What the BPHCT 135 Solved Question Paper June 2026 covers
BPHCT 135 Solved Question Paper June 2026 is our fully worked answer file for BPHCT 135 Thermal Physics and Statistical Mechanics, the IGNOU core physics course of the BSCG and BSCM programmes that runs from the kinetic theory of gases to quantum statistics.
The June 2026 paper carries 50 marks over 2 hours. It has five compulsory questions with internal choice: eight 2-mark short parts, three groups of 5-mark parts and a 10-mark choice between Planck’s law and the Boltzmann entropy relation. Our file answers all 19 printed items, so you can revise every choice, not just the ones you plan to pick.
This paper is derivation-heavy. The most probable speed, the survival equation of free paths, Mayer’s formula and the Carnot efficiency all reward a clean chain of steps far more than long theory. That is why every answer shows the starting expression, each step and a clearly marked final result.
Want the unsolved papers from earlier sessions for timed practice? Our IGNOU Previous Year Question Paper section lists them. This page is the solved version of the IGNOU BPHCT 135 Question Paper for the June 2026 session only.
BPHCT 135 sample answers: two June 2026 derivations solved in full
Below are two complete answers from the June 2026 paper, Q2(a) on the most probable speed and Q4(a) on Carnot efficiency, written the way an IGNOU examiner marks a 5-mark derivation.
Q2 (a) Show that the most probable speed of a Maxwellian gas molecule is given by: vp = √(2kBT/m) [5 marks]
Q2 (a) Show that the most probable speed of a Maxwellian gas molecule is given by: vp = √(2kBT/m) [5 marks]
Answer
Answer: According to Maxwell’s law, out of N molecules of mass m at temperature T, the number whose speeds lie between v and v + dv is:
dNv = 4πN (m / 2πkBT)3/2 v2 exp(−mv2 / 2kBT) dv
The most probable speed vp is the speed at which this distribution is a maximum, that is, the speed held by the largest fraction of molecules. Put α = m/2kBT and A = 4πN(α/π)3/2. The distribution function then becomes f(v) = A v2 e−αv².
Condition for maximum: at v = vp the slope df/dv must be zero.
df/dv = A[2v e−αv² − 2αv3 e−αv²] = 2A v e−αv²(1 − αv2) = 0
A is not zero, v is not zero at the peak and the exponential never vanishes, so the bracket must be zero: 1 − αvp2 = 0. This gives vp2 = 1/α = 2kBT/m, and therefore:
vp = √(2kBT / m) = √(2RT / M)
Checking it is a maximum: the second derivative is d²f/dv² = 2A e−αv²(1 − 5αv2 + 2α2v4). At αv2 = 1 this equals 2A e−1(1 − 5 + 2) = −4A/e, which is negative, so the turning point is indeed a maximum. Here R is the gas constant and M the molar mass.
Comparison with other speeds: the average speed is v̄ = √(8kBT/πm) and the rms speed is vrms = √(3kBT/m). So vp : v̄ : vrms = 1 : 1.128 : 1.225. The most probable speed is always the smallest of the three because the Maxwell curve has a long tail on the high-speed side.
Numerical check: for nitrogen at 300 K, m = 28 × 1.66 × 10−27 kg = 4.65 × 10−26 kg. Then vp = √(2 × 1.38 × 10−23 × 300 / 4.65 × 10−26) ≈ 422 m s−1.
Conclusion: since vp ∝ √T, the peak of the Maxwell curve shifts to higher speeds and flattens as temperature rises, while the area under the curve stays equal to N. A lighter gas has a higher vp at the same temperature because vp ∝ 1/√m.
Q4 (a) Show that efficiency of Carnot engine is given by: η = 1 − T2/T1 [5 marks]
Q4 (a) Show that efficiency of Carnot engine is given by: η = 1 − T2/T1 [5 marks]
Answer
Answer: A Carnot engine works between a source at temperature T1 and a sink at T2 (T1 > T2). The working substance is n moles of an ideal gas in a cylinder with a frictionless, perfectly fitting piston. On the P-V indicator diagram the cycle is A → B → C → D → A, made of two isotherms and two steeper adiabats, and every step is reversible.
