12.5 Kinetic Theory of Gases, First Law, Heat Engines & Carnot Cycle

Kinetic Theory of Gases, First Law, Heat Engines & Carnot Cycle

In the Intermediate Physics Part I syllabus and the UET ECAT entry test, heat and thermodynamics represent a core mechanical domain. Entry test questions frequently test work done in thermodynamic cycles, adiabatic relations, molar specific heat derivations ($C_p - C_v = R$), Carnot efficiency limits, refrigerator coefficient of performance, and entropy changes under timed conditions. Developing strong analytical fluency across microscopic kinetic theory and macroscopic classical thermodynamics is vital.

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1. Postulates of the Kinetic Theory of Gases

The kinetic molecular theory models macroscopic gas behavior in terms of the statistical Newtonian mechanics of microscopic particles. The fundamental postulates are:

1. Point Particles: A finite volume of gas consists of an enormous number of identical molecules whose individual molecular volumes are negligible compared to the total volume occupied by the gas.
2. Continuous Random Motion: Gas molecules are in constant, chaotic, straight-line motion in all directions, continually colliding with one another and the container walls.
3. Elastic Collisions: Collisions between molecules and against container walls are perfectly elastic; linear momentum and kinetic energy are conserved, and collision durations are negligible compared to the time between collisions.
4. Negligible Intermolecular Forces: Except during instantaneous collisions, molecules exert no attractive or repulsive intermolecular forces on one another (potential energy is zero; internal energy is purely kinetic).
5. Classical Mechanics Validity: Newton's laws of motion apply directly to molecular dynamics, and molecular velocities obey Maxwell-Boltzmann statistical distributions.

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2. Microscopic Derivation of Gas Pressure

Consider an ideal gas of $N$ identical molecules, each of mass $m$, enclosed in a cubical container of side length $L$ and volume $V = L^3$.

Let a molecule move with velocity $\mathbf{v}_1 = (v_{1x}, v_{1y}, v_{1z})$ toward a wall perpendicular to the $x$-axis:

1. Upon an elastic collision with the wall, its $x$-component of velocity reverses from $+v_{1x}$ to $-v_{1x}$, while $y$ and $z$ components remain unchanged.
2. The change in momentum per collision is:

$\Delta p_x = (-m v_{1x}) - (m v_{1x}) = -2m v_{1x}$

1. By Newton's third law, the momentum imparted to the wall is $+2m v_{1x}$.
2. The time interval between successive collisions of this molecule with the same wall is the round-trip time:

$\Delta t = \frac{2L}{v_{1x}}$

1. By Newton's second law, the time-averaged force exerted by this single molecule on the wall is:

$F_{1x} = \frac{\Delta p}{\Delta t} = \frac{2m v_{1x}}{2L / v_{1x}} = \frac{m v_{1x}^2}{L}$

1. The total force exerted on the wall by all $N$ molecules is the sum of individual contributions:

$F_x = \sum_{i=1}^N \frac{m v_{ix}^2}{L} = \frac{m}{L} \sum_{i=1}^N v_{ix}^2 = \frac{m N}{L} \langle v_x^2 \rangle$

where $\langle v_x^2 \rangle = \frac{1}{N}\sum v_{ix}^2$ is the mean-square velocity along the $x$-axis.

1. Pressure is force divided by wall area $A = L^2$:

$P = \frac{F_x}{L^2} = \frac{m N}{L^3} \langle v_x^2 \rangle = \frac{m N}{V} \langle v_x^2 \rangle$

Because molecular motion is isotropic (equally distributed along all spatial directions):

$\langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle = \frac{1}{3}\langle v^2 \rangle$

Substituting this isotropy condition yields the fundamental kinetic pressure equation:

$P = \frac{1}{3}\frac{m N}{V}\langle v^2 \rangle = \frac{1}{3}\rho \langle v^2 \rangle$

where $\rho = \frac{m N}{V}$ is the mass density of the gas.

