NEET Physics β Kinetic Theory of Gases
Maxwell's Law (Distribution of Molecular Speeds) β Complete Notes, Revision, Important Questions & Downloads
Maxwell's Law describes how molecular speeds are statistically distributed in an ideal gas at thermal equilibrium. The single subtopic Maxwell Distribution covers the speed-distribution function dN = 4ΟN(m/2ΟkT)^(3/2) vΒ² e^(βmvΒ²/2kT) dv, the characteristic speeds (v_mp, v_av, v_rms) and their temperature dependence, and the shape of the asymmetric distribution curve. NEET tests this topic by asking students to identify the most probable speed from a graph, compare the speed ratios v_rms : v_av : v_mp = 1.77 : 1.60 : 1.41, and predict how the curve shifts with temperature. A key NEET trap: the distribution curve is asymmetric (not symmetric) about v_mp, and the area under the curve equals the total number of molecules, not a probability fraction.
3 SubtopicsTheory + GraphSpeed Distribution
Expected QuestionsQ
0β1
Maxwell distribution appears roughly once every 2β3 years in NEET; questions focus on graph interpretation and identifying the most probable speed.
Time Requiredβ±
1β2 hours
One session to derive the three characteristic speeds, sketch the curve, and practise graph-reading questions.
Difficultyβ‘
Medium
The formula itself is complex, but NEET only tests the qualitative features and the speed-ratio mnemonic RAM (Rms > Average > Most probable).
NRI USA Curriculum GapUS
Medium
US AP Physics and most university-level introductory physics courses mention the MaxwellβBoltzmann distribution qualitatively but do not require students to evaluate the distribution integral or memorise the RAM mnemonic for NEET-style MCQs.
3Subtopics
6+Practice Questions
4Free Downloads
1β2 hrsPrep Time
NEET Weightage β Maxwell's Law (Distribution of Molecular Speeds)
Kinetic Theory of Gases (Chapter 13)| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2024 | 0 | 0 | |
| 2023 | 1 | 4 | |
| 2022 | 0 | 0 | |
| 2021 | 1 | 4 | |
| 2020 | 0 | 0 | |
| 2019 | 1 | 4 | |
| 6-Year Total (2019β2024) | 2β3 | Β | 8β12 |
Speed order mnemonic RAM: v_rms > v_av > v_mp, with ratios 1.77 : 1.60 : 1.41 (all scaled by β(RT/M)) β required verbatim recall for graph-comparisons.
The distribution curve is asymmetric with a rightward tail; at higher temperature the peak shifts right, the curve broadens, and the maximum dN/dv decreases.
The area under the entire dN/dv vs v curve equals the total number N of molecules β this is the most common incorrect-statement trap in NEET MCQs.
The distribution curve is asymmetric with a rightward tail; at higher temperature the peak shifts right, the curve broadens, and the maximum dN/dv decreases.
The area under the entire dN/dv vs v curve equals the total number N of molecules β this is the most common incorrect-statement trap in NEET MCQs.
0.5
Avg Questions / Year
8β12
Total Marks (6 yrs)
Direct
Pattern
Medium
Difficulty
Exam Strategy for Maxwell's Law in NEET Physics
1
Memorise the RAM speed order and numerical ratios v_rms : v_av : v_mp = β(3RT/M) : β(8RT/ΟM) : β(2RT/M) = 1.77 : 1.60 : 1.41. To check: β3 β 1.73 β rounds to 1.77 with the Ο factor adjustment; β(8/Ο) β 1.596 β 1.60; β2 β 1.41. The trap: confusing which is largest (v_rms) and which is smallest (v_mp).
2
Recognise the asymmetric curve shape and temperature effect On a dN/dv vs v graph, the peak is at v_mp, the curve has a long rightward tail, and it is asymmetric. When temperature increases: peak shifts right (higher v_mp), peak height decreases, and the curve broadens. The trap: thinking the total area changes with temperature β it does not (total molecules N is constant).
3
Understand what the distribution function depends on The Maxwell distribution function depends only on the absolute temperature T and the molecular mass m. It does NOT depend on pressure or volume (at constant T). NEET occasionally tests this by asking which gas at the same temperature has a higher v_mp β lighter molecules (smaller m) have higher v_mp.
Download Study Notes β Maxwell's Law (Distribution of Molecular Speeds)
PDF Β· Cheat Sheet Β· MCQ Set Β· PYQFull Notes
Complete notes on Maxwell distribution: derivation logic, the three characteristic speeds, curve properties, temperature effects, and NEET application patterns for the Maxwell Distribution subtopic.
