Thermodynamics revision notes

Thermal Equilibrium & The Zeroth Law

Thermodynamics is a macroscopic science that deals with bulk systems and the inter-conversion of heat and other forms of energy. Unlike mechanics, which focuses on the motion of particles under forces, thermodynamics is concerned with the internal macroscopic state of a body.

• Thermal Equilibrium: A system is in an equilibrium state if its macroscopic variables (like pressure, volume, temperature, mass, and composition) do not change over time. Equilibrium in thermodynamics fundamentally differs from mechanics, where it means net force and torque are zero.
• Microscopic vs. Macroscopic Equilibrium: In a state of thermodynamic equilibrium, the system is macroscopically stable, but its microscopic constituents (molecules) are in constant random motion and collision. Therefore, the microscopic constituents are not in equilibrium in the mechanical sense.
• Types of Walls:
  • Adiabatic Wall: An insulating wall that strictly prevents the flow of heat energy between systems.
  • Diathermic Wall: A conducting wall that allows heat energy to flow until systems reach thermal equilibrium.
• Zeroth Law of Thermodynamics: Formulated by R.H. Fowler in 1931, it states that "two systems in thermal equilibrium with a third system separately are in thermal equilibrium with each other".
• Temperature: The Zeroth Law formally introduces the concept of temperature ($T$). If systems A and B are in thermal equilibrium with C, then $T_A = T_C$ and $T_B = T_C$, mathematically proving $T_A = T_B$. Temperature acts as the marker of "hotness" and dictates the direction of heat flow.

Heat, Internal Energy, and Work

To deeply understand thermodynamics, one must distinguish between state variables and modes of energy transfer.

• Historical Context of Heat (Caloric vs. Kinetic Theory): Initially, heat was thought to be an invisible fluid called "caloric" that flowed from hot to cold bodies. Benjamin Thomson (Count Rumford) disproved this in 1798 by showing that the boring of a brass cannon generated heat proportional to the work done, not the sharpness of the drill, proving heat is a form of energy conversion.
• Internal Energy ($U$): This is the sum of the kinetic and potential energies of a system's molecular constituents. It is evaluated in a frame of reference where the center of mass of the system is at rest. Thus, the bulk kinetic energy of the system is not included in $U$. → [JEE TIP] A moving bullet's kinetic energy does not contribute to its temperature until it impacts a target, converting bulk kinetic energy into internal (disordered) energy. Internal energy is a state variable; it depends solely on the current state (P, V, T), not on the path taken to reach it.
• Heat ($Q$): Heat is energy in transit transferred across a boundary due to a temperature difference between the system and its surroundings.
• Work ($W$): Work is energy transfer brought about by mechanical means, such as moving a piston, which does not require a temperature difference.

The First Law of Thermodynamics

The First Law is the general principle of conservation of energy applied to thermodynamic systems.

• Equation: $\Delta Q = \Delta U + \Delta W$.
  • The energy supplied to a system ($\Delta Q$) goes partly into increasing its internal energy ($\Delta U$) and partly into doing work on the environment ($\Delta W$).
• Path Independence: While $\Delta Q$ and $\Delta W$ are path-dependent, the combination $\Delta Q - \Delta W = \Delta U$ is path-independent because internal energy is a state variable.
• Work Done by a Gas: For a gas expanding against a constant pressure $P$, the work done is $\Delta W = P \Delta V$. Thus, the First Law becomes $\Delta Q = \Delta U + P \Delta V$.

Specific Heat Capacity

• Specific Heat Capacity ($s$): Defined as the heat required per unit mass to raise the temperature by $\Delta T$. Formula: $s = \frac{\Delta Q}{m\Delta T}$, with SI unit $\text{J kg}^{-1}\text{K}^{-1}$.
• Molar Specific Heat Capacity ($C$): Defined as the heat required per mole ($\mu$) to raise the temperature by $\Delta T$. Formula: $C = \frac{\Delta Q}{\mu\Delta T}$, with SI unit $\text{J mol}^{-1}\text{K}^{-1}$.
• Specific Heats of Gases: Molar specific heat depends on the conditions of heating.
  • Mayer's Relation: For an ideal gas, $C_p - C_v = R$, where $C_p$ is heat capacity at constant pressure, $C_v$ is heat capacity at constant volume, and $R$ is the universal gas constant.
• Specific Heat of Solids (Dulong-Petit Law): Using equipartition of energy, a 3D atomic oscillator has an average energy of $3 k_B T$. For one mole of a solid ($N_A$ atoms), $U = 3 k_B T \times N_A = 3RT$. Since volume expansion of solids is negligible, $\Delta V \approx 0$, making $C = \frac{\Delta Q}{\Delta T} \approx \frac{\Delta U}{\Delta T} = 3R$. This predicted value generally agrees with experiments at room temperature (except for Carbon) but breaks down at low temperatures.
• Specific Heat of Water & The Calorie: Historically, 1 calorie was the heat needed to raise 1g of water by 1°C. Because water's specific heat varies slightly with temperature, the calorie is strictly defined using the interval 14.5 °C to 15.5 °C. 1 cal = 4.186 J. The term "mechanical equivalent of heat" is merely a conversion factor between calorie and joule, which is superfluous in SI units.