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3. Kinetic Interpretation of Temperature & RMS Speed

Multiplying the pressure equation by volume $V$:

$P V = \frac{2}{3} N \left(\frac{1}{2}m \langle v^2 \rangle\right) = \frac{2}{3} N \langle K.E. \rangle$

Comparing this with the macroscopic Ideal Gas Law $P V = N k_B T$ (where $k_B = 1.38 \times 10^{-23}\text{ J/K}$ is the Boltzmann constant):

$N k_B T = \frac{2}{3} N \langle K.E. \rangle \implies \langle K.E. \rangle = \frac{1}{2}m \langle v^2 \rangle = \frac{3}{2} k_B T$

ECAT Foundational Insight: The absolute thermodynamic temperature $T$ of an ideal gas is directly proportional to the average translational kinetic energy of its molecules: $T \propto \langle K.E. \rangle$ Absolute zero ($T = 0\text{ K} = -273.15^\circ\text{C}$) corresponds to the theoretical state where all molecular translational motion ceases.

Root-Mean-Square (RMS) Speed

The root-mean-square velocity $v_{\text{rms}}$ is defined as the square root of the mean-square velocity:

$v_{\text{rms}} = \sqrt{\langle v^2 \rangle} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 P}{\rho}}$

where $M = m N_A$ is the molar mass of the gas and $R = N_A k_B = 8.314\text{ J}/(\text{mol}\cdot\text{K})$ is the universal gas constant.

• Dependence on Temperature: $v_{\text{rms}} \propto \sqrt{T}$ (quadrupling absolute temperature doubles RMS speed).
• Dependence on Molar Mass: $v_{\text{rms}} \propto \frac{1}{\sqrt{M}}$ (at identical temperature, light hydrogen molecules $\text{H}_2$ move four times faster on average than oxygen molecules $\text{O}_2$).

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4. The First Law of Thermodynamics

The First Law of Thermodynamics is the universal principle of conservation of energy applied to thermodynamic systems:

$Q = \Delta U + W$

In differential form: $dQ = dU + dW = dU + P dV$.

Standard Sign Conventions

• Heat ($Q$):
  • $Q > 0$ (positive): Heat is absorbed/added to the system from surroundings.
  • $Q < 0$ (negative): Heat is released/rejected by the system to surroundings.
• Work ($W$):
  • $W > 0$ (positive): Work is performed by the system on surroundings (expansion: $dV > 0$).
  • $W < 0$ (negative): Work is performed on the system by surroundings (compression: $dV < 0$).
• Internal Energy Change ($\Delta U$):
  • $\Delta U > 0$: Temperature increases.
  • $\Delta U < 0$: Temperature decreases.

Internal Energy of an Ideal Gas

For an ideal gas with zero intermolecular forces, internal energy $U$ depends strictly on temperature: $U = U(T)$ (Joule's Law). For $n$ moles of an ideal gas:

$\Delta U = n C_v \Delta T$

Regardless of the path taken between two thermodynamic states, $\Delta U$ is identical because internal energy is a state function.

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5. The Four Fundamental Thermodynamic Processes

1. Isothermal Process ($T = \text{constant}$)

A process occurring at constant temperature, typically carried out very slowly in a container with perfectly heat-conducting walls immersed in a thermal reservoir.

• Governing Law: Boyle's Law ($P V = \text{constant} \implies P_1 V_1 = P_2 V_2$).
• Internal Energy Change: Because $\Delta T = 0$, $\Delta U = 0$.
• First Law Formulation: $Q = W$.
• Work Done:

$W = \int_{V_1}^{V_2} P dV = \int_{V_1}^{V_2} \frac{nRT}{V} dV = nRT\ln\left(\frac{V_2}{V_1}\right) = nRT\ln\left(\frac{P_1}{P_2}\right)$

• Slope of $P$-$V$ Curve:

$P V = C \implies P dV + V dP = 0 \implies \left(\frac{dP}{dV}\right)_{\text{iso}} = -\frac{P}{V}$

2. Isobaric Process ($P = \text{constant}$)

A process occurring at constant pressure (e.g., heating gas inside a cylinder beneath a freely movable, weighted piston).