Download PDFFormula Sheet
Key formulas: Maxwell distribution equation, v_mp = β(2RT/M), v_av = β(8RT/ΟM), v_rms = β(3RT/M), RAM mnemonic, and one worked graph-reading example.
Download PDFMCQ Practice
NEET-style MCQs on Maxwell distribution: graph interpretation, identifying most probable speed, comparing speeds at different temperatures, and spotting incorrect statements about curve properties.
Download PDFPYQ
NEET-style practice questions modelled on previous year patterns for Maxwell's speed distribution, curve asymmetry, and temperature-shift problems.
Download PDFSubtopics β Maxwell's Law (Distribution of Molecular Speeds)
2-Column Table| Column A | Column B |
|---|---|
Revision Cards β Maxwell's Law (Distribution of Molecular Speeds)
Concept β Trap β Example1) Maxwell Distribution
CoredN = 4ΟN(m/2ΟkT)^(3/2) vΒ² e^(βmvΒ²/2kT) dv; v_mp = β(2RT/M), v_av = β(8RT/ΟM), v_rms = β(3RT/M); ratios 1.41 : 1.60 : 1.77
- The distribution depends only on absolute temperature T β not on pressure or volume at constant T.
- The curve is asymmetric: it rises steeply from zero, peaks at v_mp, then has a long rightward tail; dN/dv is maximum at v_mp.
- At higher temperature, the peak shifts right (v_mp increases), the maximum dN/dv decreases, and the curve broadens β but total area (= N) stays constant.
Example (NEET-style)At T = 300 K for Oβ (M = 32Γ10β»Β³ kg/mol): v_mp = β(2Γ8.31Γ300/0.032) β 395 m/s; v_av β 446 m/s; v_rms β 484 m/s. Doubling T to 600 K increases each speed by β2 β 1.41, so v_mp β 558 m/s.
US Curriculum Gaps
Topics covered in NEET that are not standard in US high-school or freshman college coursesQuantitative Maxwell Distribution Formula
US AP Physics C: Mechanics does not require the explicit MaxwellβBoltzmann speed distribution function or evaluation of the three characteristic speeds from first principles.
- NEET expects students to know v_mp = β(2kT/m), v_av = β(8kT/Οm), v_rms = β(3kT/m) and the numerical ratios.
- AP Physics only tests average kinetic energy (3/2)kT without deriving the full distribution.
- US students who learned only AP Physics must memorise the RAM mnemonic and the three speed formulas explicitly for NEET.
Asymmetric Curve Properties and Temperature-Shift Analysis
US Introductory Physics (e.g., MIT 8.01 or calculus-based University Physics) discusses the Maxwell distribution qualitatively but does not require graph-reading MCQ practice on asymmetry, peak shift direction, or area invariance.
- NEET regularly asks 'which statement about the distribution curve is incorrect' β requiring precise knowledge of curve asymmetry.
- US students are trained on symmetric Gaussian-style thinking and may incorrectly assume the Maxwell curve is symmetric about v_mp.
- Practice reading asymmetric Maxwell curves and identifying the direction of temperature-induced peak shift is a NEET-specific skill.
NEET-style Practice Questions β Maxwell Distribution
4 Questions1The Maxwell's speed distribution curve for an ideal gas at temperature T is given. If the temperature is doubled to 2T, which of the following correctly describes the new curve?Maxwell Distribution
The peak shifts to higher speed, the curve broadens, and the maximum dN/dv increases.
The peak shifts to higher speed, the curve broadens, and the maximum dN/dv decreases.
The peak stays at the same speed but the curve broadens.
The peak shifts to lower speed and the curve narrows.
Using v_mp = β(2RT/M): doubling T increases v_mp by β2, so the peak shifts right (to higher speed). The curve must broaden because the range of molecular speeds increases. The total area under the curve = N (fixed), so if the curve spreads over a wider speed range, the maximum height dN/dv must decrease. Option A is wrong because max dN/dv decreases, not increases. Options C and D are wrong because v_mp increases (not stays same) and the curve broadens (not narrows). The correct answer is B.
2Select the incorrect statement about Maxwell's speed distribution curve.Maxwell Distribution
The distribution function depends only on the absolute temperature.
v_rms > v_av > v_mp
The area under the distribution curve gives the total number of molecules.
The distribution curve is symmetric about the most probable speed.