Thermodynamic State Variables & Equation of State

Equilibrium states are completely defined by macroscopic variables, connected via an equation of state. For an ideal gas, this is $PV = \mu RT$. Non-equilibrium states (like free expansion into a vacuum or explosive reactions) lack uniform pressure and temperature, hence cannot be described by simple state variables.

Thermodynamic variables are classified into two types:

1. Extensive Variables: Indicate the "size" or mass of the system. Their values halve if the system is divided into two equal parts. Examples: Volume ($V$), Internal Energy ($U$), Mass ($M$).
2. Intensive Variables: Independent of the system's size. They remain unchanged if the system is divided. Examples: Pressure ($P$), Temperature ($T$), Density ($\rho$).

• Entropy and Enthalpy as State Variables: While P, V, T, U, and mass are primary, Entropy (a measure of disorderness) and Enthalpy (a measure of total heat content) are also explicitly macroscopic thermodynamic state variables.
• → [JEE TIP] Consistency Check: The product of an intensive variable and an extensive variable yields an extensive variable (e.g., $P \Delta V$ is extensive). Thermodynamic equations must balance extensive/intensive properties on both sides.

Thermodynamic Processes

To mathematically model thermodynamic changes, we use a quasi-static process: a hypothetical, infinitely slow process where the system remains in thermal and mechanical equilibrium with its surroundings at every instant. In such a process, system pressure and temperature differ from the surroundings only by infinitesimally small amounts.

1. Isothermal Process: Temperature is constant ($T$ = fixed).
   • Equation of state: $PV = \text{constant}$ (Boyle's Law).
   • Internal energy of an ideal gas depends only on $T$, so $\Delta U = 0$.
   • First Law reduces to: $Q = W$.
   • Work Done: $W = \int_{V_1}^{V_2} P dV = \mu RT \ln\left(\frac{V_2}{V_1}\right)$.
2. Adiabatic Process: No heat flows between the system and surroundings ($Q = 0$).
   • First Law reduces to: $\Delta W = -\Delta U$. If the gas expands (does work), its internal energy drops, causing a temperature decrease.
   • Equation of state (Ideal Gas): $PV^\gamma = \text{constant}$, where $\gamma = \frac{C_p}{C_v}$. Other forms: $TV^{\gamma-1} = \text{constant}$ and $P^{1-\gamma}T^\gamma = \text{constant}$.
   • Work Done: $W = \int_{V_1}^{V_2} P dV = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} = \frac{\mu R (T_1 - T_2)}{\gamma - 1}$.
3. Isochoric Process: Volume is constant ($V$ = fixed).
   • Work done is zero ($W = 0$).
   • First Law reduces to: $\Delta Q = \Delta U = \mu C_v \Delta T$.
4. Isobaric Process: Pressure is constant ($P$ = fixed).
   • Work Done: $W = P(V_2 - V_1) = \mu R(T_2 - T_1)$.
   • Heat absorbed goes partially into internal energy and partially into work.
5. Cyclic Process: The system returns to its initial state.
   • Since $U$ is a state variable, $\Delta U = 0$ over the cycle.
   • First Law reduces to: Total heat absorbed = Total work done ($\Delta Q = \Delta W$).

Second Law of Thermodynamics & Reversibility

While the First Law demands energy conservation, it doesn't dictate the direction of processes. The Second Law limits processes that are perfectly allowed by energy conservation.

• Kelvin-Planck Statement: No process is possible whose sole result is the absorption of heat from a reservoir and the complete conversion of the heat into work. (Implies no heat engine can have 100% efficiency).
• Clausius Statement: No process is possible whose sole result is the transfer of heat from a colder object to a hotter object. (Implies heat pumps/refrigerators require external work; Coefficient of Performance cannot be infinite).
• Reversible Processes: A process is reversible if it can be turned back such that both the system and surroundings return exactly to their initial states, leaving no footprint in the universe. A process must be quasi-static and devoid of dissipative effects to be reversible.
• Irreversible Processes: Spontaneous natural processes are irreversible. Irreversibility arises from two main causes:
  1. The system passing through non-equilibrium states (e.g., free expansion, explosive reactions).
  2. Presence of dissipative effects like friction, viscosity, or finite temperature gradients.

