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Title Name	IGNOU BSCM BPHCT 135 Thermal Physics and Statistical Mechanics Solved Assignment ENGLISH
Type	Soft Copy (E-Assignment) .pdf
University	IGNOU
Degree	BACHELOR DEGREE PROGRAMMES
Course Code	BSCM
Course Name	Four Year Under Graduate Programmes/Bachelor of Science
Subject Code	BPHCT 135
Subject Name	Thermal Physics and Statistical Mechanics
Year	2026
Session	-
Language	ENGLISH
Assignment Code	BPHCT 135/Assignment-1/2026
Product Description	Assignment of BSCM (Four Year Under Graduate Programmes/Bachelor of Science) 2026. Latest BPHCT 135 2026 Solved Assignment Solutions
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BPHCT 135 (January 2026 - July 2026) - ENGLISH

Tutor Marked Assignment
THERMAL PHYSICS AND STATISTICAL MECHANICS

Course Code: BPHCT-135
Assignment Code: BPHCT-135/TMA/2026
Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Calculate v_rms for helium atoms at 300K. At what temperature will oxygen molecules have the same value of v_rms? Take $m_{He} = 6.67 \times 10^{-27}\text{ kg}$ .

b) What is most probable speed? Using the expression of the number of molecules in a Maxwellian gas in the range v to v + dv as
$$dN_v = 4\pi N \left( \frac{m}{2\pi k_BT} \right)^{3/2} v^2 \exp \left[ - \left( \frac{mv^2}{2k_BT} \right) \right] dv$ $
Obtain an expression for most probable speed.

c) Define mean free path? Obtain an expression of mean free path for zeroth order approximation.

d) What is Brownian motion? Write its two examples. Show that during sedimentation, particles concentration decreases exponentially as height increases.

e) Write the expressions of an ideal gas and van der Waals’ equation. 10 moles of nitrogen gas occupy a volume of $5 \times 10^{-3}\text{ m}^3$ in a vessel at $27^\circ\text{C}$ . Calculate the pressure exerted by the molecules, if it is assumed to obey van der Waals’ equation. Compare this Value with that obtained using ideal gas law.
Given: $a = 1.39 \times 10^{-6}\text{ atm m}^6\text{ mol}^{-2}$ ,
$b = 39.1 \times 10^{-6}\text{ m}^3\text{ mol}^{-1}$
and $R = 8.2 \times 10^{-5}\text{ atm m}^3\text{ K}^{-1}\text{ mol}^{-1}$ .

2. a) A mass of an ideal gas $(\gamma = 1.5)$ at $27^\circ\text{C}$ is suddenly compressed to four times its original pressure. Calculate the final temperature of the gas.

b) i) What are intensive and extensive variables for thermodynamic systems? Write two examples of each.
ii) Represent an (a) isobaric (b) isochoric (c) isothermal and (d) a cyclic process on p-V indicator diagram.

c) State the zeroth law of thermodynamics. Explain how the zeroth law introduces the concept of temperature.

d) Obtain the expressions of work done for an ideal gas (i) in an isothermal expansion (ii)) in an adiabatic process.

e) Obtain an expression of coefficient of volume expansion for one mole of van der Waals’ gas.

PART B

3. a) Consider that two ideal gases having n₁ moles are n₂ moles are enclosed in two separate containers at constant temperate T and pressure P. It these two gases mix, obtain an expression for the entropy of mixing per mole of mixture.

b) Draw p-V diagram of the Carnot Cycle and hence obtain an expression of efficiency.

c) Using the Maxwell's relations of thermodynamics, obtain the first and second TdS-equations.

d) Derive Clausius-Clayperon equation of phase transition. Draw phase diagram of water.

e) What is Joule-Thomson effect? Write any four findings of Joule- Thomson experiment.

