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NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation and Matter

NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation and Matter

Edited By Vishal kumar | Updated on Sep 12, 2023 12:10 PM IST | #CBSE Class 12th

NCERT Solutions for Class 12 Physics Chapter 11 – Free PDF Download

NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation and Matter: Welcome to the updated NCERT solutions for Class 12. On this Careers360 page, you'll find comprehensive class 12 physics chapter 11 exercise solutions written by subject experts in straightforward English. These solutions are also available in PDF format, allowing students to access them offline at their convenience.

Particles and the wave nature of matter are discussed NCERT Class 12 Physics chapter 11. In the NCERT CBSE Class 12 Physics chapter 11 solutions, you will study questions related to both the particle and wave nature of light. Experimental study of the photoelectric effect is an important topic of NCERT for the CBSE board exam. Dual Nature of Matter and Radiation Class 12 discuss this topic.

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Try to solve all the questions in NCERT Class 12 Physics book yourself. If you are getting any doubt while solving questions, you can refer to chapter 11 physics class 12 ncert solutions. Understand all the graphs in NCERT syllabus for Class 12 Physics chapter 11. This will help to solve the questions from Dual Nature of Matter and Radiation Class 12 Physics NCERT chapter and to build an interest in the chapter 'Dual Nature of Radiation and Matter' .

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NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation and Matter

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NCERT Solutions for Class 12 Physics Chapter 11 Dual Nature of Radiation and Matter - Exercise Solutions


Q: 11.1 (a) Find the maximum frequency of X-rays produced by \small 30 \hspace{1mm}kV electrons.

Answer:

The X-Rays produced by electrons of 30 keV will have a maximum energy of 30 keV.

By relation,

\\eV_{0}=h\nu \\ \nu =\frac{eV_{0}}{h}\\ \nu =\frac{1.6\times 10^{-19}\times 30\times 10^{3}}{6.62\times 10^{-34}}\\ \nu =7.25\times 10^{18}\ Hz

Q: 11.1 (b) Find the minimum wavelength of X-rays produced by \small 30 \hspace{1mm}kV electrons.

Answer:

From the relation eV_{0}=h\nu , we have calculated the value of frequency in the previous questions, using that value and the following relation

\\\lambda =\frac{c}{\nu }\\ \lambda =\frac{3\times 10^{8}}{7.25\times 10^{18}}\\ \lambda =0.04\ nm

Q: 11.2 (b) The work function of caesium metal is 2.14\hspace{1mm}eV When light of frequency 6\times 10^1^4\hspace{1mm}Hz is incident on the metal surface, photoemission of electrons occurs. What is the stopping potential

Answer:

The stopping potential depends on the maximum Kinetic Energy of the emitted electrons. Since maximum Kinetic energy is equal to 0.34 eV, stopping potential is the maximum kinetic energy by charge equal to 0.34 V.

Q : 11.3 The photoelectric cut-off voltage in a certain experiment is 1.5\hspace{1mm}V . What is the maximum kinetic energy of photoelectrons emitted?

Answer:

Since the photoelectric cut-off voltage is 1.5 V. The maximum Kinetic Energy (eV) of photoelectrons emitted would be 1.5 eV.

KE max =1.5 eV

KE mac =2.4 \times 10 -19 J

Q: 11.4 (a) Monochromatic light of wavelength 632.8\hspace{1mm} nm is produced by a helium-neon laser. The power emitted is 9.42\hspace{1mm} mW . Find the energy and momentum of each photon in the light beam.

Answer:

The energy of photons is given by the relation

\\E=h\nu \\ E=\frac{hc}{\lambda }\\ E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{632\times 10^{-9}}\\ E=3.14\times 10^{-19}\ J

Momentum is given by De Broglie's Equation

\\p=\frac{h}{\lambda }\\ p=\frac{6.62\times 10^{-34}}{632.8\times 10^{-9}}\\ p=1.046\times 10^{-27}\ kg\ m\ s^{-1}

The energy of the photons in the light beam is 3.14 \times 10 -19 J and the momentum of the photons is 1.046 \times 10 -27 kg m s -1 .

Q: 11.4 (c) Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW. How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?

Answer:

Mass of Hydrogen Atom (m)=1.67 \times 10 -27 kg.

The speed at which hydrogen atom must travel to have momentum equal to that of the photons in the beam is v given by

\\v=\frac{p}{m}\\ v=\frac{1.05\times 10^{-27}}{1.67\times 10^{-27}}\\ v=0.628\ ms^{-1}

Q: 11.5 The energy flux of sunlight reaching the surface of the earth is 1.388\times 10^3 \hspace{1mm}W/m^2 . How many photons (nearly) per square metre are incident on the Earth per second? Assume that the photons in the sunlight have an average wavelength of 550 \hspace{1mm}nm ?