Carnot Engine & Carnot Cycle

Sadi Carnot (1824) conceptualized the maximum efficiency heat engine, which must utilize entirely reversible steps. Since any finite temperature difference between the system and reservoirs causes irreversibility, the heat exchange must happen isothermally, and temperature changes must happen adiabatically.

• The Carnot Cycle (4 Steps for an Ideal Gas):
  1. Isothermal Expansion ($1 \rightarrow 2$): Gas absorbs heat $Q_1$ from the hot reservoir at $T_1$. Work $W_{1\rightarrow2} = Q_1 = \mu R T_1 \ln(V_2/V_1)$.
  2. Adiabatic Expansion ($2 \rightarrow 3$): Gas expands, cooling from $T_1$ to $T_2$. Work $W_{2\rightarrow3} = \frac{\mu R (T_1 - T_2)}{\gamma - 1}$.
  3. Isothermal Compression ($3 \rightarrow 4$): Gas rejects heat $Q_2$ to the cold reservoir at $T_2$. Work done on gas $W_{3\rightarrow4} = Q_2 = \mu R T_2 \ln(V_3/V_4)$.
  4. Adiabatic Compression ($4 \rightarrow 1$): Gas is compressed, heating from $T_2$ to $T_1$. Work done on gas $W_{4\rightarrow1} = \frac{\mu R (T_1 - T_2)}{\gamma - 1}$.
• Efficiency ($\eta$): Efficiency is $\eta = \frac{W}{Q_1} = 1 - \frac{Q_2}{Q_1}$. Using adiabatic relations $\frac{V_2}{V_1} = \frac{V_3}{V_4}$, we get the final Carnot efficiency: $\eta = 1 - \frac{T_2}{T_1}$.
• Carnot's Theorem:
  1. No engine operating between two given temperatures can be more efficient than a Carnot engine operating between the same temperatures.
  2. The efficiency of a Carnot engine is absolutely independent of the nature of the working substance.
• Universal Temperature Scale: Because the relation $\frac{Q_1}{Q_2} = \frac{T_1}{T_2}$ is universal and independent of the working substance in a Carnot cycle, it is used to mathematically define the absolute thermodynamic temperature scale.

Key Concepts & Definitions

Thermodynamics:

Branch of physics dealing with heat, temperature, and inter-conversion of heat and other forms of energy; macroscopic in nature.

Macroscopic Variables:

Quantities like P, V, T, mass, and composition that are felt by sense perceptions and measurable without considering molecular distribution.

Free Expansion:

Expansion of a gas against a vacuum. It is a non-equilibrium, irreversible process where state variables are temporarily undefined.

Heat Pump / Refrigerator:

A Carnot cycle run in reverse. It extracts heat Q2Q_2Q2​ from a cold sink at T2T_2T2​, requires external work WWW, and rejects heat Q1Q_1Q1​ to a hot source at T1T_1T1​.

Conditions & Limitations

• $\Delta Q = \Delta U + P \Delta V$: This specific form of the First Law assumes that the only work done is expansion/compression work against a constant external pressure.
• $C_p - C_v = R$: This relation strictly holds true ONLY for an ideal gas.
• $\Delta U = 0$ in Isothermal Processes: This assumption is strictly valid ONLY for an ideal gas, because real gases have intermolecular potential energy that depends on volume, meaning their internal energy is not exclusively a function of temperature.
• $PV^\gamma = \text{constant}$: This relation is strictly valid ONLY for an ideal gas undergoing a reversible adiabatic process.
• Carnot Theorem Limitation: The Carnot engine is an idealized hypothetical construct; perfectly reversible processes do not exist due to inevitable dissipative forces like friction and viscosity.

