Find the best tutors and institutes for Class 11 Tuition
Q12.6:
Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity (25.0 \text{ m}^{3}) at a temperature of (27 °\text{C}) and 1 atm pressure.
Solution :
Step 1: Given Variables and Initial Setup
To estimate the total number of air molecules, we utilize the Ideal Gas Equation. The state variables are provided as follows:
- Volume ($V$) = $25.0 \text{ m}^3$
- Temperature ($T$) = $27 \text{ °C} = 27 + 273.15 = 300.15 \text{ K} \approx 300 \text{ K}$ [Per the thermodynamic conversion to Kelvin]
- Pressure ($P$) = $1 \text{ atm} = 1.013 \times 10^5 \text{ Pa}$ [Standard atmospheric pressure definition]
- Boltzmann Constant ($k_B$) = $1.38 \times 10^{-23} \text{ J/K}$ [Physical constant]
Step 2: Selecting the Governing Equation
The Ideal Gas Law in terms of the number of molecules ($N$) is expressed as: $$PV = Nk_BT$$ where $P$ is pressure, $V$ is volume, $N$ is the number of molecules, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. [Reference: Kinetic Theory of Gases, Ideal Gas Equation derivation]
Step 3: Algebraic Manipulation
To solve for $N$, we rearrange the equation: $$N = \frac{PV}{k_BT}$$
Step 4: Numerical Calculation
Substitute the given values into the rearranged equation:
$$N = \frac{(1.013 \times 10^5 \text{ Pa}) \times (25.0 \text{ m}^3)}{(1.38 \times 10^{-23} \text{ J/K}) \times (300 \text{ K})}$$
Perform the calculation in the denominator: $$(1.38 \times 10^{-23}) \times 300 = 4.14 \times 10^{-21} \text{ J}$$
Perform the calculation in the numerator: $$(1.013 \times 10^5) \times 25.0 = 25.325 \times 10^5 = 2.5325 \times 10^6 \text{ J}$$
Complete the division: $$N = \frac{2.5325 \times 10^6}{4.14 \times 10^{-21}} \approx 0.6117 \times 10^{27}$$ $$N \approx 6.12 \times 10^{26} \text{ molecules}$$
Visual Representation of Kinetic Distribution
Step 5: Analysis of Results
The calculated value represents the total ensemble of gas molecules within the enclosed volume. The result is independent of the chemical composition (Oxygen, Nitrogen, etc.) because, per Avogadro’s Hypothesis and the Kinetic Theory of Gases, all gases behave identically under the assumption of ideal gas behavior at these standard conditions. [Justification: Kinetic theory assumes negligible molecular volume and elastic collisions, which holds true for air at room temperature and pressure.]
Final Solution: The total number of air molecules in the room is approximately $6.12 \times 10^{26}$.
More Questions from Class 11 Physics Kinetic Theory EXERCISES
- Q12.1: Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.
- Q12.10: Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature (17 °\text{C}). Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of (N_{2}) = 28.0 u).
- Q12.2: Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP : 1 atmospheric pressure, (0 °\text{C})). Show that it is 22.4 litres.
- Q12.3 (a): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify?
- Q12.3 (b): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (b) Which is true: (T_{1} > T_{2}) or (T_{1} < T_{2})?
- Q12.3 (c): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (c) What is the value of (\frac{PV}{T}) where the curves meet on the y-axis?
- Q12.3 (d): Figure 12.8 shows plot of (\frac{PV}{T}) versus (P) for (1.00 \times 10^{-3}) kg of oxygen gas at two different temperatures. (d) If we obtained similar plots for (1.00 \times 10^{-3}) kg of hydrogen, would we get the same value of (\frac{PV}{T}) at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of (\frac{PV}{T}) (for low pressure high temperature region of the plot) ? (Molecular mass of (H_{2}) = 2.02 u, of (O_{2}) = 32.0 u, (R) = 8.31 J (\text{mol}^{-1}) (\text{K}^{-1}).)
- Q12.4: An oxygen cylinder of volume 30 litre has an initial gauge pressure of 15 atm and a temperature of (27 °\text{C}). After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to (17 °\text{C}). Estimate the mass of oxygen taken out of the cylinder ((R) = 8.31 J (\text{mol}^{-1}) (\text{K}^{-1}), molecular mass of (O_{2}) = 32 u).
- Q12.5: An air bubble of volume (1.0 \text{ cm}^{3}) rises from the bottom of a lake 40 m deep at a temperature of (12 °\text{C}). To what volume does it grow when it reaches the surface, which is at a temperature of (35 °\text{C}) ?
