate the total energy required to heat a substance AND change its state. You must use both formulae in stages.

Example: Heating Ice to Steam

Heating the ice: Temperature change (e.g., from $-10^\circ\text{C}$ to $0^\circ\text{C}$). Use $Q_1 = mc_{ice}\Delta\theta$.
Melting the ice: Phase change at $0^\circ\text{C}$. Use $Q_2 = mL_f$. (No $\Delta\theta$!)
Heating the water: Temperature change (e.g., from $0^\circ\text{C}$ to $100^\circ\text{C}$). Use $Q_3 = mc_{water}\Delta\theta$.
Boiling the water: Phase change at $100^\circ\text{C}$. Use $Q_4 = mL_v$. (No $\Delta\theta$!)
Total Energy: $Q_{total} = Q_1 + Q_2 + Q_3 + Q_4$.

Mnemonic Aid: Look at the question. If you see a change in temperature ($\Delta\theta$), use the $c$ equation. If you see a change in state (fusion/vaporisation), use the $L$ equation. You can never use both in the same step!

Quick Review: Specific Latent Heat

When the state of a substance changes (melting, boiling, freezing, condensing), use: $Q = mL$.

Section 3: Key Concepts and Practical Measurements

3.1 Definition of Heat Capacity (C) (Non-Specific)

Sometimes, you will encounter the term Heat Capacity ($C$) without the word 'specific'.

Heat capacity is the energy required to raise the temperature of an entire object by 1 K or $1^\circ\text{C}$.

$C$ depends on the mass of the object.
$C$ is related to specific heat capacity by: $C = mc$.
The formula for energy change is: $Q = C \Delta\theta$.
Unit: $J\ K^{-1}$.

Always check if the question asks for Specific Heat Capacity ($c$, unit includes mass) or just Heat Capacity ($C$, unit does not include mass).

3.2 Distinguishing $L_f$ and $L_v$ in Practical Contexts

The principle for measuring $L_f$ (e.g., for ice) or $L_v$ (e.g., for water) is the same: supply a known amount of electrical energy ($Q = VIt$) and measure the mass ($m$) that melts or vaporises.

Practical Tip: When measuring $L_f$ of ice, water is usually already melting or melting slowly due to ambient heat transfer. We measure the mass of water collected ($m_0$) during a time when the heater is off, and the mass ($m_H$) collected when the heater is on for the same time. The mass melted purely by the electrical energy is $m = m_H - m_0$. This technique is called the Method of Compensation, and it helps remove systematic errors from heat loss.

3.3 Common Mistakes to Avoid

Mistake 1: Unit Confusion
Always use SI units! Mass must be in kg (not grams), and time in seconds (not minutes). If SHC units are given in $J\ kg^{-1}\ ^\circ\text{C}^{-1}$, then the temperature change must be in $^\circ\text{C}$.

Mistake 2: Mixing the Formulae
Students often accidentally use $Q = mcL\Delta\theta$. Remember they are strictly separate:

Temperature Change: $Q = mc\Delta\theta$
State Change: $Q = mL$

Mistake 3: Forgetting the Total Process
If a question asks for the energy to go from solid X at $-5^\circ\text{C}$ to liquid X at $20^\circ\text{C}$, you must calculate the energy for heating the solid PLUS the energy for melting PLUS the energy for heating the liquid.

Chapter Summary: Key Takeaways

This chapter introduced two fundamental ways heat interacts with matter:

Changing Temperature: Governed by Specific Heat Capacity, $Q = mc\Delta\theta$. The energy increases the random kinetic energy of molecules.
Changing State: Governed by Specific Latent Heat, $Q = mL$. The energy increases the random potential energy (breaking bonds), keeping temperature constant.

You’ve got this! Practice applying the correct formula for the correct stage of the process, and you will find these calculations straightforward.