Q.Calculate the energy equivalent of 1 g of substance.
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Mass Energy Equivalence: From Intuition to the Formula
Imagine you have a lump of coal. You know you can burn it to get heat, and that heat can run a steam engine. The energy you get out seems to come from the chemical bonds in the coal. But what if I told you that the coal itself — just sitting there, not burning — already contains a staggering amount of energy locked inside its very mass? That is the core idea of mass-energy equivalence.
The Intuition: Mass is Frozen Energy
Think of mass as a kind of "frozen" or "stored" energy. When you burn coal, you are only releasing a tiny fraction of this stored energy — the energy in the chemical bonds. The rest of the mass remains as matter. But if you could somehow completely convert that lump of coal into pure energy, you would get an unimaginable amount — enough to power a city for years.
This is not a metaphor. Mass and energy are not two separate things that can be converted into each other like dollars and rupees. They are the same fundamental thing, just in different forms. Mass is a highly concentrated form of energy. Energy, when concentrated enough, behaves like mass.
The Precise Statement
The relationship is given by the most famous equation in physics:
E=mc2
Where:
- E is the energy equivalent of the mass (in joules, J)
- m is the mass (in kilograms, kg)
- c is the speed of light in vacuum (3×108 m/s)
The speed of light is a huge number. Squaring it makes it enormous. This is why a tiny amount of mass corresponds to a colossal amount of energy.
What This Equation Actually Means
The equation tells you exactly how much energy is "stored" inside any object with mass m. If you could annihilate that mass completely, you would get E joules of energy.
Example: A 1 kg mass (like a litre of water) contains:
E=1×(3×108)2=9×1016 J
That is 90 quadrillion joules — roughly the energy released by a 20-megaton nuclear bomb. This is not energy you can normally access; it is locked inside the nucleus of atoms.
Where Does This Show Up in Real Life?
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Nuclear Reactions: In nuclear fission (splitting atoms) or fusion (joining atoms), a tiny fraction of the mass of the nucleus is converted into energy. The mass of the products is slightly less than the mass of the reactants. The "missing" mass has become energy — exactly as E=mc2 predicts. This is how the Sun works and how nuclear power plants generate electricity.
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Particle Physics: When a particle and its antiparticle meet, they annihilate completely into pure energy (usually gamma rays). The energy produced equals mc2 for the two particles. …
Why this formula?
Why E=mc2 — The Reasoning Behind Mass-Energy Equivalence
The formula E=mc2 is not a random guess. Einstein arrived at it by thinking deeply about what happens to energy when you move an object. The core insight: if an object gains energy, it must behave as if it has gained mass.
The Starting Point: Relativistic Momentum
In special relativity, the momentum of a particle is not simply p=mv. Instead, it is:
p=1−v2/c2m0v
where m0 is the rest mass (mass measured when the object is at rest). This formula already tells us something strange: as speed approaches c, momentum shoots toward infinity — no object with mass can reach the speed of light.
The Energy-Momentum Relation
Einstein then asked: what is the correct expression for kinetic energy that matches this new momentum? In classical physics, kinetic energy is K=21mv2. But that formula fails at high speeds.
The relativistic kinetic energy turns out to be:
K=1−v2/c2m0c2−m0c2
This looks odd — why subtract m0c2? Because when v=0, the first term becomes m0c2, and we want K=0 at rest. So the subtraction gives zero kinetic energy when the object is stationary.
The term m0c2 appears naturally as a rest energy — energy that an object has simply because it has mass, even when completely at rest.
The Crucial Step: What Happens When You Add Energy?
Now consider a box that emits light (photons) in opposite directions. The light carries away energy. Classical physics says the box loses energy but its mass stays the same. Einstein showed this cannot be true.
The argument (simplified): if the box emits a pulse of light with energy E, the light carries momentum p=E/c. By conservation of momentum, the box recoils. But after the light is absorbed by the opposite wall, the box stops. The net effect: the box has moved slightly. Its center of mass has shifted — unless the energy carried by the light also carried mass.