- Q12.7 (i): Estimate the average thermal energy of a helium atom at (i) room temperature ((27 °\text{C})),
- Q12.7 (ii): Estimate the average thermal energy of a helium atom at (ii) the temperature on the surface of the Sun (6000 K),
- Q12.7 (iii): Estimate the average thermal energy of a helium atom at (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
- Q12.8: Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules ? Is the root mean square speed of molecules the same in the three cases? If not, in which case is (v_{rms}) the largest ?
- Q12.9: At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at (– 20 °\text{C}) ? (atomic mass of Ar = 39.9 u, of He = 4.0 u).
- Q13.1: Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3 Å.
- Q13.10: Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 0C. Take the radius of a nitrogen molecule to be roughly 1.0 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of $N_2$ = 28.0 u).
- Q13.2: Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP : 1 atmospheric pressure, 0 °C). Show that it is 22.4 litres.
- Q13.3(a): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (a) What does the dotted plot signify?
- Q13.3(b): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (b) Which is true: $T_1 > T_2$ or $T_1 < T_2$?
- Q13.3(c): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (c) What is the value of $frac{PV}{T}$ where the curves meet on the y-axis?
- Q13.3(d): Figure 13.8 shows plot of $frac{PV}{T}$ versus $P$ for $1.00 \times 10^{-3}$ kg of oxygen gas at two different temperatures. (d) If we obtained similar plots for $1.00 \times 10^{-3}$ kg of hydrogen, would we get the same value of $frac{PV}{T}$ at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of $frac{PV}{T}$ (for low pressure high temperature region of the plot) ? (Molecular mass of $H_2$ = 2.02 u, of $O_2$ = 32.0 u, $R = 8.31$ J $mol^{-1} K^{-1}$.)
- Q13.4: An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder ($R = 8.31$ J $mol^{-1} K^{-1}$, molecular mass of $O_2$ = 32 u).
- Q13.5: An air bubble of volume 1.0 $cm^3$ rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C ?
- Q13.6: Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 $m^3$ at a temperature of 27 °C and 1 atm pressure.
- Q13.7(i): Estimate the average thermal energy of a helium atom at (i) room temperature (27 °C),
- Q13.7(ii): Estimate the average thermal energy of a helium atom at (ii) the temperature on the surface of the Sun (6000 K),
- Q13.7(iii): Estimate the average thermal energy of a helium atom at (iii) the temperature of 10 million kelvin (the typical core temperature in the case of a star).
- Q13.8: Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules ? Is the root mean square speed of molecules the same in the three cases? If not, in which case is $v_{rms}$ the largest ?
- Q13.9: At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the rms speed of a helium gas atom at – 20 °C ? (atomic mass of $Ar$ = 39.9 u, of $He$ = 4.0 u).
CBSE Solutions for Class 11 Physics Kinetic Theory
Chapters in CBSE - Class 11 Physics
Top Tutors who teach Kinetic Theory
Super Tutor
Panvel, Mumbai
14 yrs of
Exp
30 students
3301 hrs of classes completed
500
per hour
Online Classes
I provide private tuition for Physics and Mathematics for Class XI and XII (All Boards) Physics for Engineering Aspirants Physics for Medical Aspirants Class IX and X Mathematics and Science.
Devendra sir is really an amazing mentor and a down to heart pearson who is always ready to give you a clear cut idea about the concepts and is an expert in getting the concepts really into my head. With him physics will be made easy and you will not have to struggle at all especially for a student who shifted from 10th to 11th I can tell he is just amazing!!
Top Tutor
Sector 39 Charmwood Village, Faridabad
6 yrs of
Exp
3 students
112 hrs of classes completed
500
per hour
Online Classes
In my experience providing tuition for Class 11 students, I aimed to create a supportive and engaging learning environment tailored to their unique needs. I focused on key subjects such as English, Social Studies which are crucial for their board examinations. Teaching Methodology: I began each session by assessing the students’ current understanding and identifying specific areas where they needed support. This allowed me to customize my lesson plans to address their weaknesses while reinforcing their strengths. I employed a variety of teaching methods, including: Interactive Discussions: I encouraged students to participate actively by asking questions and engaging in discussions around key concepts, which helped to deepen their understanding. Problem-Solving Sessions: In subjects like Mathematics and Science, I emphasized problem-solving techniques. I walked them through complex problems step-by-step, encouraging them to think critically and develop their reasoning skills. Practice Tests: I conducted regular quizzes and mock tests to familiarize them with the exam format and to track their progress. This also helped reduce anxiety leading up to their actual exams. Student Engagement: To keep students motivated, I incorporated real-life applications of the concepts we covered. For example, in Science, I would relate topics to everyday phenomena, making the material more relatable and interesting. I also encouraged peer collaboration, allowing students to work together on challenging problems, fostering teamwork and communication skills. Feedback and Support: I provided constructive feedback on their work, highlighting both areas of improvement and their successes. I made myself available for additional support outside of regular sessions, whether through email or online messaging, to answer questions and clarify doubts. Results and Impact: Many of my students showed significant improvement in their confidence and academic performance. Witnessing their growth, especially as they prepared for their board exams, was incredibly rewarding. I emphasized the importance of a positive mindset, helping them develop not just academically, but also personally. Overall, my experience teaching Class 11 tuition was fulfilling, as I was able to make a meaningful impact on my students' educational journeys while fostering a love for learning.