Step 1, isothermal expansion A → B (V1 to V2 at T1): internal energy of an ideal gas depends only on T, so ΔU = 0 and all heat taken from the source becomes work:
Q1 = W1 = nRT1 ln(V2/V1)
Step 2, adiabatic expansion B → C (V2 to V3): no heat flows, the gas cools from T1 to T2 and does work W2 = nCV(T1 − T2) at the cost of its internal energy.
Step 3, isothermal compression C → D (V3 to V4 at T2): work is done on the gas and an equal amount of heat is rejected to the sink:
Q2 = W3 = nRT2 ln(V3/V4)
Step 4, adiabatic compression D → A: the gas is compressed back to V1 and warms from T2 to T1. Work done on it is W4 = nCV(T1 − T2), which exactly cancels W2. So the net work in one cycle is W = Q1 − Q2.
Linking the volumes: for a reversible adiabatic change TVγ−1 is constant, so for B → C and D → A:
T1V2γ−1 = T2V3γ−1 and T1V1γ−1 = T2V4γ−1
Dividing the first by the second gives (V2/V1)γ−1 = (V3/V4)γ−1, so V2/V1 = V3/V4. The two logarithms are therefore equal, and Q2/Q1 = T2/T1.
Efficiency: efficiency is useful work divided by heat taken from the source:
η = W/Q1 = (Q1 − Q2)/Q1 = 1 − Q2/Q1 = 1 − T2/T1
Key points for full marks: η depends only on the two reservoir temperatures, not on the working substance. It can equal 1 only if T2 = 0 K, which the third law rules out. For example, an engine working between 500 K and 300 K has η = 1 − 300/500 = 0.40, that is 40%. Always draw the labelled P-V diagram first; it carries marks of its own.
Questions answered only in the full file
The other 17 items from the June 2026 paper are listed below exactly as printed. Their answers are not shown on this page; they sit only in the full solved file.
- Q1(i) Plot van der Waals’ equation for a real gas on P-V diagram. [2 marks]
- Q1(ii) Write van der Waals’ assumptions for a real gas. [2 marks]
- Q1(iii) Name the intensive and extensive variables of a surface film. [2 marks]
- Q1(iv) A bottle of water at room temperature is cooled by putting it in a refrigerator. What type of system do the bottle and water form? [2 marks]
- Q1(v) Plot Carnot cycle on T-S diagram. [2 marks]
- Q1(vi) What is Joule-Thomson effect? [2 marks]
- Q1(vii) Draw the Fermi function versus energy at T = 0 K and T > 0 K. [2 marks]
- Q1(viii) Write the difference between Bosons and Fermions. [2 marks]
- Q2(b) Define mean free path. Obtain an expression for survival equation of free paths. [5 marks]
- Q2(c) If coefficient of viscosity of oxygen is 19.6 × 10−6 kg m−2 s−1, at 15°C and v̄ = 436 m s−1, calculate the radius of an oxygen molecule, if n = 2.7 × 1025 m−3. Assume that the gas obeys Maxwell’s law of distribution of speeds. [5 marks]
- Q3(a) Show that for an ideal gas the isothermal compressibility (βT) and volume expansion coefficient (α) are given by: βT = 1/p and α = 1/T [5 marks]
- Q3(b) Distinguish between adiabatic and isothermal processes. Calculate the work done for an ideal gas expanding adiabatically from volume V1 to V2. [5 marks]
- Q3(c) State first law of thermodynamics. Prove Mayer’s formula for an ideal gas and discuss its importance in physics. [5 marks]
- Q4(b) State Kelvin-Planck and Clausius’ statements of 2nd law of thermodynamics. Establish their equivalence. [5 marks]
- Q4(c) Calculate the entropy of mixing per mole of a mixture of two gases. [5 marks]
- Q5(a) By treating electromagnetic radiation as a system of indistinguishable particles derive Planck’s law. [10 marks]
- Q5(b) Obtain the relation: S = kB ln W where S is entropy and W is thermodynamic probability. [10 marks]
Thermal Physics and Statistical Mechanics: BPHCT 135 block and unit map
BPHCT 135 is taught through 4 blocks and 14 units on eGyanKosh, moving from the kinetic theory of gases through the laws of thermodynamics to classical and quantum statistics.