For the center of mass of the entire system (box + light) to remain stationary, the light must behave as if it has an effective mass m=E/c2. Therefore, energy itself has inertia.
The Full Formula
The total energy of any object — moving or at rest — is:
E=1−v2/c2m0c2
For an object at rest (v=0), this reduces to:
E=m0c2
For a moving object, the total energy is the sum of rest energy and kinetic energy:
E=m0c2+K
E=mc2
where m is the relativistic mass m=1−v2/c2m0, or equivalently:
E2=(pc)2+(m0c2)2
Why It's Not Just a "Conversion" …
The key idea is mass-energy equivalence: mass can be converted into energy according to E=mc2.
Step 1: Convert mass to kilograms.
1 g=1×10−3 kg.
Step 2: Use c=3×108 m/s.
E=(1×10−3)×(3×108)2.
Step 3: Compute. …
The energy equivalent of mass is given by Einstein's mass-energy equivalence relation E=mc2. For 1 g of substance, this yields 9×1013 J.
The core idea here is Einstein's mass-energy equivalence — one of the most profound results in physics. It tells us that mass is not just a measure of inertia or gravitational pull; it is a concentrated form of energy. The relation E=mc2 means that even a tiny amount of mass contains an enormous amount of energy because c2 is a huge number (c=3×108 m/s).
Why does this matter? In nuclear reactions (fission, fusion) or particle-antiparticle annihilation, a small fraction of mass converts into energy. The problem asks for the complete conversion of 1 g — that is, all its mass turns into energy. This is the maximum possible energy you could extract from that mass.
Let's work through the calculation step by step.
- Write down the formula The energy equivalent of mass m is:
E=mc2
where c=3×108 m/s is the speed of light in vacuum.
- Convert mass to SI units The mass is given as 1 g. In the SI system, the standard unit of mass is the kilogram. So:
m=1 g=1×10−3 kg
- Plug in the numbers
E=(1×10−3 kg)×(3×108 m/s)2
First square the speed of light:
c2=(3×108)2=9×1016 m2/s2
Then multiply:
E=10−3×9×1016=9×1013 J
- Interpret the result …
Method: Einstein's Mass-Energy Equivalence (Direct Formula)
The core idea is that mass and energy are two forms of the same thing, linked by the speed of light squared. For any amount of mass m, the equivalent energy E is given by:
E=mc2
where c=3×108 m/s (the speed of light in vacuum).
Steps
Step 1: Convert mass to SI units.
The formula uses mass in kilograms.
1 g=1×10−3 kg.
Step 2: Substitute into E=mc2.
E=(10−3 kg)×(3×108 m/s)2
Step 3: Square the speed of light first.
(3×108)2=9×1016 m2/s2
Step 4: Multiply by mass. …
Common Mistakes with Mass–Energy Equivalence
The question "Calculate the energy equivalent of 1 g of substance" looks deceptively simple — and that's exactly where students slip up. Here are the most frequent errors and how to avoid each.
Mistake 1: Forgetting to convert grams to kilograms
E=mc2 requires mass in kilograms, not grams. Using m=1 instead of m=10−3 gives an answer that is 1000 times too large.
How to avoid: Before you write anything, convert every mass to kg. For 1 g, write m=1×10−3 kg explicitly. Make it a habit: mass in kg first, then plug into E=mc2.
Mistake 2: Using the wrong value of c
Some students use c=3×108 m/s (which is correct) but then square it incorrectly — either forgetting to square or squaring only the coefficient and not the power of ten.
How to avoid: Write c2 step by step:
c2=(3×108)2=9×1016 m2/s2
Do not skip this step. A common slip is writing 3×1016 instead of 9×1016.
A frequent error: c2=3×1016 (wrong). The square of 3×108 is 9×1016, not 3×1016.
Mistake 3: Getting the units wrong
The energy comes out in joules (J) when mass is in kg and c in m/s. Some students report the answer in eV or forget to state units entirely.
How to avoid: Always write the unit after your numerical answer. For this problem, the answer is in joules. If the question asks for energy equivalent, the standard SI unit is the joule unless specified otherwise.