• Governing Law: Charles's Law ($\frac{V}{T} = \text{constant}$).
• Work Done:

$W = P\Delta V = P(V_2 - V_1) = nR\Delta T = nR(T_2 - T_1)$

• Heat Transferred: $Q = n C_p \Delta T$.
• Internal Energy Change: $\Delta U = Q - W = n C_p \Delta T - n R \Delta T = n(C_p - R)\Delta T = n C_v \Delta T$.

3. Isochoric (Isovolumetric) Process ($V = \text{constant}$)

A process occurring at constant volume (e.g., heating gas inside a rigid closed container).

• Governing Law: Gay-Lussac's Law ($\frac{P}{T} = \text{constant}$).
• Work Done: Because $dV = 0$, $W = 0$.
• First Law Formulation:

$Q = \Delta U = n C_v \Delta T$

All heat supplied goes entirely into raising internal energy and temperature.

4. Adiabatic Process ($Q = 0$)

A process in which no heat enters or leaves the system ($Q = 0$). Realized either by encasing the gas in perfectly thermally insulated walls or by executing expansion/compression extremely rapidly.

• First Law Formulation:

$0 = \Delta U + W \implies W = -\Delta U = -n C_v \Delta T$

• Adiabatic Expansion: The gas does positive work ($W > 0$) strictly at the expense of its own internal energy ($\Delta U < 0$); hence, the gas cools ($T_2 < T_1$).

• Adiabatic Compression: Work is done on the gas ($W < 0$), increasing internal energy ($\Delta U > 0$); hence, the gas heats up ($T_2 > T_1$).

• Adiabatic Equations of State:

$P V^\gamma = \text{constant}, \quad T V^{\gamma - 1} = \text{constant}, \quad T^\gamma P^{1-\gamma} = \text{constant}$

• Work Done in Adiabatic Expansion:

$W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} = \frac{nR(T_1 - T_2)}{\gamma - 1}$

• Slope of $P$-$V$ Curve:

$P V^\gamma = C \implies \gamma P V^{\gamma-1} dV + V^\gamma dP = 0 \implies \left(\frac{dP}{dV}\right)_{\text{adia}} = -\gamma\frac{P}{V}$

ECAT Slope Rule: The adiabatic curve on a $P$-$V$ indicator diagram is $\gamma$ times steeper than the isothermal curve passing through the same state point: $\left(\frac{dP}{dV}\right)_{\text{adia}} = \gamma \left(\frac{dP}{dV}\right)_{\text{iso}}$.

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6. Molar Heat Capacities: Mayer's Relation ($C_p - C_v = R$)

The molar heat capacity of a gas is the amount of heat energy required to raise the temperature of one mole of gas by $1\text{ K}$ ($1^\circ\text{C}$):

$C = \frac{Q}{n \Delta T}$

1. Molar Heat Capacity at Constant Volume ($C_v$): Heat is added while volume is clamped ($W = 0$). All heat increases internal energy: $Q_v = \Delta U = n C_v \Delta T$.
2. Molar Heat Capacity at Constant Pressure ($C_p$): Heat is added while the gas expands at constant pressure, performing boundary work against external atmosphere ($W = P\Delta V$). Heat must supply both the internal energy rise and the boundary work: $Q_p = \Delta U + W$.

Setting $Q_p = n C_p \Delta T$, $\Delta U = n C_v \Delta T$, and $W = P\Delta V = n R \Delta T$:

$n C_p \Delta T = n C_v \Delta T + n R \Delta T \implies C_p = C_v + R$

$C_p - C_v = R$

This fundamental equation is Mayer's Relation. Because work is done during constant pressure expansion, $C_p$ is always strictly greater than $C_v$.

