One mole of hydrogen gas burns in excess of oxygen to give 290 kJ of heat. What is the amount of heat produced when 4g of hydrogen gas is burnt under the same conditions ?

1. The Mole Concept and Avogadro's Number (basic)

In the world of physics and chemistry, we often need to bridge the gap between the tiny world of atoms and the visible world of grams and kilograms. The Mole is the fundamental SI unit that acts as this bridge. Think of a 'mole' just like you think of a 'dozen'—it is simply a counting unit. While a dozen represents 12 items, one mole represents 6.022 × 10²³ particles (atoms, molecules, or ions). This staggering figure is known as Avogadro’s Number (Nₐ).

Why this specific number? It was chosen so that the mass of one mole of a substance in grams is numerically equal to its atomic or molecular mass in atomic mass units (u). For example, a single Hydrogen atom has an atomic mass of 1 u. Therefore, one mole of Hydrogen atoms has a mass of exactly 1 gram. If we look at the molecular level, a molecule of Hydrogen (H₂) consists of two atoms; since each Hydrogen atom has an atomic mass of 1 u Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p. 66, the molar mass of H₂ is 2 grams per mole (g/mol).

Understanding this relationship allows us to calculate the 'amount of substance' by simply weighing it. This is crucial in thermal physics because properties like pressure and heat capacity often depend on the number of particles present rather than just their total weight. To find the number of moles (n) in a sample, we use the simple ratio: n = Mass of substance (m) / Molar mass (M). As we see in laboratory measurements, mass is a tangible way to quantify the matter we are dealing with Science, Class VIII, NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p. 141.

Term	Definition	Units
Atomic Mass	Mass of a single atom	u (unified mass unit)
Molar Mass	Mass of 6.022 × 10²³ particles	g/mol
Avogadro’s Number	The number of particles in one mole	6.022 × 10²³

Sources: Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.66; Science, Class VIII, NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.141

2. Molar Mass of Elements and Diatomic Molecules (basic)

To understand thermal physics, we must first master how to count particles in a substance. While we measure mass in grams, chemical reactions and thermal properties depend on the number of particles (moles). The molar mass is the mass of one mole of a substance, and it is numerically equal to its atomic or molecular mass but expressed in grams per mole (g/mol). For instance, the atomic mass of Carbon is 12 u, meaning its molar mass is exactly 12 g/mol Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.66. This allows us to bridge the gap between the microscopic world of atoms and the macroscopic world we can weigh on a scale.

Nature often prefers stability through partnership. In the case of Hydrogen, a single atom has only one electron in its K shell and needs another to become stable. Consequently, two hydrogen atoms share their electrons to form a diatomic molecule, H₂ Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.59. Because one hydrogen atom has an atomic mass of 1 u, a molecule of H₂ has a mass of 2 u (1 + 1 = 2). Therefore, the molar mass of H₂ is 2 g/mol. This is a critical distinction: if you are given 4 grams of hydrogen gas, you aren't dealing with 4 moles; you are dealing with 4 ÷ 2 = 2 moles of molecules.

This principle applies to many gases found in our atmosphere, such as Nitrogen (N₂) and Oxygen (O₂) Physical Geography by PMF IAS, Earth's Atmosphere, p.270. When calculating moles for these substances, you must always account for their diatomic nature. To find the number of moles (n) in any sample, we use the fundamental formula: n = Given Mass (m) / Molar Mass (M). Mastering this conversion is the 'secret sauce' for solving more complex thermal energy and enthalpy problems later in your preparation.

Sources: Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.66; Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.59; Physical Geography by PMF IAS, Earth's Atmosphere, p.270

3. Chemical Thermodynamics: Enthalpy and Heat of Reaction (intermediate)

In the study of thermal physics, we often focus on how energy moves between a system and its surroundings. When this energy exchange occurs during a chemical reaction at constant pressure (like in an open test tube in a lab), we refer to the heat involved as Enthalpy (H). Since it is difficult to measure the absolute enthalpy of a substance, scientists focus on the Enthalpy of Reaction (ΔH), which is the difference between the enthalpy of the products and the reactants. This value tells us whether a reaction will warm up its surroundings or cool them down.

Chemical reactions generally fall into two categories based on their heat exchange. In exothermic reactions, energy is released into the surroundings, often making the container feel hot. A classic example is respiration, which provides the energy needed for life Science, class X (NCERT 2025 ed.), Chemical Reactions and Equations, p.15. Conversely, endothermic reactions absorb energy from the surroundings. For instance, mixing barium hydroxide with ammonium chloride absorbs heat, causing the temperature of the reaction mixture to drop significantly Science, class X (NCERT 2025 ed.), Chemical Reactions and Equations, p.10. Understanding these shifts is crucial for calculating the energy efficiency of fuels or the thermal stability of compounds.

