To which one of the following is the average speed of molecules of a gas directly proportional ?

1. Matter and Thermal Expansion (basic)

To master thermal physics, we must first look at the very 'building blocks' of everything around us: matter. All matter is composed of tiny particles that are never truly still. The way these particles behave—how they are packed and how they move—determines whether a substance is a solid, a liquid, or a gas. In a solid, particles are closely packed with very strong attractive forces, meaning they can only vibrate in fixed positions. In contrast, particles in a liquid have slightly more space and can move past each other, while in a gas, they have enough energy to overcome almost all attractive forces and move freely in all directions Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p.112.

The defining difference between these states lies in the interparticle distance and the strength of their attractive forces. As the internal energy of a substance increases, these particles move more vigorously. For instance, when ice (solid) melts into water (liquid), the interparticle attractions decrease, allowing the particles to break away from their fixed positions Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p.113. This relationship between energy and motion is the foundation of thermal physics.

At a molecular level, temperature is essentially a measure of the average kinetic energy of these particles. According to the Kinetic Theory, as the absolute temperature (T) of a substance increases, the average speed of its molecules also increases. Specifically, in gases, this speed is directly proportional to the square root of the absolute temperature (v_avg ∝ √T). As particles move faster and collide more forcefully, they tend to push further apart, which explains the fundamental reason why most materials undergo thermal expansion when heated.

State of Matter	Particle Arrangement	Movement
Solid	Closely packed, fixed positions	Vibration only
Liquid	Less closely packed	Move past each other
Gas	Far apart, large spaces	Free movement in all directions

Sources: Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p.112; Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p.113

2. Fundamental Gas Laws (basic)

To understand thermal physics, we must first grasp how gases behave under different conditions. At a macroscopic level, the state of a gas is defined by three interconnected variables: Pressure (P), Volume (V), and Temperature (T). The fundamental gas laws describe how these variables influence one another. For example, if you keep the volume of a container constant, the pressure of the gas inside will increase as you raise the temperature. This is a practical reality for vehicle tires; as friction with the road heats the air inside a tube, the pressure rises, which can lead to a burst if it exceeds a certain threshold Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.296.

While the macroscopic laws tell us what happens, the Kinetic Theory of Gases explains why it happens at a molecular level. According to this theory, temperature is simply a measure of the average kinetic energy of the gas molecules. When we heat a gas, we are essentially giving its molecules more energy to move. The relationship is precise: the average speed (v_avg) of gas molecules is directly proportional to the square root of the absolute temperature (√T). This means that as temperature goes up, the molecules hit the walls of their container more frequently and with greater force, which we observe as an increase in pressure.

Gas Law	Constant Variable	Relationship
Boyle’s Law	Temperature (T)	Pressure is inversely proportional to Volume (P ∝ 1/V)
Charles’s Law	Pressure (P)	Volume is directly proportional to Temperature (V ∝ T)
Gay-Lussac’s Law	Volume (V)	Pressure is directly proportional to Temperature (P ∝ T)

Sources: Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.296

3. The Ideal Gas Equation (intermediate)

To understand the behavior of gases in our atmosphere or within a laboratory, we use a fundamental mathematical model known as the Ideal Gas Equation: PV = nRT. This equation acts as a bridge between the macroscopic properties we can measure (like pressure and volume) and the microscopic behavior of individual molecules. While no 'perfect' ideal gas exists in nature, most atmospheric gases like Nitrogen (78%) and Oxygen (21%) behave very closely to this model under standard conditions Physical Geography by PMF IAS, Earths Atmosphere, p.271.
Let’s break down the components of the equation:

Symbol	Property	Description
P	Pressure	The force exerted by gas molecules colliding with the container walls.
V	Volume	The physical space the gas occupies.
n	Amount	The number of moles (quantity) of the gas.
R	Gas Constant	A universal value (≈ 8.314 J/mol·K) that links these properties together.
T	Temperature	The absolute temperature measured in Kelvin (K).

The most critical insight for a student of physics is the relationship between Temperature and Kinetic Energy. According to the Kinetic Theory of Gases, temperature is a direct measure of the average kinetic energy of the molecules. Specifically, the average speed (v_avg) of gas molecules is directly proportional to the square root of the absolute temperature (v_avg ∝ √T). This means that as the atmosphere warms, gas molecules move faster, which explains why warmer air expands and becomes less dense—a concept vital for understanding the Adiabatic Lapse Rate and how air parcels rise in the atmosphere Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.296.

