11. Thermodynamics

1. Meaning of Thermodynamics

In the context of physics, Thermodynamics is the branch of science that deals with the relationship between heat and other forms of energy (such as mechanical, electrical, or chemical energy). At its core, it is the study of how thermal energy (heat) is converted into mechanical work and vice versa. It governs the laws that dictate how energy flows and changes form, setting the limits on how efficient engines, refrigerators, and biological systems can be.

2. Origin of the Term

The word "Thermodynamics" is derived from two Greek words that perfectly describe its original purpose:

• Thermo (from Greek thérmē): Meaning Heat.
• Dynamics (from Greek dynamis): Meaning Power or Force. Historical Context:
• Who coined it? The term was popularized by William Thomson (Lord Kelvin) in 1849 (originally spelled as thermo-dynamics).
• Why? In the 19th century, during the Industrial Revolution, scientists were obsessed with understanding steam engines. They wanted to know how to turn the heat from burning coal into the "motive power" (motion) to drive trains and factory machinery.
• Evolution: Originally, it specifically meant "the motive power of heat." Over time, the definition broadened to include all forms of energy transfer, not just in engines, but in chemical reactions, stars, and even black holes.

Heat

Heat (Q) is energy in transit. It is the energy moving from one body to another due to a temperature difference. Once heat enters a system, it is no longer "heat"—it becomes part of the system's Internal Energy (U).

Internal energy

Internal Energy (U) is the sum of the kinetic and potential energies of the molecules inside the system.

• Heat is like a deposit or withdrawal (the act of moving money).
• Internal Energy is like the balance in the account.

What happens in Thermodynamic Equilibrium?

A system is in thermodynamic equilibrium when its macroscopic variables (P, V, T) do not change with time.

• Mechanical Equilibrium: Pressure (P) is constant (no unbalanced forces).
• Thermal Equilibrium: Temperature (T) is constant (no net flow of heat).
• Chemical Equilibrium: Composition is constant

The "Missing Link" in Thermodynamics

By the 19th century, scientists had already established the First Law (Conservation of Energy) and the Second Law (Entropy and Direction of Heat). However, these laws assumed that we already understood what "Temperature" was.
Scientists eventually realized there was a logical gap:
They were using thermometers to measure temperature, but they hadn't actually proven why a thermometer works. They needed a law that defined Temperature as a valid, measurable physical property.

Why is it called the "Zeroth" Law?

• When Fowler formulated this law, the First and Second Laws were already famous and widely used in textbooks globally.
• The Dilemma: This new law was arguably more basic and fundamental than the First Law. Logic dictated it should be "Law #1."
• The Solution: Renumbering the existing famous laws would have caused massive confusion. So, instead of moving everything up, they gave it the number Zero (0) to place it before the First Law in terms of importance, without changing history.

Thermal equilibrium

"If two systems (A and B) are essentially in thermal equilibrium with a third system (C), then they are in thermal equilibrium with each other."

• conducting wall is a boundary that allows heat to pass through it freely.

• The Zeroth Law relies entirely on conducting walls.

• When you use a thermometer to check your fever, the glass bulb of the thermometer acts as a conducting wall.

• It allows the heat from your body to pass into the mercury, ensuring the mercury reaches the exact same temperature as your body so you get an accurate reading.

Physics Term Analogy Component Explanation

• Internal Energy (U) The Account Balance This is the total amount of money stored in the bank right now. It belongs to the account. You can ask, "What is my balance?"
• Heat (Q) Deposit via Cash This is a way to transfer money into the account. It is energy moving due to a "temperature difference" (direct contact).
• Work (W) Deposit via Check This is a different way to transfer money into the account. It is energy moving due to mechanical force (piston moving).

Heat and Work are "Processes," not "Things"

You cannot say, "My account has $500 of deposit in it." "Deposit" is the act of moving money, not the money itself. Similarly, Heat and Work are the act of transferring energy. They only exist during the process.

The Difference: State vs. Path

• State Variable (Point Function): A property that depends only on the current condition of the system.

  • Examples: Pressure ($P$), Temperature ($T$), Volume ($V$), Internal Energy ($U$).
  • Analogy: Altitude. If you are at the top of a mountain, your altitude is 8,000 ft. It doesn't matter if you took a helicopter or hiked up; your altitude is the same.

• Path Variable (Process Function): A quantity that depends on the history or the specific process used to change the state.

  • Examples: Heat ($Q$), Work ($W$).
  • Analogy: Distance Traveled. To get to the top of that mountain, the helicopter took a short straight path (5 km), but the hiker took a winding long path (20 km). The destination (State) is the same, but the distance (Path) is very different.

Force –> Pressure

This chain of equations is the mathematical bridge that connects Mechanics (Force and Distance) to Thermodynamics (Pressure and Volume).

It explains how a gas actually manages to "do work" on the outside world.

The Core Difference

• Force is the total "push" or "pull" applied to an object. It answers: How hard are you pushing?
• Pressure is the Force spread out over an Area. It answers: How concentrated is that push? Here is the step-by-step breakdown of how we get from Force to Pressure:

1. The Setup: A Gas in a Cylinder

Imagine a gas trapped in a cylinder fitted with a movable piston.

• $A$ = The Cross-sectional Area of the piston (the flat surface the gas pushes against).
• $P$ = The Pressure of the gas inside.
• $F$ = The Force the gas exerts on the piston.
• $\Delta x$ = The small distance the piston moves outwards.