4. a) Derive an expression of the partition function for a single particle monatomic ideal gas. Write the expression for a monatomic gas consisting of N-identical yet distinguishable and non-interacting particles.

b) N particles obey MB distribution. They are distributed among three states with energies $E_1 = 0, E_2 = k_BT$ and $E_3 = 3k_BT$ . If the equilibrium energy of the system is 2000 k_BT, calculate the total number of particles, N.

c) Using the Planck's radiation law:
$$u_\lambda d\lambda = \frac{8\pi hc}{\lambda^5} \left( \frac{1}{\exp \left( \frac{hc}{\lambda k_B T} \right) - 1} \right) d\lambda \text{ , deduce (i) Rayleigh-Jean's law, (ii) Wien's law}$
and (iii) Stefan's law. (10)

d) What is Fermi-energy? Obtain an expression for Fermi energy for a system of N Fermions enclosed in a volume V. What will be the value of Fermi-energy (in eV) of gold at Fermi-temperature $6.42 \times 10^4 K$ ?

BPHCT 135 (January 2026 - July 2026) - HINDI

अध्यापक जांच सत्रीय कार्य
ऊष्मीय भौतिकी और सांख्यिकीय यांत्रिकी

पाठ्यक्रम कोड : BPHCT-135
सत्रीय कार्य कोड : BPHCT-135/TMA/2026
अधिकतम अंक : 100

नोट : सभी प्रश्न हल करें। प्रत्येक प्रश्न के अंक उसके सामने दिए गए हैं।

भाग क

1. क) $300\text{ K}$ तापमान पर हीलियम परमाणुओं के लिए v_rms का मान परिकलित कीजिए। आक्सीजन के अणुओं के लिए v_rms का यह मान किस तापमान पर प्राप्त होगा? $m_{He} = 6.67 \times 10^{-27}\text{ kg}$ लें।

ख) प्रायिकतम चाल क्या होती है? मैक्सवेली गैस में चाल परिसर v से v+dv में अणुओं की संख्या के व्यंजक $dN_v = 4\pi N \left( \frac{m}{2\pi k_BT} \right)^{3/2} v^2 \exp \left[ -\left( \frac{mv^2}{2k_BT} \right) \right] dv$ का उपयोग कर प्रायिकतम चाल का व्यंजक प्राप्त कीजिए।

ग) माध्य मुक्त पथ को परिभाषित कीजिए। शून्य कोटि सन्निकटन के लिए माध्य मुक्त पथ का व्यंजक प्राप्त कीजिए।

घ) ब्राउनी गति क्या होती है? इसके दो उदाहरण लिखिए। सिद्ध कीजिए कि अवसादन के दौरान ऊँचाई के साथ कण के संकेंद्रण में चरघातांकी कमी आती है।

ड़) किसी पात्र में $27\text{ }^\circ\text{C}$ तापमान पर नाइट्रोजन गैस कें 10 मोलों का आयतन $5 \times 10^{-3}\text{ m}^3$ है। यदि यह गैस वाण्डरवाल्स समीकरण का अनुपालन करती हो तो इसके अणुओं द्वारा आरोपित दाब परिकलित किजिए। इस मान की तुलना आदर्श गैस समीकरण के आधार पर प्राप्त मान से कीजिए।
दिया है : $a = 1.39 \times 10^{-6}\text{ atm m}^6\text{ mol}^{-2}$ ,
$b = 39.1 \times 10^{-6}\text{ m}^3\text{ mol}^{-1}$
और $R = 8.2 \times 10^{-5}\text{ atm m}^3\text{ K}^{-1}\text{ mol}^{-1}$ .

2. क) $27\text{ }^\circ\text{C}$ तापमान पर आदर्श गैस ( $\gamma = 1.5$ ) का कुछ द्रव्यमान अचानक प्रारंभिक दाब से चार गुणा तक संपीड़ित किया जाता है। गैस का अन्तिम तापमान परिकलित कीजिए।

ख) i) ऊष्मागतिक तंत्रो के लिए अविस्तारात्मक और विस्तारात्मक चर क्या होते हैं? प्रत्येक के दो उदाहरण लिखें।
ii) (क) समदाबी, (ख) समआयतनिक, (ग) समतापी और (घ) चक्रीय प्रक्रमों का $p\text{-}V$ सूचक आरेखों पर निरूपण कीजिए।

ग) ऊष्मागतिकी के शून्य कोटि नियम का कथन लिखिए। शून्य कोटि नियम से तापमान की संकल्पना कैसे प्रस्तुत होती है इसका विवरण दीजिए।