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.271; Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.296

4. Heat, Temperature, and Internal Energy (intermediate)

To master thermal physics, we must first distinguish between three often-confused terms: Heat, Temperature, and Internal Energy. At a fundamental level, every substance is made of particles in constant motion. Heat represents the total energy (kinetic and potential) resulting from this molecular movement, while Temperature is simply the measurement of how hot or cold a substance is Fundamentals of Physical Geography, Solar Radiation, Heat Balance and Temperature, p.70. Think of it this way: Heat is the 'total energy' contained in the system, whereas temperature is the 'average kinetic energy' per molecule. In the atmosphere, this is why we see a shift in 'heat belts' as solar radiation interacts with the Earth's surface, leading to varying temperature recordings across different latitudes Contemporary India-I, Climate, p.30.

According to the Kinetic Theory of Gases, the temperature of a gas is a direct reflection of the speed of its molecules. Specifically, the average speed (v_avg) of gas molecules is directly proportional to the square root of the absolute temperature (√T). The formula is expressed as v_avg = √(8RT/πM). This means if you increase the temperature, the molecules dance faster, increasing the system's internal energy. However, adding heat doesn't always raise the temperature. During a phase change (like ice melting into water), the temperature remains constant because the heat supplied—known as latent heat—is used to break molecular bonds rather than increase molecular speed Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.295.

Finally, we must understand how this energy moves. Heat transfer occurs from a hotter part to a colder part through three primary methods: Conduction (in solids, where particles vibrate but stay in place), Convection (in liquids and gases, where particles move physically), and Radiation (which requires no medium) Science-Class VII, Heat Transfer in Nature, p.101.

Feature	Heat	Temperature
Definition	Total energy of molecular motion.	Average kinetic energy of molecules.
Unit	Joules (J) or Calories (cal).	Celsius (°C), Kelvin (K), or Fahrenheit (°F).
Property	An extensive property (depends on the amount of matter).	An intensive property (independent of the amount of matter).

Sources: Fundamentals of Physical Geography, Solar Radiation, Heat Balance and Temperature, p.70; Contemporary India-I, Climate, p.30; Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.295; Science-Class VII, Heat Transfer in Nature, p.101

5. Speed of Sound in Gases (intermediate)

To understand why sound travels at a specific speed through a gas, we must look at the Kinetic Theory of Gases. Imagine gas molecules as tiny, energetic billiard balls constantly colliding. Sound is a mechanical wave that propagates through these collisions—it is essentially a "relay race" of energy. The faster the molecules move individually, the faster they can pass the sound vibration to their neighbors. According to the Maxwell-Boltzmann distribution, the average speed of gas molecules (v_avg) is directly linked to the absolute temperature (T) of the gas. Specifically, the speed of sound in an ideal gas is directly proportional to the square root of its absolute temperature (v ∝ √T).

A common misconception is that increasing the pressure of a gas will speed up sound. However, in an ideal gas of constant composition, the speed of sound is independent of pressure and density. This is because if you increase the pressure, the density increases proportionally, and these two effects cancel each other out. Thus, temperature remains the primary master of sound speed Physical Geography by PMF IAS, Earths Atmosphere, p.274. As we move through different layers of the atmosphere—from the Troposphere to the Thermosphere—the speed of sound fluctuates precisely because the temperature profile changes at different altitudes Physical Geography by PMF IAS, Earths Atmosphere, p.274.

Beyond temperature, the composition of the gas also plays a role. Heavier molecules (high molar mass) move more sluggishly than lighter ones at the same temperature. This is why sound travels faster in Hydrogen than in Oxygen. In our atmosphere, humidity also has a subtle effect. Because water vapor molecules are lighter than the Nitrogen and Oxygen molecules they replace, moist air is actually less dense than dry air Physical Geography by PMF IAS, Hydrological Cycle (Water Cycle), p.326. This reduction in the average molar mass of the air allows sound to travel slightly faster on a humid day than on a dry one.

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.274; Physical Geography by PMF IAS, Hydrological Cycle (Water Cycle), p.326

6. Atmospheric Adiabatic Processes (exam-level)

In atmospheric science, an adiabatic process refers to a change in the temperature of a gas (like a parcel of air) without any heat being exchanged with its surroundings. This is a fundamental concept for understanding how clouds form and why it is colder at the top of a mountain. The term "adiabatic" literally means that all temperature changes are internal to the system—no heat enters or leaves the parcel (Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.296).

To understand why this happens, we look at the relationship between pressure, volume, and temperature. As a parcel of air rises, it moves into regions of lower atmospheric pressure because the density of air decreases with altitude (NCERT Class XI Fundamentals of Physical Geography, Atmospheric Circulation and Weather Systems, p.76). According to the Gas Law, as the external pressure drops, the air parcel expands. To expand, the gas molecules must do work against the surrounding atmosphere. Since no outside heat is provided, the energy for this work is taken from the parcel's own internal kinetic energy. From a molecular perspective, since the average speed of molecules (v_avg) is directly proportional to the square root of the absolute temperature (√T), a loss in kinetic energy results in a drop in temperature.