Satisfactory, command on subject. Very well described the topic it was just a demo class , rest depends on other topics, though I found mam is sweet n goo in her subject.
Top Tutor
Powai, Mumbai
5 yrs of
Exp
5 students
160 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I have been teaching Physics to students of Class 11 and 12 for the past five years. During this time, I have guided students through board exams as well as competitive exams like IIT-JEE, helping them strengthen their understanding of concepts, develop problem-solving skills, and improve their overall academic performance. Watching students succeed and gain confidence in Physics has been incredibly rewarding. I am currently pursuing my M.Sc. in Physics, which allows me to stay updated with advanced concepts and recent developments in the field. This academic background enables me to teach complex topics in a structured and understandable way, making them accessible to students at different levels. Over the past five years, I have worked with a diverse group of students, including those preparing for highly competitive exams. This experience has helped me develop effective teaching strategies, advanced problem-solving techniques, and exam-oriented approaches that prepare students for both boards and competitive assessments. My classes are designed to be interactive and engaging. I follow a structured lesson plan that ensures systematic coverage of topics while encouraging active student participation. I use PPTs and the whiteboard to simplify difficult concepts and make learning more visual and practical. I conduct both online and offline sessions, offering flexibility to students while maintaining a high level of interaction. Whether in-person or virtual, I ensure that students receive personalized attention, have their doubts clarified promptly, and develop a strong understanding of each topic. Regular assessments, practice sessions, and feedback are an integral part of my teaching approach. By tracking each student’s progress, I can identify areas that need reinforcement and provide targeted guidance to help them improve steadily and achieve consistent results. My goal as a Physics teacher is to make the subject both approachable and enjoyable. I strive to instill curiosity, analytical thinking, and confidence in students, helping them not only excel academically but also develop problem-solving skills that will benefit them throughout their educational journey.
Sir, helped prepare me for my physics exam and made sure I prepare well by giving extra time and efforts.
Top Tutor
Bapu Nagar, Jaipur
3 yrs of
Exp
1 student
8 hrs of classes completed
550
per hour
Online Classes
Allow me to introduce myself—I am deeply passionate about biology, particularly in the context of CBSE curriculum for classes 10th, 11th, and 12th. With a solid educational background including a BSc Honors and MSc in Microbiology, I bring a wealth of knowledge and expertise to my teaching endeavors. My journey into the world of education began two years ago, driven by my desire to share my love for biology with students and help them excel in their academic pursuits. Having qualified as a CSIR UGC NET and GATE trainer, I possess the necessary credentials and skills to guide students through their studies effectively. During my tenure at Apex School, I had the opportunity to make a meaningful impact on the lives of students, imparting knowledge and nurturing their curiosity about the wonders of biology. My dedication to my students' success was further validated when I achieved CSIR UGC NET JRF in 2019, as well as clearing the GATE exam, showcasing my proficiency in the subject matter. In my online classes, which typically last for one hour, I strive to create an interactive and engaging learning environment. Through carefully curated lesson plans and PowerPoint presentations, I aim to make complex biological concepts accessible and understandable to students of all levels. My teaching philosophy revolves around fostering critical thinking skills and a deep understanding of biological principles. I encourage active participation and discussion, allowing students to explore concepts in depth and develop a holistic understanding of the subject. As a mentor, I am committed to supporting my students' academic growth and helping them achieve their full potential. Whether they aspire to pursue further studies in biology or pursue careers in related fields, I am dedicated to providing them with the knowledge, skills, and guidance they need to succeed. In conclusion, my passion for biology and dedication to teaching make me a valuable asset to any student seeking to excel in their CBSE biology studies. With my expertise and support, I am confident that students can unlock their academic potential and achieve their goals in the fascinating world of biology.
Top Tutor
Electronic City Electronic City Phase II, Bangalore
14 yrs of
Exp
800
per hour
Student's Home
Tutor's home