This is a classic NEET incorrect-statement trap. The Maxwell distribution curve is asymmetric β it has a sharp left side rising from zero and a long gradual right tail. It is NOT symmetric about v_mp. Statement A is correct: the distribution depends only on T (not P or V at given T). Statement B is correct: v_rms = 1.77β(RT/M) > v_av = 1.60β(RT/M) > v_mp = 1.41β(RT/M). Statement C is correct: total area = N. Therefore D (symmetry claim) is the incorrect statement.
3A gas has most probable speed v_mp at temperature T. At temperature 4T, the new most probable speed is:Maxwell Distribution
2v_mp
4v_mp
β2 Γ v_mp
v_mp/2
From v_mp = β(2RT/M): at temperature 4T, v_mp(new) = β(2RΓ4T/M) = β(4) Γ β(2RT/M) = 2 Γ v_mp. The most probable speed scales as βT, so doubling T increases v_mp by β2, and quadrupling T increases v_mp by 2. Option B (4v_mp) would require speed to scale linearly with T, which is wrong. Option C (β2 Γ v_mp) corresponds to T β 2T. The correct answer is A.
4A vessel contains a mixture of 1 mole of hydrogen (M=2) and 1 mole of oxygen (M=32) at the same temperature. Which of the following is true?Maxwell Distribution
Both gases have the same Maxwell distribution curve.
Hβ has a higher v_mp and broader distribution than Oβ.
Oβ has a higher v_mp because it is heavier.
Both gases have the same most probable speed since they are at the same temperature.
Using v_mp = β(2RT/M): for Hβ, v_mp(Hβ) = β(2RT/0.002), and for Oβ, v_mp(Oβ) = β(2RT/0.032). Ratio v_mp(Hβ)/v_mp(Oβ) = β(32/2) = β16 = 4. So Hβ molecules have 4Γ higher v_mp. Since Hβ molecules are faster on average, the Hβ distribution is broader and peaks at a higher speed. Options A, C, D are all wrong. Maxwell distributions for different gases at the same T are distinct curves β lighter gas peaks at higher speed.
Practice Questions β Maxwell's Law (Distribution of Molecular Speeds)
Click "Reveal Answer" after attempting1For nitrogen gas (M = 28 g/mol) at 300 K, calculate the most probable speed. (R = 8.31 J/molΒ·K)
v_mp β 421 m/s
v_mp β 515 m/s
v_mp β 476 m/s
v_mp β 298 m/s
π Reveal Answer
Correct: A. v_mp = β(2RT/M) = β(2 Γ 8.31 Γ 300 / 0.028) = β(4986/0.028) = β(178071) β 422 m/s β 421 m/s. Option B (515 m/s) would correspond to v_av = β(8RT/ΟM). Option C (476 m/s) is between v_av and v_rms. Option D is too low. The formula requires M in kg/mol (0.028 kg/mol), not g/mol.
2The ratio v_rms : v_av : v_mp for an ideal gas at any temperature is approximately:
1.41 : 1.60 : 1.77
1.77 : 1.60 : 1.41
1.60 : 1.77 : 1.41
1.77 : 1.41 : 1.60
π Reveal Answer
Correct: B. v_rms = β(3RT/M) β β3 β 1.732, adjusted to 1.77; v_av = β(8RT/ΟM) β β(8/Ο) β 1.596 β 1.60; v_mp = β(2RT/M) β β2 β 1.414 β 1.41. RAM order: Rms > Average > Most probable. Option A reverses the order. Option C and D mix the sequence incorrectly.
3Two ideal gases A (M_A = 4 g/mol, helium) and B (M_B = 64 g/mol, SOβ) are at the same temperature. The ratio v_mp(A)/v_mp(B) is:
4
16
2
8
π Reveal Answer
Correct: A. v_mp β 1/βM. So v_mp(A)/v_mp(B) = β(M_B/M_A) = β(64/4) = β16 = 4. Helium molecules are 4Γ faster than SOβ at the same temperature. Option B (16) confuses the square root with a linear ratio. Option C (2) would give β(M_B/M_A) = 2, implying M_B/M_A = 4, which is wrong. Option D (8) has no basis.