| Block | Unit | Key topics | June 2026 items |
|---|---|---|---|
| Block 1: Kinetic Theory of Gases | Unit 1 Ideal and Real Gases | van der Waals’ equation and assumptions; Andrews’ curves for CO2; critical constants | Q1(i), Q1(ii) |
| Block 1 | Unit 2 Molecular Velocity Distribution Function | Maxwell’s distribution of molecular speeds; most probable, average and rms speeds; mean energy per degree of freedom | Q2(a) |
| Block 1 | Unit 3 Mean Free Path and Transport Phenomena | Mean free path and the survival equation; viscosity, thermal conduction and diffusion in a Maxwellian gas | Q2(b), Q2(c) |
| Block 1 | Unit 4 Brownian Motion | Observed characteristics; Einstein’s displacement result and Perrin’s check; sedimentation | None this session |
| Block 2: The Zeroth and the First Laws of Thermodynamics | Unit 5 Thermodynamic Description of a System | Open, closed and isolated systems; intensive and extensive variables of films and dielectrics; βT and α of a pVT system | Q1(iii), Q1(iv), Q3(a) |
| Block 2 | Unit 6 The Zeroth Law | Thermal equilibrium; the idea of temperature; empirical temperature scales | None this session |
| Block 2 | Unit 7 The First Law and its Applications | Differential form of the first law; Cp − CV (Mayer’s formula); isothermal and adiabatic work, Laplace’s speed of sound | Q3(b), Q3(c) |
| Block 3: Second and Third Laws of Thermodynamics | Unit 8 Carnot Cycle | Carnot indicator diagram; efficiency η = 1 − T2/T1; the cycle on the T-S plane | Q1(v), Q4(a) |
| Block 3 | Unit 9 Entropy and the Laws of Thermodynamics | Kelvin-Planck and Clausius statements; entropy change and entropy of mixing; third law | Q4(b), Q4(c) |
| Block 3 | Unit 10 The Thermodynamic Potentials | Enthalpy, Helmholtz and Gibbs functions; Maxwell’s relations and TdS equations; Clausius-Clapeyron equation | Q1(vi) |
| Block 3 | Unit 11 Theory of Radiation | Spectral energy density of a blackbody; Stefan-Boltzmann law; Wien’s and Rayleigh-Jeans laws | None this session |
| Block 4: Statistical Mechanics | Unit 12 Basic Concepts of Statistical Mechanics | Microstates and macrostates; thermodynamic probability W; Boltzmann relation S = kB ln W | Q5(b) |
| Block 4 | Unit 13 Classical Statistics | Maxwell-Boltzmann distribution; single-particle partition function of a monoatomic gas; heat capacities from Z | None this session |
| Block 4 | Unit 14 Quantum Statistics | Bose-Einstein and Fermi-Dirac distributions; Fermi function and Fermi energy; Planck’s law from photon statistics | Q1(vii), Q1(viii), Q5(a) |
Block 3 and Block 4 together supplied 9 of the 19 June 2026 items, so they deserve most of your last week. For the unit-wise theory behind each derivation, our BPHCT 135 Study Material follows the same block order as this table.
The official self-learning material for this course is on IGNOU eGyanKosh, where each block and unit can be read as a PDF.
BPHCT 135 question paper pattern and marks split
The BPHCT 135 question paper is a 50-mark, 2-hour paper with five compulsory questions and an internal choice in every one.
| Question | What you attempt | Marks | June 2026 focus |
|---|---|---|---|
| Q1 | Any 5 of 8 short parts | 5 × 2 = 10 | Plots and one-line definitions: van der Waals’ curve, Carnot T-S cycle, Fermi function |
| Q2 | Any 2 of 3 parts | 2 × 5 = 10 | Kinetic theory: most probable speed, free paths, a viscosity numerical |
| Q3 | Any 2 of 3 parts | 2 × 5 = 10 | First law block: βT and α, adiabatic work, Mayer’s formula |
| Q4 | Any 2 of 3 parts | 2 × 5 = 10 | Second law block: Carnot efficiency, statement equivalence, entropy of mixing |
| Q5 | Any 1 of 2 parts | 1 × 10 = 10 | Long derivation: Planck’s law or S = kB ln W |
The pattern shifts slightly between sessions. In June 2025 and December 2025, Q4 and Q5 were each a single 10-mark question with an either-or choice. June 2026 went back to three 5-mark parts in Q4, as in December 2024. Prepare both shapes, because a long TdS or Clausius-Clapeyron derivation can still appear as a full 10-marker.