घ) आदर्श गैस के लिए (i) समतापी प्रक्रम और (ii) रूद्धोष्म प्रक्रम में किये गये कार्य के व्यंजक प्राप्त कीजिए।

ड़) एक मोल वाण्डर वाल्स गैस के लिए आयतन प्रसार गुणांक का व्यंजक प्राप्त कीजिए।

भाग ख

3. क) मान लें कि दो अलग-अलग पात्रों में अचर तापमान T और दाब P पर दो आदर्श गैसों के क्रमशः n₁ मोल और n₂ मोल हैं। यदि इन गैसों को मिश्रित किया जाए, तो मिश्रण की प्रतिमोल मिश्रण एन्ट्रॉपी का व्यंजक प्राप्त कीजिए।

ख) कार्नो चक्र का $p\text{-}V$ आरेख पर निरूपण किजिए और दक्षता का व्यंजक व्युत्पन्न कीजिए।
ग) ऊष्मागतिकी के मैक्सवेल संबंधों का उपयोग कर प्रथम और द्वितीय TdS संबंध प्राप्त कीजिए।
घ) प्रवस्था संक्रमण की क्लासियस-क्लैपेरॉन समीकरण व्युत्पन्न कीजिए। जल का प्रावस्था आरेख खींचिए।
ड़) जूल-टॉमसन प्रभाव क्या होता है? जूल-टॉमसन प्रयोग के परिणामों से निकले किन्हीं चार निष्कर्षों को लिखिए।

4. क) एक परमाणुक आदर्श गैस के एकल कण के लिए संवितरण फलन का व्यंजक व्युत्पन्न कीजिए। N सर्वसम तथापि विभेद्य अन्योन्य क्रियाहीन कणो वाली एक-परमाणुक गैस के लिए संवितरण फलन लिखिए।
ख) N कण मैक्सवेल बोल्ट्मान (MB) बंटन का अनुपालन करते हैं। वे $E_1 = 0, E_2 = k_B T$ तथा $E_3 = 3k_B T$ इन तीन ऊर्जा अवस्थाओं में वितरित हैं। यदि तंत्र की साम्यावस्था ऊर्जा 2000 k_B T हो तो कणों की कुल संख्या N परिकलित कीजिए। (5)

ग) प्लांक के विकिरण नियम $u_\lambda d\lambda = \frac{8\pi hc}{\lambda^5} \left( \frac{1}{\exp \left( \frac{hc}{\lambda k_B T} \right) - 1} \right) d\lambda$ , का उपयोग कर (i) रैले-जीन का नियम, (ii) वीन का नियम, और (iii) स्टीफ़न का नियम व्युत्पन्न कीजिए।
घ) फर्मी ऊर्जा क्या होती है? आयतन V में परिबद्ध N फर्मिऑनों के एक तंत्र के लिए फर्मी ऊर्जा का व्यंजक प्राप्त कीजिए। फर्मी-तापमान $6.42 \times 10^4\text{ K}$ पर सोने की फर्मी ऊर्जा ( $\text{eV}$ ) का मान क्या होगा?

BPHCT 135 2025 - English

Tutor Marked Assignment

THERMAL PHYSICS AND STATISTICAL MECHANICS

Course Code: BPHCT-135

Assignment Code: BPHCT-135/TMA/2025

Max. Marks: 100

Note: Attempt all questions. The marks for each question are indicated against it.

PART A

1. a) Discuss Regnault's experiments on Hydrogen, nitrogen and carbon dioxide for 273 K. Also discuss the Andrews experiments for CO₂ on the p-V diagram at various temperatures.

b) Write the assumptions made by Maxwell to derive the expression for distribution function of velocities. Hence derive the expression of Maxwellian distribution function for molecular speeds. Plot Maxwellian distribution function as a function of molecular speed.

c) The average speed of hydrogen molecules is 1850 ms-1. The radius of a hydrogen molecule is 1.40x10^-10m. Calculate (i) Collision cross-section, (ii) collision frequency, and (iii) mean free path. Take n = 3x10²⁵m^-3.