Conversely, when air descends, it is compressed by the higher pressure of the lower atmosphere (Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305). This compression does work on the air parcel, increasing its internal kinetic energy and causing it to warm up. We categorize these changes using the Adiabatic Lapse Rate (ALR), which is the specific rate at which temperature changes with height during these vertical movements.

Process	Movement	Pressure Change	Temperature Change
Adiabatic Cooling	Rising Air	Decreases (Expansion)	Cools down
Adiabatic Warming	Sinking Air	Increases (Compression)	Warms up

Sources: Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.296; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Atmospheric Circulation and Weather Systems, p.76; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.305

7. Postulates of Kinetic Theory of Gases (exam-level)

The Kinetic Theory of Gases (KTG) provides a microscopic lens to understand macroscopic properties like pressure and temperature. At its heart, the theory assumes that a gas consists of a vast number of tiny particles (atoms or molecules) in constant, random motion. These particles are so small compared to the distances between them that their individual volume is considered negligible. This explains why gases are highly compressible and can easily spread to fill any available space Science, Class VIII NCERT, Particulate Nature of Matter, p.115.

A fundamental postulate of KTG is that these molecules are in a state of perfectly elastic collisions with each other and the walls of their container. When you heat a gas, you are essentially adding energy to these particles. This manifests as an increase in their Average Kinetic Energy, which is directly proportional to the Absolute Temperature (T). This is why a balloon expands when placed in hot water; the molecules gain energy, move faster, and strike the balloon's walls with greater force Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.84.

To quantify this motion, we look at the Average Speed (v_avg) of the molecules. Derived from the Maxwell-Boltzmann distribution, the formula is v_avg = √(8RT/πM), where R is the gas constant, T is the absolute temperature, and M is the molar mass. This tells us two critical things: first, the speed of molecules is directly proportional to the square root of the absolute temperature (v ∝ √T). Second, speed is inversely proportional to the square root of the mass; thus, lighter gases like Hydrogen move much faster than heavier gases like Oxygen at the same temperature Physical Geography by PMF IAS, Earth's Atmosphere, p.271.

Postulate	Physical Reality
Constant Random Motion	Gases diffuse and mix spontaneously.
Elastic Collisions	No energy is lost as heat during molecular impacts; pressure remains constant if T is constant.
KE ∝ Temperature	Heating a gas increases molecular velocity and internal pressure.

Sources: Science, Class VIII NCERT, Particulate Nature of Matter, p.115; Science, Class VIII NCERT, Pressure, Winds, Storms, and Cyclones, p.84; Physical Geography by PMF IAS, Earth's Atmosphere, p.271

8. Molecular Speeds: Root Mean Square and Average (exam-level)

Concept: Molecular Speeds: Root Mean Square and Average

9. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamentals of the Kinetic Theory of Gases, this question allows you to apply the core principle that thermal energy is essentially the kinetic energy of molecular motion. You have learned that as a substance absorbs heat, its molecules move more vigorously. The "building blocks" here are the concepts of Absolute Temperature (Kelvin) and the mathematical derivation of kinetic energy ($KE = \frac{1}{2}mv^2$). This specific question tests your ability to move from a general qualitative understanding—that gases move faster when hot—to the precise quantitative relationship defined by the Maxwell-Boltzmann distribution.

To arrive at the correct answer, think like a physicist: the formula for average speed is $v_{avg} = \sqrt{\frac{8RT}{\pi M}}$. Notice that the temperature variable ($T$) is nested inside a square root sign. This means that if you were to increase the temperature fourfold, the speed would only double. Therefore, the average speed is directly proportional to the square root of the absolute temperature. This relationship is a fundamental pillar of thermal physics and atmospheric science, as it explains how energy is distributed across molecules in our atmosphere, a concept touched upon in Physical Geography by PMF IAS when discussing the structure and thermal properties of the Earth's atmosphere.

UPSC often includes "decoy" options to reward precision over general knowledge. Option (A) is a classic trap; while speed increases with temperature, it is not a linear (1:1) relationship. Option (B) reverses the logic, and Option (C) mentions pressure, which is a result of molecular collisions rather than the primary variable determining individual molecular velocity. By focusing on the square root relationship, you avoid the common pitfall of assuming all direct proportionality is linear. Always remember: in the world of gas kinetics, temperature drives speed through the lens of a square root.