A scientific calculator is allowed, which matters for the Q2(c) viscosity numerical. If you are collecting all physics papers of your programme, start from our BSCG Question Paper listing.
Topics that keep returning in BPHCT 135 papers
Across the four official BPHCT 135 papers we checked, December 2024 to June 2026, Maxwell’s speed distribution, van der Waals’ gases, the first law with Cp − CV, Fermi-Dirac statistics and blackbody radiation appeared in every one.
| Topic | Where it appeared | Sessions | Unit |
|---|---|---|---|
| Maxwell speed distribution and characteristic speeds | Dec 2024 Q1(b), Q2(a); Jun 2025 Q1(c), Q2(b), Q2(c); Dec 2025 Q1(c); Jun 2026 Q2(a) | 4 of 4 | Unit 2 |
| van der Waals’ gas and Andrews’ curves | Dec 2024 Q4(c); Jun 2025 Q1(b), Q3(b); Dec 2025 Q1(b), Q1(e); Jun 2026 Q1(i), Q1(ii) | 4 of 4 | Units 1, 10 |
| First law, Cp − CV and Mayer’s formula | Dec 2024 Q1(c), Q3(c); Jun 2025 Q3(a); Dec 2025 Q1(e); Jun 2026 Q3(c) | 4 of 4 | Unit 7 |
| Fermi-Dirac distribution, Fermi function, Fermi energy | Dec 2024 Q5(b); Jun 2025 Q1(g), Q5; Dec 2025 Q5 (Or); Jun 2026 Q1(vii) | 4 of 4 | Unit 14 |
| Blackbody radiation and Planck’s law | Dec 2024 Q1(f); Jun 2025 Q4 (Or); Dec 2025 Q1(g); Jun 2026 Q5(a) | 4 of 4 | Units 11, 14 |
| Mean free path and survival equation | Dec 2024 Q2(c); Dec 2025 Q2(a); Jun 2026 Q2(b) | 3 of 4 | Unit 3 |
| Brownian motion | Dec 2024 Q2(b); Jun 2025 Q1(d); Dec 2025 Q2(c) | 3 of 4 | Unit 4 |
| Carnot cycle and efficiency | Dec 2024 Q4(a); Dec 2025 Q4 (Or); Jun 2026 Q1(v), Q4(a) | 3 of 4 | Unit 8 |
| βT and α for a pVT system | Dec 2024 Q3(b); Jun 2025 Q3(b); Jun 2026 Q3(a) | 3 of 4 | Unit 5 |
| Types of system and thermodynamic variables | Dec 2024 Q3(a); Jun 2025 Q1(e); Jun 2026 Q1(iii), Q1(iv) | 3 of 4 | Unit 5 |
| Bose-Einstein statistics, bosons and fermions | Jun 2025 Q1(g); Dec 2025 Q5; Jun 2026 Q1(viii) | 3 of 4 | Unit 14 |
| Adiabatic process: equation of state, work, speed of sound | Jun 2025 Q3(c); Dec 2025 Q3(c); Jun 2026 Q3(b) | 3 of 4 | Unit 7 |
| Entropy change and entropy of mixing | Dec 2024 Q4(b); Jun 2026 Q4(c) | 2 of 4 | Unit 9 |
| Kelvin-Planck and Clausius statements | Jun 2025 Q1(f); Jun 2026 Q4(b) | 2 of 4 | Unit 9 |
| Maxwell’s relations and TdS equations | Dec 2024 Q4(c); Jun 2025 Q4 | 2 of 4 | Unit 10 |
| Joule-Thomson effect | Dec 2025 Q1(f); Jun 2026 Q1(vi) | 2 of 4 | Unit 10 |
| Viscosity of a gas | Dec 2025 Q2(b); Jun 2026 Q2(c) | 2 of 4 | Unit 3 |
| Boltzmann relation S = kB ln W | Dec 2025 Q1(h); Jun 2026 Q5(b) | 2 of 4 | Unit 12 |
| Partition function of a monoatomic ideal gas | Dec 2024 Q5(a); Jun 2025 Q5 (Or) | 2 of 4 | Unit 13 |
We counted a topic once per session even when it appeared in two parts of the same paper. Only papers we could read in full from ignou.ac.in were used, so this is a four-session picture, not a five-year one.