d) What is Brownian motion? Discuss Perrin's method for determination of Avogadro number in Brownian motion. How can this method be used to estimate the mass of molecule

2. a) Explain the classification of boundaries in a thermodynamic system.

b) State Zeroth law of thermodynamics. How does this law introduces the concept of temperature. Write parametric as well as exact equation of state for one mole of a Ideal gas and stretched wire.

c) Show that for an ideal gas

$a=frac{1}{T}$ and $eta _{T}=frac{1}{p}$

where ẞ_T is the isothermal compressibility and a is isobaric coefficient of volume expansion. 1

d) Derive Mayer's formula: C_p-C_v = R where C_p and C_v are the molar heat capacity at constant pressure and constant volume respectively.

e) Obtain an expression for work done in expanding a gas from volume V_ito V_f in an Isobaric process.

PART B

3. a) With the help of entropy – temperature diagram of Carnot cycle, obtain an expression of efficiency of a Carnot engine. A Carnot engine has an efficiency of 50%. It operates between reservoirs of constant temperature with temperature difference of 80 K. Calculate the temperature of the low-temperature reservoir in Celsius.

b) Define thermodynamic potentials. Derive Maxwell’s relations from thermodynamic potentials.

c) When two phases of a substance coexist in equilibrium at constant temperature and pressure, their specific Gibb’s free energies are equal. Using this fact, obtain Clausius-Clapeyron equation.

d) Derive Planck’s law of radiation and hence obtain Rayleigh-Jeans law and Wien’s law.

4. a) Consider a classical ideal gas consisting of N particles. The energy $varepsilon$ of a particle is given by $varepsilon$ =cp where c is a constant and p is the magnitude of the momentum. Calculate (i) the partition function of the system, (ii) internal energy, and (iii) C_V .

b) 5.4x10 ²¹electrons are confined in a box of volume 1cm³. Calculate their Fermi wavelength and Fermi energy.

c) Define thermodynamic probability of a macrostate. Establish the Boltzmann relation between entropy and thermodynamic probability.

S = k_B lnW.

d) Obtain an expression of Fermi-Dirac distribution function. Plot Fermi function versus energy at different temperatures.

BPHCT 135 2025 - Hindi

अध्यापक जांच सत्रीय कार्य ऊष्मीय भौतिकी और सांख्यिकीय यांत्रिकी

पाठ्यक्रम कोड: BPHCT-135

सत्रीय कार्य कोड: BPHCT-135/TMA/2025

अधिकतम बैंक : 100

नोट: प्रथन हल करें। प्रत्येक प्रश्न के अंक उसके सामने दिए गए हैं।

भाग क

1. क) हाइड्रोजन, नाइट्रोजन एवं कार्बन डाइऑक्साइड के लिए 273 K पर रेना के प्रयोगों का विवरण दीजिए। CO₂ के लिए, p-V आरेख पर विभिन्न तापमानों पर एण्ड्रयूज प्रयोग का भी विवरण दीजिए।

ख) वेग बंटन फलन का व्यंजक व्युत्पन्न करने के लिए मैक्सवेल द्वारा दी गई अभिधारणाओं को लिखिए। अतएव, आण्विक चालों के मैक्सवेलीय बंटन फलन का व्यंजक व्युत्पन्न कीजिए। मैक्सवेलीय बंटन फलन तथा आण्विक चालों के बीच ग्राफ खींचिए I

ग) हाइड्रोजन अणुओं की माध्य चाल 1850 ms^-1है। हाइड्रोजन अणु की त्रिज्या 1.40×10^-10m है। (i) संघट्टन अनुप्रस्थ परिच्छेद (ii) संघट्टन-आवृत्ति तथा (iii) माध्य मुक्त पथ परिकलित कीजिए। n=3x10²⁵m^-3मान लें।

घ) ब्राऊनी गति क्या है? ब्राऊनी गति में आयोगाद्वो संख्या के निर्धारण के लिए पैरों विधि की चर्चा कीजिए। इस विधि का उपयोग अणु के द्रव्यमान का आकलन करने के लिए किस प्रकार किया जा सकता है?