Important Graphs & Diagrams

• Specific Heat of Water vs. Temperature:
  • Curve Description: The specific heat drops from 0°C, reaches a minimum around 35°C-40°C, and then increases towards 100°C. It is not a flat line, which is why the calorie was strictly redefined for the 14.5 °C to 15.5 °C interval.
• P-V Indicator Diagrams for Isothermal and Adiabatic Processes:
  • Curve Description: In a P-V plane, isotherms ($T=\text{const}$) are hyperbolas ($P \propto 1/V$). Adiabats ($PV^\gamma = \text{const}$) are steeper than isotherms intersecting at the same point because $\gamma > 1$.
  • Area: The area under a P-V curve represents the work done by the gas ($W = \int P dV$).
• Carnot Cycle P-V Diagram:
  • A closed loop formed by two isotherms bounded by two steeper adiabats. The net work done in one cycle is the area completely enclosed inside this loop.

Standard Derivations & Step-by-Step Problem Solving

Derivation of $C_p - C_v = R$ for an Ideal Gas:

1. Apply First Law at Constant Volume: $Q = \Delta U + P \Delta V$. Since $\Delta V=0$, $Q = \Delta U$. By definition, heat capacity at constant volume is $C_v = \left(\frac{\Delta Q}{\mu \Delta T}\right)_v = \frac{\Delta U}{\mu \Delta T}$.
2. Apply First Law at Constant Pressure: $\Delta Q = \Delta U + P \Delta V$. Divide by $\mu \Delta T$: $\left(\frac{\Delta Q}{\mu \Delta T}\right)_p = \frac{\Delta U}{\mu \Delta T} + P \left(\frac{\Delta V}{\mu \Delta T}\right)$. This gives: $C_p = C_v + P \left(\frac{\Delta V}{\mu \Delta T}\right)$.
3. Use Ideal Gas Equation: $PV = \mu RT$. At constant P, a change in T causes a change in V: $P \Delta V = \mu R \Delta T \implies P \left(\frac{\Delta V}{\mu \Delta T}\right) = R$.
4. Substitute back: $C_p = C_v + R \implies C_p - C_v = R$.

Previous Year JEE Topics

• Carnot Engine Efficiency: Frequent numerical calculations involving changing sink/source temperatures to achieve a target efficiency percentage.
• P-V Graphs (Work Done): Calculating area under cyclic P-V diagrams or comparing the work done in Isobaric vs Isothermal vs Adiabatic expansions between identical initial and final volumes.
• First Law in Cyclic Processes: Identifying that $\Delta U = 0$ for a cycle, making the total heat supplied equal to the area enclosed by the cyclic P-V graph.
• Molar Heat Capacities & Mixtures: Utilizing the equipartition theorem results ($C=3R$ for solids, applying Mayer's relation for gases) to determine $\gamma$ or final mixture temperatures.
• Free Expansion: Conceptual traps asking for heat exchange or final temperature during vacuum expansion.

JEE Traps

• [JEE TIP] Trap 1 - The Caloric Fluid Illusion:

  • Misconception: Heat is a physical fluid substance stored inside a hot body that drains out and flows into a cold body upon contact.
  • Correct Understanding: Heat is not a fluid, material substance, or contained entity. It is strictly defined as energy in transit across a boundary driven exclusively by a temperature gradient or thermodynamic work. A body contains internal energy ($U$), but it never contains "heat" or "work".

• [JEE TIP] Trap 2 - Heat and Work State Variable Fallacy:

  • Misconception: A gas resting in a specific, fixed equilibrium state on a $P\text{-}V$ indicator diagram possesses a designated, trackable quantity of heat ($Q$) and work ($W$).
  • Correct Understanding: Heat and work are path functions, not state functions. They describe transient processes of energy exchange and cannot be written as differentials of a state property ($\oint dQ \neq 0$). A system contains a definite amount of internal energy ($U$), but the values of $Q$ and $W$ depend entirely on the specific coordinate path taken to reach that state.

• [JEE TIP] Trap 3 - Free Expansion Equation Hijacking:

  • Misconception: Because a gas expanding into an insulated vacuum exchange zero heat, it must follow the standard reversible adiabatic state equation $PV^\gamma = \text{constant}$.
  • Correct Understanding: Free expansion is a violent, rapid, and highly irreversible process that passes through non-equilibrium states where macro-variables like pressure and temperature are completely undefined. Because the external resisting pressure is zero ($P_{\text{ext}} = 0$), the expansion work is zero ($W = 0$). Since $Q = 0$, the First Law dictates $\Delta U = 0$, meaning $T_f = T_i$ for an ideal gas. The intermediate steps cannot be plotted on a $P\text{-}V$ diagram, and the equation $PV^\gamma = \text{constant}$ does not apply.