Values of Heat Capacities for Gases

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7. Heat Engines and the Second Law of Thermodynamics

A heat engine is a mechanical device that operates in a thermodynamic cycle, absorbing heat $Q_H$ from a high-temperature reservoir ($T_H$), converting a fraction into net mechanical work $W$, and discharging waste heat $Q_C$ to a low-temperature sink ($T_C$).

Because the engine returns to its initial thermodynamic state after completing one cycle, cyclic internal energy change is zero: $\Delta U_{\text{cycle}} = 0$. By the First Law:

$W = Q_H - Q_C$

Thermal Efficiency ($\eta$)

The thermal efficiency $\eta$ of a heat engine is defined as the ratio of net work output to total heat input:

$\eta = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$

The Second Law of Thermodynamics

While the First Law asserts energy conservation, the Second Law establishes the physical direction of spontaneous thermal processes:

1. Kelvin-Planck Statement (Engine Formulation): It is impossible to construct a cyclic heat engine whose sole effect is to absorb heat from a single thermal reservoir and convert it completely into mechanical work without rejecting any heat to a lower-temperature sink.
   • Implication: No heat engine can ever achieve $100\%$ thermal efficiency ($\eta < 1$). A cold sink is physically indispensable for continuous cyclic work production.
2. Clausius Statement (Refrigerator Formulation): It is impossible to construct a cyclic device whose sole effect is the transfer of heat from a lower-temperature body to a higher-temperature body without the consumption of external mechanical work.
   • Implication: Heat never flows spontaneously from cold to hot.

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8. The Carnot Cycle and Carnot Engine

In 1824, the French engineer Sadi Carnot conceived an idealized, perfectly reversible cyclic heat engine that operates between two constant temperature reservoirs at $T_H$ and $T_C$. The Carnot cycle consists of four successive reversible thermodynamic processes:

1. Process 1 $\to$ 2: Reversible Isothermal Expansion at $T_H$
   • The cylinder is placed on the hot reservoir at $T_H$.
   • Gas expands slowly from volume $V_1$ to $V_2$, absorbing heat $Q_H = n R T_H \ln\left(\frac{V_2}{V_1}\right)$.
   • Temperature remains constant at $T_H$; $\Delta U_{12} = 0$; positive work $W_1 = Q_H$ is done by the gas.
2. Process 2 $\to$ 3: Reversible Adiabatic Expansion ($T_H \to T_C$)
   • The cylinder is placed on a perfectly insulating base ($Q = 0$).
   • Gas expands from $V_2$ to $V_3$, cooling from $T_H$ to $T_C$.
   • Positive work $W_2 = -\Delta U_{23} = n C_v (T_H - T_C)$ is performed at the expense of internal energy.
3. Process 3 $\to$ 4: Reversible Isothermal Compression at $T_C$
   • The cylinder is placed on the cold sink at $T_C$.
   • Work is done on the gas, compressing it from $V_3$ to $V_4$ while rejecting heat $Q_C = n R T_C \ln\left(\frac{V_3}{V_4}\right)$.
   • $\Delta U_{34} = 0$; negative work $W_3 = -Q_C$ is performed.
4. Process 4 $\to$ 1: Reversible Adiabatic Compression ($T_C \to T_H$)
   • The cylinder is placed on the insulating base ($Q = 0$).
   • Gas is compressed from $V_4$ to initial volume $V_1$, heating from $T_C$ back to $T_H$.
   • Negative work $W_4 = -\Delta U_{41} = -n C_v (T_H - T_C)$ restores the initial state.

Carnot Thermal Efficiency Formula

From the adiabatic state relations along paths $2 \to 3$ and $4 \to 1$:

$\frac{V_2}{V_1} = \frac{V_3}{V_4} \implies \frac{Q_C}{Q_H} = \frac{T_C}{T_H}$

Substituting this thermodynamic temperature ratio into the efficiency formula yields the Carnot Efficiency:

$\eta_C = 1 - \frac{T_C}{T_H} = \frac{T_H - T_C}{T_H}$

where $T_H$ and $T_C$ must be expressed strictly in Kelvin.