2. क) ऊष्मागतिक तंत्र में परिसीमाओं का वर्गीकरण समझाइए।

ख) ऊष्मागतिकी का शून्य नियम लिखिए। यह नियम ताप की संकल्पना कैसे प्रस्तुत करता है। एक मोल आदर्श गैस एवं तानित तार के लिए प्राथलिक एवं यथार्थ अवस्था समीकरण लिखिए।

ग) आदर्श गैस के लिए सिद्ध करें किA

$a=frac{1}{ au}$ और $eta au=frac{1}{p}$

जहां $eta au$ समतापी संपीड्‌यता और $alpha$ आयतन-प्रसार का समदाबी गुणांक है।

घ) मेयर सूत्र C_p-C_v=R व्युत्पन्न कीजिए जहां C_p अचर दाब मोलीय ऊष्माधारिता तथा C_vअचर आयतन मोलीय ऊष्माधारिता है।

ङ) समदाबी प्रक्रम में एक गैस का आयतन V_iसे V_fतक बढ़ाने में किए गए कार्य का व्यंजक व्युत्पन्न कीजिए।

भाग ख

3.क) कार्नो चक्र के लिए एन्ट्रॉपी-तापमान आरेख की सहायता से कार्नो इंजन की दक्षता का व्यंजक प्राप्त कीजिए। एक कार्नो इंजन की दक्षता 50 प्रतिशत है। स्थिर तापमान वाले जलाशयों - जिनके बीच तापमान अंतर 80K के साथ यह इंजन संचालित होता है। निम्न-तापमान जलाशय का तापमान ℃ में परिकलित कीजिए।

ख) ऊष्मागतिक विभव को परिभाषित कीजिए। ऊष्मागतिक विभवों का उपयोग कर मैक्सवेल संबंध व्युत्पन्न कीजिए।

ग) अचर ताप और दाब पर जब किसी पदार्थ की दो प्रावस्थाएँ साम्यावस्था में सह अस्तित्व करती है, जिनकी विशिष्ट गिब्ज मुक्त ऊर्जाएँ बराबर है। इस तथ्य का उपयोग कर क्लासियस-क्लैपेरॉन समीकरण प्राप्त करें।

घ) विकिरणों के प्लांक नियम को व्युत्पन्न कीजिए। इस नियम का उपयोग कर रैले जीन्स नियम और वीन नियम प्राप्त कीजिए।

4. क) N कणों से बनी चिरप्रतिष्ठित आदर्श गैस पर विचार कीजिए। इसमें एक कण की ऊर्जा का व्यंजक $varepsilon$ = cp है, जहाँ एक नियतांक है और p संवेग का परिमाण है। तंत्र के (i) संवितरण फलन (ii) आंतरिक ऊर्जा

(iii) C_v का परिकलन कीजिए।

ख) 1cm³ आयतन के बॉक्स में 5.4×10²¹ इलेक्ट्रॉन परिबद्ध हैं। इनकी फर्मी तरंगदैर्ध्य और फर्मी ऊर्जा परिकलित कीजिए।

ग) स्थूल अवस्था की ऊष्मागतिक प्रायिकता को परिभाषित कीजिए। एन्ट्रापी और ऊष्मागतिक प्रायिकता के बीच बोल्ट्मान संबंध S = k_B In W स्थापित कीजिए।

घ) फर्मी-डिरैक बंटन फलन का व्यंजक प्राप्त कीजिए। विभिन्न तापों पर फर्मी-फलन का ऊर्जा के सापेक्ष विचरण का आरेख खीचिए।

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Course: Four Year Under Graduate Programmes/Bachelor of Science (BSCM)
Subject: BPHCT 135 - Thermal Physics and Statistical Mechanics
Session: January 2026 - December 2026
Language: ENGLISH

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Course Name	Four Year Under Graduate Programmes/Bachelor of Science
Course Code	BSCM
Subject Code	BPHCT 135
Subject	Thermal Physics and Statistical Mechanics
Programme	BACHELOR DEGREE PROGRAMMES Courses
Language	ENGLISH
Material Type	Solved Assignment – Soft Copy / PDF

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