2. The Step-by-Step Derivation

Step 1: The Definition of Mechanical Work

$\Delta W = F \Delta x$ From basic mechanics, Work is defined as Force $\times$ Displacement.

• The gas pushes the piston outwards with a force $F$.

• The piston moves a small distance $\Delta x$.

• Therefore, the work done by the gas is $F \Delta x$.

Step 2: Converting Force to Pressure

$F = P \times A$

We know that Pressure is Force per unit Area ($P = F/A$).

• Therefore, the total Force pushing the piston is simply the Pressure times the Area ($PA$).

• We substitute this into our work equation: $\Delta W = (PA) \Delta x$

Step 3: Identifying the Change in Volume

$A \Delta x = \Delta V$

Look at the geometry of the cylinder.

• If you take the Area of the base ($A$) and multiply it by the height change ($\Delta x$), you get the Volume of the cylinder slice that the piston just swept through.

• This represents the Change in Volume ($\Delta V$) of the gas.

Step 4: The Final Thermodynamic Formula

Substituting $A \Delta x$ with $\Delta V$, we arrive at the final form: $\Delta W = P \Delta V$

Ideal Gases

• Isothermal means the Temperature ($T$) is constant ($\Delta T = 0$).
• For an Ideal Gas, the Internal Energy ($U$) depends only on Temperature.
• Therefore, if $T$ doesn't change, the Internal Energy doesn't change:$\Delta U = 0$

First Law of Thermodynamics ($\Delta Q$ = $\Delta U$ + $\Delta W$) essentially says:

Change in Balance = (Cash Deposits) - (Check Withdrawals) If you add heat (Deposit Cash), the Internal Energy (Balance) goes up. If the system does work (Writes a Check), the Internal Energy (Balance) goes down.

Motion

The molecules inside a system are never at rest. They possess kinetic energy due to three types of motion:

• Translational Motion: The molecules moving in straight lines from one place to another. (This is the dominant form in gases).
• Rotational Motion: The molecules spinning around their axis (like a top). This happens in multi-atomic molecules (like O_2 or H_2O).
• Vibrational Motion: The atoms within a molecule vibrating back and forth as if connected by springs. (This usually happens at higher temperatures).

Molar specific heat capacity

Molar Specific Heat Capacity at Constant Pressure (Cp) is defined as the amount of heat energy required to raise the temperature of 1 mole of a gas by 1°C (or 1 K), while keeping the pressure constant.
Where:

• $\Delta Q$ = Heat supplied
• $\Delta T$ = Change in temperature
• n = Number of moles

When you heat a gas at constant pressure, the gas is free to expand (like a piston moving up). Because of this expansion, the heat you supply has to do two jobs:

Increase Internal Energy: It makes the molecules move faster (raises temperature).

Do Mechanical Work: It pushes the surroundings back to expand the volume (P$\Delta V$).

for any ideal gas: Cp > Cv

Mayer’s Relation

Cp - Cv = R Where R is the Universal Gas Constant (8.314 {J mol}^{-1} {K}^{-1}).

PV = RT means

• P Pressure The force the gas exerts on the walls of its container.
• V Molar Volume The volume occupied by exactly 1 mole of the gas.
• R Universal Gas Constant A fixed number that relates energy to temperature scale. (8.314 {J mol}^{-1} {K}^{-1})
• T Temperature The absolute temperature of the gas in Kelvin

$P$ = Pressure

• What it is: The force exerted by the gas molecules colliding with the walls of the container.

• Physics View: It is the "push" of the gas.

• Standard Unit (SI): Pascal (Pa) or $N/m^2$.

• Other common units: Atmosphere (atm), Bar.

  • Note: In calculations, always convert to Pascal ($1 \text{ atm} \approx 10^5 \text{ Pa}$).

2. $V$ = Volume

• What it is: The total space occupied by the gas.

• Physics View: Since gases expand to fill their container, $V$ is simply the internal volume of the container.

• Standard Unit (SI): Cubic meter ($m^3$).

• Other common units: Liters ($L$).

  • Warning: Chemistry often uses Liters, but Physics usually demands $m^3$. ($1 m^3 = 1000 \text{ Liters}$).

3. $\mu$ (Mu) = Number of Moles

• What it is: The measure of the amount of substance.

• Symbol Note: In Chemistry, this is usually written as $n$. In Physics (NCERT), it is often written as $\mu$. They mean exactly the same thing.

• Definition: One mole contains Avogadro's number of particles ($6.022 \times 10^{23}$).

• How to calculate it: $\mu = \frac{\text{Mass of gas (m)}}{\text{Molar Mass (M)}}$

4. $R$ = Universal Gas Constant

• What it is: A fundamental physical constant that relates the energy scale to the temperature scale. It is the same for all ideal gases, regardless of whether it is Oxygen, Hydrogen, or Helium.
• Value: $R = 8.314 \, J \cdot mol^{-1} \cdot K^{-1}$
• Why it's there: It acts as a "converter" to make the units on the left side (Pressure $\times$ Volume = Energy) match the units on the right side (Moles $\times$ Temperature).

5. $T$ = Temperature

• What it is: A measure of the average kinetic energy of the gas particles.
• Crucial Rule: You MUST use the Kelvin (K) scale.
• The Trap: If a question gives you temperature in Celsius ($^\circ C$), the equation will fail. You must convert it first.
  • Formula: $T_{(Kelvin)} = T_{(Celsius)} + 273.15$

Summary Equation View

$\underbrace{P \times V}_{\text{Mechanical Energy}} = \underbrace{\mu \times R \times T}_{\text{Thermal Energy}}$