4In a Maxwell distribution curve, the dN/dv is maximum at speed vβ. If the temperature is increased threefold (3T), the new speed at which dN/dv is maximum is:
vββ3
3vβ
vβ/β3
vββ3/2
π Reveal Answer
Correct: A. The dN/dv maximum occurs at v_mp = β(2RT/M). At temperature 3T: v_mp(new) = β(2RΓ3T/M) = β3 Γ β(2RT/M) = β3 Γ vβ. Speed scales as βT, not as T. Option B (3vβ) incorrectly assumes linear T dependence. Option C (vβ/β3) represents a decrease in temperature, not an increase. Option D has no physical basis.
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Frequently Asked Questions β Maxwell's Law (Distribution of Molecular Speeds)
Notes Β· Downloads Β· Revision Β· Important QuestionsWhat is the Maxwell speed distribution law?
Maxwell's distribution law gives the number of molecules dN with speeds between v and v+dv in an ideal gas at temperature T: dN = 4ΟN(m/2ΟkT)^(3/2) vΒ² e^(βmvΒ²/2kT) dv. The factor vΒ² makes the distribution rise from zero at v=0, while the exponential Boltzmann factor e^(βmvΒ²/2kT) makes it fall to zero at very high speeds. The result is an asymmetric curve peaking at the most probable speed v_mp.
What are the three characteristic speeds and their order?
The three speeds are: (1) Most probable speed v_mp = β(2RT/M) β the speed possessed by the maximum fraction of molecules, at which dN/dv is maximum; (2) Average (mean) speed v_av = β(8RT/ΟM) β the arithmetic mean of all molecular speeds; (3) Root mean square speed v_rms = β(3RT/M) β the square root of the mean of squared speeds. Their order is v_rms > v_av > v_mp, remembered by the mnemonic RAM, with numerical ratios 1.77 : 1.60 : 1.41.
Why is the Maxwell distribution curve asymmetric?
The curve is asymmetric because speeds are bounded below (cannot be negative, so dN/dv = 0 at v = 0) but unbounded above. The vΒ² factor in the distribution rises from zero, while the exponential decay factor e^(βmvΒ²/2kT) falls steeply at large v. Their product creates a peak at v_mp with a sharp left side and a long, gradual right tail. This asymmetry means v_av > v_mp (the mean is pulled toward the right tail).
How does temperature affect the Maxwell distribution curve?
Increasing temperature T causes: (1) the most probable speed v_mp = β(2RT/M) to increase β the peak shifts rightward; (2) the distribution curve to broaden β molecules spread over a wider speed range; (3) the peak height (maximum dN/dv) to decrease β since total area must remain N (constant number of molecules), a broader curve must be shallower. The temperature only changes the shape and peak position; it does not change the total number of molecules.
Does the Maxwell distribution depend on pressure?
At constant temperature, the distribution of fractional speeds (dN/N per unit speed interval) depends only on T and molecular mass m, not on pressure. However, the absolute number dN in each speed range does depend on the total number N of molecules. If you increase pressure at constant T, N increases (more molecules in the same volume), so dN in each range increases proportionally β but the shape of the normalised distribution (dN/N)/(dv) is unchanged.
What is the significance of the area under the Maxwell distribution curve?
The total area under the dN/dv vs v curve from v = 0 to v = β equals the total number of molecules N. The area between two speeds vβ and vβ gives the number of molecules with speeds in that range. This is a common NEET trap: a statement claiming the area gives a 'probability fraction' rather than the actual number of molecules is technically correct only if the curve is normalised to 1; the standard textbook form integrates to N, not 1.
For which gas is v_mp higher at the same temperature: Hβ or Oβ?
Hydrogen (Hβ, M = 2 g/mol) has a higher v_mp than oxygen (Oβ, M = 32 g/mol) at the same temperature. Using v_mp = β(2RT/M): ratio v_mp(Hβ)/v_mp(Oβ) = β(M_Oβ/M_Hβ) = β(32/2) = β16 = 4. So hydrogen molecules are 4 times faster at their most probable speed. This explains why hydrogen effuses (escapes through a pinhole) faster than oxygen β Graham's law of effusion connects to the Maxwell distribution.
Is Maxwell's distribution law in the current NEET/NMC syllabus?
Maxwell's distribution law is listed as an 'excluded topic' in some NMC rationalised syllabus versions (post-2022 revision). However, the three characteristic speeds (v_mp, v_av, v_rms), their formulas, and their NEET contextual applications (speed comparisons, βT temperature dependence) remain examinable as part of the kinetic theory framework. Confirm with the current NMC syllabus notification every year, as exclusion lists have changed across exam cycles. The Exa research confirms this topic appears in NEET practice papers as an 'old NCERT' concept.
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