A vital principle to remember is that enthalpy is an extensive property, meaning its value depends on the amount of substance involved. If burning one mole of a fuel releases a specific amount of heat (known as the Molar Enthalpy of Combustion), then burning two moles will release exactly twice that amount. To calculate the total heat of a reaction for a specific sample, you must first determine the number of moles present using the formula: moles = mass / molar mass. This quantitative approach allows us to scale laboratory-scale observations up to industrial-scale energy calculations.

Feature	Exothermic Reaction	Endothermic Reaction
Energy Direction	Released to surroundings	Absorbed from surroundings
ΔH Sign	Negative (ΔH < 0)	Positive (ΔH > 0)
Example	Combustion, Respiration	Decomposition of Zinc Carbonate

Sources: Science, class X (NCERT 2025 ed.), Chemical Reactions and Equations, p.10; Science, class X (NCERT 2025 ed.), Chemical Reactions and Equations, p.15

4. Calorific Value of Fuels (intermediate)

At its heart, the Calorific Value (CV) of a fuel represents its energy efficiency. It is defined as the amount of heat energy released during the complete combustion of a unit quantity (mass or volume) of a fuel. In scientific terms, when we burn a substance, the chemical energy stored in its bonds is released as thermal energy. The higher the calorific value, the more efficient the fuel is, as a smaller amount of it is needed to produce a specific amount of heat. This value is typically measured in kilojoules per gram (kJ/g), kilocalories per kilogram (kcal/kg), or for gases, per cubic meter.

When comparing fuels, we often look at their hydrocarbon content. For example, in the case of coal, Anthracite is considered the premium variety because it has the highest hydrocarbon content and the highest calorific value. It burns with great heat, is practically smokeless, and leaves minimal ash, making it ideal for industrial steam-raising Certificate Physical and Human Geography, GC Leong, Fuel and Power, p.264. On the other hand, modern gaseous fuels like LPG (Liquefied Petroleum Gas) and CNG (Compressed Natural Gas) are preferred in households and transport because they have significantly higher calorific values than solid biomass like wood or charcoal Environment, Shankar IAS Academy, Environmental Pollution, p.69.

The concept of calorific value is also a critical metric in waste management and environmental policy. For instance, non-recyclable waste is not simply dumped if it has energy potential; if its calorific value is 1500 kcal/kg or more, it must be utilized for energy generation (Refuse Derived Fuel) rather than being disposed of in landfills Environment, Shankar IAS Academy, Environmental Pollution, p.88. This transition from simple combustion to calculated energy recovery is a hallmark of modern thermal physics applications.

Comparison of Fuel Efficiency

Fuel Category	Example	Relative Calorific Value
Solid (Low)	Fuel-wood / Peat	Low to Moderate
Solid (High)	Anthracite Coal	High (Best among coals)
Liquid	Petrol / Kerosene	Very High
Gaseous	Hydrogen / LPG	Highest

Sources: Certificate Physical and Human Geography, GC Leong, Fuel and Power, p.264; Environment, Shankar IAS Academy, Environmental Pollution, p.69; Environment, Shankar IAS Academy, Environmental Pollution, p.88

5. Hydrogen as a Clean Fuel & National Green Hydrogen Mission (exam-level)

To understand hydrogen as a fuel, we must first look at its chemistry. Hydrogen (H₂) is an energy carrier with the highest energy content by weight of any common fuel. When hydrogen undergoes combustion or is used in a fuel cell, it combines with oxygen (O₂) to release energy, with the only byproduct being water vapor (H₂O). This lack of carbon dioxide (CO₂) or particulate matter at the point of use makes it a revolutionary 'clean fuel' Shankar IAS Academy, Renewable Energy, p.296. From a thermal physics perspective, the enthalpy of combustion for hydrogen is quite high—burning just one mole of H₂ (about 2 grams) releases approximately 290 kJ of energy. Therefore, if you burn 4 grams of hydrogen (2 moles), you generate double that energy: 580 kJ. However, hydrogen doesn't exist freely on Earth; it must be extracted. The environmental 'cleanliness' of hydrogen depends entirely on its production method, often categorized by colors:

• Grey Hydrogen: Produced from fossil fuels (like natural gas) through Steam Methane Reformation (SMR). This is carbon-intensive.
• Blue Hydrogen: Produced from fossil fuels, but the resulting CO₂ is trapped using Carbon Capture and Storage (CCS) technology.
• Green Hydrogen: Produced through the electrolysis of water (splitting H₂O into H₂ and O₂) using electricity derived from renewable sources like solar or wind. This is the only truly carbon-neutral pathway Shankar IAS Academy, Renewable Energy, p.298.