Five topics hit 4 of 4 sessions, which makes them the safest revision bets for December 2026. Our IGNOU Important Questions collection groups such high-return questions course by course.
How our BPHCT 135 answers are written and checked
Each BPHCT 135 answer is derived step by step from the IGNOU block text, checked by a second reviewer, and sized to the marks printed against the question.
Our physics writer works from the eGyanKosh units, so symbols match what your counsellor uses: kB for the Boltzmann constant, βT for isothermal compressibility and T1, T2 for source and sink. That avoids losing marks to notation the examiner does not expect.
Answers for 2-mark parts stay within 4 to 6 lines plus a labelled sketch where the part asks to plot or draw. 5-mark derivations show every intermediate line, and 10-mark answers add a short physical interpretation at the end.
Numericals such as the oxygen viscosity problem are solved with units carried through each line and the final value rounded sensibly. Sheetal Kirola then reviews the file for arithmetic, sign errors and missing diagram labels before release.
BPHCT 135 solved file index: all 19 items mapped to units
The full file answers all 19 items of the June 2026 paper; 2 are open on this page and 17 are in the full file only.
| Item | Topic | Marks | Unit | Answer |
|---|---|---|---|---|
| Q1(i) | van der Waals’ isotherms on P-V | 2 | Unit 1 | Full file only |
| Q1(ii) | van der Waals’ assumptions | 2 | Unit 1 | Full file only |
| Q1(iii) | Surface film variables | 2 | Unit 5 | Full file only |
| Q1(iv) | Type of system: bottle in a refrigerator | 2 | Unit 5 | Full file only |
| Q1(v) | Carnot cycle on T-S plane | 2 | Units 8 and 9 | Full file only |
| Q1(vi) | Joule-Thomson effect | 2 | Unit 10 | Full file only |
| Q1(vii) | Fermi function at 0 K and above | 2 | Unit 14 | Full file only |
| Q1(viii) | Bosons versus fermions | 2 | Unit 14 | Full file only |
| Q2(a) | Most probable speed | 5 | Unit 2 | On this page |
| Q2(b) | Mean free path and survival equation | 5 | Unit 3 | Full file only |
| Q2(c) | Oxygen molecule radius from viscosity | 5 | Unit 3 | Full file only |
| Q3(a) | βT = 1/p and α = 1/T | 5 | Unit 5 | Full file only |
| Q3(b) | Adiabatic versus isothermal, adiabatic work | 5 | Unit 7 | Full file only |
| Q3(c) | First law and Mayer’s formula | 5 | Unit 7 | Full file only |
| Q4(a) | Carnot efficiency | 5 | Unit 8 | On this page |
| Q4(b) | Equivalence of second-law statements | 5 | Unit 9 | Full file only |
| Q4(c) | Entropy of mixing | 5 | Unit 9 | Full file only |
| Q5(a) | Planck’s law from photon statistics | 10 | Unit 14 | Full file only |
| Q5(b) | S = kB ln W | 10 | Unit 12 | Full file only |
Programmes that carry BPHCT 135 Thermal Physics and Statistical Mechanics
BPHCT 135 Thermal Physics and Statistical Mechanics is a core physics course in two IGNOU undergraduate science programmes, BSCG and BSCM.
- BSCG, Bachelor of Science (General): taken by learners who choose physics as one of their disciplines.