Carnot's Theorem

1. No real heat engine operating between two specified heat reservoirs can be more efficient than a reversible Carnot engine operating between the same two reservoirs: $\eta_{\text{real}} < \eta_{\text{Carnot}}$.
2. All reversible engines operating between the same two temperatures possess identical efficiency, completely independent of the working substance.

ECAT Strategy Note: To maximize Carnot efficiency, increase $T_H$ as high as possible and decrease $T_C$ as low as possible. In practice, reducing sink temperature $T_C$ by a given number of degrees increases efficiency more than raising source temperature $T_H$ by the identical number of degrees.

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9. Refrigerators, Heat Pumps and Coefficient of Performance (COP)

A refrigerator is a heat engine operated in reverse: external mechanical work $W$ is supplied to extract heat $Q_C$ from a cold space and dump heat $Q_H = Q_C + W$ into warmer surroundings.

Coefficient of Performance (COP)

• Refrigerator COP ($\beta$): Ratio of desired cooling to required work input:

$\text{COP}_{\text{ref}} = \frac{Q_C}{W} = \frac{Q_C}{Q_H - Q_C}$

For an ideal Carnot refrigerator:

$\text{COP}_{\text{Carnot, ref}} = \frac{T_C}{T_H - T_C}$

• Heat Pump COP: Ratio of heating delivered to required work input:

$\text{COP}_{\text{HP}} = \frac{Q_H}{W} = \frac{T_H}{T_H - T_C} = \text{COP}_{\text{ref}} + 1$

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10. Concept of Entropy

In 1865, Rudolf Clausius introduced the state function entropy ($S$) to quantify the unavailability of thermal energy for mechanical work and the microscopic degree of molecular disorder.

Mathematical Formulation

For an infinitesimal reversible transfer of heat $dQ_{\text{rev}}$ at absolute temperature $T$:

$dS = \frac{dQ_{\text{rev}}}{T} \implies \Delta S = \int_1^2 \frac{dQ_{\text{rev}}}{T}$

• SI Unit of Entropy: $\text{J}/\text{K}$ (Joules per Kelvin).
• For an isothermal heat transfer: $\Delta S = \frac{Q}{T}$.
• For an adiabatic reversible process ($dQ_{\text{rev}} = 0$): $\Delta S = 0$ (isentropic process).

Principle of Entropy Increase

In any real, spontaneous (irreversible) process, the total entropy of the universe (system plus surroundings) strictly increases:

$\Delta S_{\text{universe}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} > 0$

Entropy remains constant only for idealized reversible cycles. This principle defines the thermodynamic arrow of time: natural macroscopic processes proceed unidirectionally toward states of greater probability, higher molecular chaos, and degraded energy availability.

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11. Thermodynamic Processes Comparison Table

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12. Petrol and Diesel Engines

Both are internal-combustion heat engines, so both are limited by the second law.

The diesel's higher compression raises the air temperature enough to ignite the fuel and also raises the efficiency. Neither engine can reach the Carnot limit, because its working cycle is irreversible and heat must be rejected in the exhaust.

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13. The Thermodynamic (Kelvin) Temperature Scale

Because the efficiency of a reversible (Carnot) engine depends only on the reservoir temperatures, Kelvin used the ratio of heats to define temperature independently of any thermometric substance:

$\frac{Q_c}{Q_h} = \frac{T_c}{T_h}$

The scale was fixed by assigning $273.16\text{ K}$ to the triple point of water; since 2019 the kelvin has been defined by fixing the Boltzmann constant at $k = 1.380649 \times 10^{-23}\text{ J/K}$. Absolute zero is the temperature at which a reversible engine would reject no heat, which is why it can be approached but never reached.