India has launched the National Green Hydrogen Mission to position itself as a global hub for this technology. The mission aims to develop a production capacity of at least 5 Million Metric Tonnes (MMT) per annum by 2030, supported by an addition of 50 GW of renewable energy capacity Shankar IAS Academy, Renewable Energy, p.297. This is vital for decarbonizing 'hard-to-abate' sectors like heavy industry (steel and chemicals) and long-haul transport, where batteries are often too heavy or inefficient to be used Nitin Singhania, Sustainable Development and Climate Change, p.605.

Feature	Battery Electric Vehicles (BEV)	Hydrogen Fuel Cell Vehicles (FCEV)
Energy Source	Stored electricity in Lithium-ion cells	Compressed Hydrogen gas
Refueling Time	Long (minutes to hours)	Short (3-5 minutes, similar to petrol)
Best Use Case	Light passenger cars, short commutes	Heavy trucks, buses, ships, and industry

Sources: Shankar IAS Academy, Renewable Energy, p.296-298; Nitin Singhania, Sustainable Development and Climate Change, p.605

6. Stoichiometry: Relating Mass to Energy Release (exam-level)

In chemistry, we often talk about stoichiometry as the math behind chemical reactions—the recipe that tells us how much of each ingredient we need. However, stoichiometry isn't just about the mass of the substances; it also accounts for the energy exchanged. Every chemical bond contains energy, and when we break or form these bonds, energy is either absorbed or released. When a reaction releases heat into the surroundings, we call it an exothermic reaction Science, Chapter 1: Chemical Reactions and Equations, p. 15. The amount of energy released is directly proportional to the amount of substance consumed.

To relate mass to energy release, we must first pass through the mole. Think of the mole as the "universal translator" between the macro world of grams and the micro world of molecules. For example, if we are burning hydrogen gas (H₂), we need to know its molar mass. Since a hydrogen molecule consists of two hydrogen atoms (each with an atomic mass of approximately 1 u), the molar mass of H₂ is 2 g mol⁻¹. This means that 2 grams of hydrogen is exactly one mole of hydrogen. If the combustion of 1 mole of hydrogen is known to release a specific amount of heat, burning 4 grams (which is 2 moles) would release double that amount.

This scaling principle is vital for calculating the efficiency of fuels. In professional chemical equations, we often see the heat change listed at the end of the reaction, which is based on the molar ratios shown in the balanced chemical equation Science, Chapter 1: Chemical Reactions and Equations, p. 3. For instance, if the equation shows 1 mole of fuel reacting, the energy value provided applies to exactly that quantity. If you double the fuel, you double the energy. This linear relationship allows engineers and scientists to predict exactly how much heat a specific mass of fuel will produce in a furnace or an engine.

Sources: Science, Chapter 1: Chemical Reactions and Equations, p.15; Science, Chapter 1: Chemical Reactions and Equations, p.3

7. Solving the Original PYQ (exam-level)

This question effectively synthesizes your understanding of stoichiometry and thermochemistry. To solve it, you must apply the fundamental concept of molar mass—specifically that a hydrogen molecule (H₂) consists of two atoms—to convert the provided mass into its molar equivalent. As established in NCERT Class X Science, Chapter 4, chemical reactions involve specific proportions of matter; because the energy released is an extensive property, the enthalpy of combustion scales linearly with the number of moles consumed.

Walking through the reasoning, your first step is to determine how many moles are in 4g of hydrogen. Since the atomic mass of H is 1, the molar mass of the H₂ molecule is 2 g/mol. Dividing the given 4g by this molar mass reveals we are burning 2 moles of gas. Next, apply the unitary method: if burning 1 mole releases 290 kJ, then burning 2 moles must produce 2 × 290 kJ, leading us directly to (C) 580 kJ. This logical progression from mass to moles to energy is a core competency for UPSC science questions.

UPSC often uses distractors to punish common conceptual slips. Option (A) 145 kJ is a trap for students who might divide the energy by 2 instead of multiplying, while (B) 290 kJ is for those who fail to account for the change in mass entirely. Option (D) 1160 kJ is a classic error where one incorrectly multiplies the heat by the mass (4g) rather than the number of moles (2 mol). Always remember: energy values in these contexts are typically given "per mole," making the conversion to moles the most critical step in your calculation.