- BSCM, Bachelor of Science (Multidisciplinary): the June 2026 paper header prints this programme alongside BSCG.
If you are in the IGNOU Bachelor of Science (General) programme, BPHCT 135 sits with the other BPHCT physics courses, and carries 4 theory credits. Its 2-credit lab component, BPHCL-136, is assessed separately from this 50-mark theory paper.
Earlier papers printed only BSCG on the cover; June 2026 is the first of the papers we checked to name both programmes. The paper itself is the same for both groups.
How to get the full BPHCT 135 Solved Question Paper
Send “BPHCT 135 June 2026” on WhatsApp 9899436384 and our team shares the full solved file with all 19 answers.
The file holds 8 short answers, 9 five-mark answers and both 10-mark derivations, that is 19 answers in all. 2 of them are on this page and the other 17 are inside the file only. It comes as a single PDF that opens on any phone.
Look through the preview pages of the IGNOU BPHCT 135 Solved Question Paper below to see the layout, the labelled diagrams and the way each derivation is set out before you message us. If you also need answers for an earlier session, mention the session in the same message.
BPHCT 135 FAQs
These are the questions BPHCT 135 learners ask us most often about the June 2026 paper and the solved file.
What is the BPHCT 135 Solved Question Paper June 2026?
It is a complete answer file for the June 2026 IGNOU BPHCT 135 Thermal Physics and Statistical Mechanics paper. Our physics writer solved all 19 printed items, from the 2-mark plots to both 10-mark derivations, using the symbols of the eGyanKosh blocks. A second reviewer checked every step. Two answers are open on this page as samples, and the full file comes on WhatsApp.
How do I get the full BPHCT 135 solved file on WhatsApp?
Send the message BPHCT 135 June 2026 to WhatsApp 9899436384. Our team replies with the details and shares the full solved file as one PDF. It contains 19 answers in total: 8 short answers, 9 five-mark answers and both 10-mark derivations. Two of these are already shown on this page, so the file adds the other 17 answers.
Which BPHCT 135 June 2026 questions are solved on this page?
Two answers are shown in full. Q2(a) derives the most probable speed of a Maxwellian gas molecule, v_p = √(2k_BT/m), and checks that it is a maximum. Q4(a) proves that the efficiency of a Carnot engine is 1 − T_2/T_1 using the four steps of the cycle. The remaining 17 questions are listed as text only.
How many questions must I attempt in the BPHCT 135 exam?
All five questions are compulsory, but each has an internal choice. In June 2026 you attempted any five of eight short parts in Q1, any two of three parts in Q2, Q3 and Q4, and any one of two parts in Q5. That adds up to 50 marks in 2 hours, so a 5-mark derivation should take about 10 to 12 minutes.
Which BPHCT 135 topics repeat most often in exams?
In the four official papers from December 2024 to June 2026, five topics appeared every time: Maxwell’s speed distribution, van der Waals’ gases, the first law with Cp minus Cv, Fermi-Dirac statistics and blackbody radiation. Mean free path, Brownian motion and the Carnot cycle followed with three appearances each. Revising these first gives the best return.
Can I use a calculator in the BPHCT 135 exam?
Yes. The June 2026 BPHCT 135 paper states in its instructions that a scientific calculator may be used. You need it mainly for numericals such as Q2(c), where the radius of an oxygen molecule is found from its viscosity, mean speed and number density. Carry units through every line, because examiners deduct marks for a final value without units.
Which programmes include BPHCT 135 Thermal Physics and Statistical Mechanics?
BPHCT 135 is a core physics course in IGNOU’s Bachelor of Science (General), code BSCG, and Bachelor of Science (Multidisciplinary), code BSCM. The June 2026 question paper prints both programme codes on its cover. Learners in either programme sit the same 50-mark theory paper, so this solved file works for both groups without any change.
How are the BPHCT 135 answers checked before release?
Every BPHCT 135 answer is first written by our physics writer from the IGNOU block text, with each derivation step shown. Sheetal Kirola then reviews the file for arithmetic, sign errors, missing units and unlabelled diagrams. Only after these corrections is the file shared, so the answers match what an IGNOU examiner expects for the marks printed.