Mistake 4: Misplacing the decimal in the final calculation
Even with correct conversion and squaring, students often mess up the powers of ten when multiplying 10−3 by 9×1016.
How to avoid: Do the powers of ten separately:
10−3×1016=1013
Then multiply by 9: 9×1013 J. Write this as a separate line — don't try to do it all in your head.
Mistake 5: Confusing mass defect with mass of a substance …
- KEAM 2026Set eng-2026-04194 marksMCQQ.The energy released by 2.35 g of 235U by fission in a nuclear reactor (in MeV) is (Average energy released per fission is 200 MeV) (A) 1.2×1024 (B) 0.4×1024 (C) 0.6×1024 (D) 0.8×1024 (E) 2.4×1024
›Reveal solutionSolution
2.35g of U-235 is 0.01mol (6.02×1021 atoms); at 200MeV each, the total is ≈1.2×1024MeV.
Number of moles:
n=2352.35=0.01mol.
Number of nuclei:
N=0.01×6.022×1023=6.022×1021.
Each fission releases 200MeV: …
- KEAM 2025Set pha-2025-0429F4 marksMCQQ.Energy equivalent of mass 0.5 kg is (A) 9×1016J (B) 3×1016J (C) 2.5×1016J (D) 6×1016J (E) 4.5×1016J
›Reveal solutionSolution
Apply Einstein's mass-energy relation E=mc2 with m=0.5kg and c=3×108ms−1.
Einstein's mass-energy equivalence states that a mass m is equivalent to an energy
E=mc2.
Substituting m=0.5kg and c=3×108ms−1: …
- KEAM 2024Set eng-2024-06074 marksMCQQ.In a nuclear fusion process, the masses of the fusing nuclei are MA and MB. Then the mass of the product nucleus MC is related to MA and MB as (A) MC<MA+MB (B) MC>MA+MB (C) MC=∣MA−MB∣ (D) MC=MA+MB (E) MC=2MA+MB
›Reveal solutionSolution
Fusion releases energy from a mass defect, so the product mass is less than the sum: MC<MA+MB. …
- KEAM 2024Set eng-2024-06084 marksMCQQ.The energy equivalent of 1 g of a substance in joules is (A) 9×1013 (B) 4.5×1013 (C) 1×1013 (D) 0.5×1013 (E) 2.25×1013
›Reveal solutionSolution
Mass–energy equivalence gives E=mc2; for 1 g the energy is 9×1013 J.
Using Einstein's relation with m=1 g=10−3 kg and c=3×108 m s−1: …
- KEAM 2022Set eng-2022-P1-A14 marksMCQQ.The energy equivalent of 5 g of a substance is (A) 4.5×1012 J (B) 9×1012 J (C) 4.5×1014 J (D) 4.5×1016 J (E) 9×1016 J
›Reveal solutionSolution
Using E = mc², 5 g of matter is equivalent to 4.5 × 10¹⁴ J.
Concept and Intuition
Mass and energy are equivalent through Einstein's relation E = mc², where c = 3 × 10⁸ m/s. Even a tiny mass corresponds to an enormous energy because c² is very large.
Step-by-Step Solution
- Convert mass: m = 5 g = 5 × 10⁻³ kg.
- Apply E = mc² = (5 × 10⁻³)(3 × 10⁸)².
- c² = 9 × 10¹⁶ m²/s². …
- KEAM 2021Set eng-2021-P1-A14 marksMCQQ.1018 fissions per second is required for producing power of 300 MW in a nuclear power station. To increase the power output to 360 MW the additional number of fissions required per second is (A) 2×1018 (B) 5×1018 (C) 5×1017 (D) 6×1017 (E) 2×1017
›Reveal solutionSolution
An extra 2×1017 fissions per second are needed.
Concept and Intuition
Each fission releases a fixed energy, so the number of fissions per second is directly proportional to the power output.
Step-by-Step Solution
- Rate constant: 1018 fissions/s → 300 MW.
- Additional power =360−300=60MW.
- Additional fissions/s =1018×30060=0.2×1018=2×1017. …
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