• [JEE TIP] Trap 4 - The Constant Gaseous Specific Heat Blanket:

  • Misconception: Similar to solid elements, a specific gas species possesses a single, fixed, unalterable specific heat capacity across all thermodynamic conditions.
  • Correct Understanding: The specific heat capacity of a gas ($C = \frac{1}{\mu}\frac{dQ}{dT}$) is highly variable and depends entirely on the process constraints under which heat is supplied. It ranges along a continuous spectrum from $0$ during a reversible adiabatic process ($dQ = 0$) to $\infty$ during an isothermal process ($dT = 0$). Only the specific boundaries $C_p$ and $C_v$ represent locked, isolated process constants.

• [JEE TIP] Trap 5 - The Mechanical Joule Equivalent Identity:

  • Misconception: The mechanical equivalent of heat ($J = 4.186 \text{ J/cal}$) is a fundamental physical constant of nature on par with the Universal Gas Constant ($R$) or Newton's Gravitational Constant ($G$).
  • Correct Understanding: In modern physics, $J$ is completely superfluous and carries no physical dimensionality. It is merely an arbitrary conversion factor that bridges an obsolete historical unit of thermal energy (the calorie) with the standardized SI unit of mechanical energy (the joule). Both represent the exact same thermodynamic currency.

• [JEE TIP] Trap 6 - Carnot Efficiency Working Substance Blindspot:

  • Misconception: The maximum operational efficiency of a reversible Carnot engine changes if you swap out a monoatomic ideal gas for a diatomic or polyatomic working substance.
  • Correct Understanding: The peak efficiency of a Carnot cycle is completely universal and depends strictly and exclusively on the absolute temperatures of the hot source and cold sink ($\eta = 1 - \frac{T_2}{T_1}$). It is entirely independent of the nature, atomicity, or specific heat ratios ($\gamma$) of the working substance running inside the cylinder.

• [JEE TIP] Trap 7 - First Law Component Path Reliance:

  • Misconception: Because the individual terms representing heat ($\Delta Q$) and work ($\Delta W$) vary dynamically based on the path carved across a $P\text{-}V$ diagram, the internal energy change ($\Delta U$) must also be path-dependent.
  • Correct Understanding: This is a classic algebraic trap. While $\Delta Q$ and $\Delta W$ are individually path-dependent, the exact differential combination dictated by the First Law of Thermodynamics—$\Delta U = \Delta Q - \Delta W$—is strictly path-independent. The path histories cancel out perfectly, making $\Delta U$ depend solely on the initial and final state coordinates ($\mu C_v \Delta T$).

• [JEE TIP] Trap 8 - Real Gas Isothermal Internal Energy:

  • Misconception: If a thermodynamic process is kept perfectly isothermal ($\Delta T = 0$), the net internal energy change ($\Delta U$) of the substance must automatically equal zero.
  • Correct Understanding: The rule $\Delta U = 0$ for an isothermal process holds true exclusively for ideal gases, where internal energy is strictly a function of temperature alone ($Unit = f(T)$). For real gases and real materials, intermolecular attractive forces are non-zero. Consequently, internal energy depends on both temperature and volume; an isothermal expansion alters the molecular separations, creating a non-zero internal potential energy change ($\Delta U \neq 0$).

• [JEE TIP] Trap 9 - The Adiabatic Work Temperature Sign Inversion:

  • Misconception: The mathematical equation to calculate work done by an expanding gas during a reversible adiabatic process is written as $W = \frac{\mu R (T_2 - T_1)}{\gamma - 1}$.
  • Correct Understanding: The correct thermodynamic formulation is $W = \frac{\mu R (T_1 - T_2)}{\gamma - 1}$, where $T_1$ is the initial temperature and $T_2$ is the final temperature. Because an adiabatic expansion ($W > 0$) occurs in complete isolation from external heat sources ($Q = 0$), the gas can only perform work by exhausting its internal thermal reserve. This forces a temperature drop ($T_1 > T_2$), meaning switching the indices will yield a fatal sign error in your calculation loops.

• [JEE TIP] Trap 10 - Intensive vs. Extensive Algebraic Operations:

  • Misconception: If you weld two identical thermodynamic systems together, their intensive properties like pressure ($P$), temperature ($T$), and mass density ($\rho$) double in numerical value.
  • Correct Understanding: Intensive properties are completely invariant to system size or scaling. Combining systems leaves $P, T,$ and $\rho$ perfectly constant. Conversely, extensive properties like mass ($M$), volume ($V$), and internal energy ($U$) scale linearly and double. Note the algebraic coupling rule: the product of an intensive variable and an extensive change—such as boundary work ($W = P \cdot \Delta V$)—always yields a net extensive property.