Answer:

Average Energy(E) of the photons reaching the surface of the Earth is given by

\\E=\frac{hc}{\lambda }\\ E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{550\times 10^{-9}} \\E=3.61\times 10^{-19}\ J

Energy flux(I) reaching the Earth's surface=1.388 \times 10 3 Wm -2

Number of photons(n) incident on Earth's surface per metre square is

\\n=\frac{I}{E}\\ n=\frac{1.388\times 10^{3}}{3.61\times 10^{-19}}\\ n=3.849\times 10^{21}\ m^{-2}

Q : 11.6 In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 * 10-15 Vs . Calculate the value of Planck’s constant.

Answer:

The slope of the cut-off voltage versus frequency of incident light is given by h/e where h is Plank's constant and e is an electronic charge.

h=slope\times e

h=4.12\times10^{-15}\times1.6\times10^{-19}

h=6.59210^{-34} Js

Q: 11.8 The threshold frequency for a certain metal is 3.3\times 10^1^4\hspace{1mm}Hz . If light of frequency 8.2\times 10^1^4\hspace{1mm}Hz is incident on the metal, predict the cutoff voltage for the photoelectric emission.

Answer:

Threshold frequency of the given metal( \nu _{0} )= 3.3\times 10^1^4\hspace{1mm}Hz

The work function of the given metal is

\\\phi _{0}=h\nu _{0}\\ \phi _{0}=6.62\times 10^{-34}\times 3.3\times 10^{-14}\\ \phi _{0}=2.18\times 10^{-19}\ J

The energy of the incident photons

\\E=h\nu \\ E=6.62\times 10^{-34}\times 8.2\times 10^{14}\\ E=5.42\times 10^{-19}\ J

Maximum Kinetic Energy of the ejected photo electrons is

\\E-\phi _{0}=3.24\times 10^{-19}\ J\\ E-\phi _{0}=2.025\ eV

Therefore the cut off voltage is 2.025 eV

Q : 11.9 The work function for a certain metal is 4.2\hspace{2mm}eV . Will this metal give photoelectric emission for incident radiation of wavelength 330\hspace{1mm}nm ?

Answer:

The energy of photons having 330 nm is

\\E=\frac{hc}{\lambda } \\ E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{330\times 10^{-9}\times 1.6\times 10^{-19}}\\ E=3.7\ eV

Since this is less than the work function of the metal there will be no photoelectric emission.

Q: 11.10 Light of frequency 7.21\times 10^1^4\hspace{1mm}Hz is incident on a metal surface. Electrons with a maximum speed of 6.0\times 10^5\hspace{1mm}m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?

Answer:

The energy of incident photons is E given by

\\E=h\nu \\ E=6.62\times 10^{-34}\times 7.21\times 10^{14}\\ E=4.77\times 10^{-19}\ J

Maximum Kinetic Energy of ejected electrons is

\\KE_{max}=\frac{1}{2}mv^{2}\\ KE_{max}=\frac{9.1\times 10^{-31}\times (6\times 10^{5})^{2}}{2} \\KE_{max}=1.64\times 10^{-19}\ J

Work Function of the given metal is

\phi _{0}=E-KE_{max}=3.13\times 10^{-19}\ J

The threshold frequency is therefore given by

\\\nu _{0}=\frac{\phi _{0}}{h}\\ \nu _{0}=4.728\times 10^{14}\ Hz

Q: 11.12 (a) Calculate the momentum of the electrons accelerated through a potential difference of 56 \hspace{1mm}V .

Answer:

On being accelerated through a potential difference of 56 V the electrons would gain a certain Kinetic energy K.

The relation between Kinetic Energy and Momentum(p) is given by

\\p=\sqrt{2mK}\\ p=\sqrt{2\times 9.1\times 10^{-31}\times 56\times 1.6\times 10^{-19}}\\ p=4.038\times 10^{-24}\ kg\ m\ s^{-1}

Q: 11.12 (b) Calculate the de Broglie wavelength of the electrons accelerated through a potential difference of 56\hspace{1mm}V .

Answer:

De Broglie wavelength is given by the De Broglie relation as

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{4.038\times 10^{-24}}\\ \lambda =0.164\ nm

the wavelength is 0.164 nm

Q: 11.13 (a) What is the momentum of an electron with kinetic energy of 120\hspace{1mm}eV .

Answer:

The relation between momentum and kinetic energy is

\\p=\sqrt{2mK}\\ p=\sqrt{2\times 9.1\times 10^{-31}\times 120\times 1.6\times 10^{-19}}\\ p=5.911\times 10^{-24}\ kg\ m\ s^{-1}

Q: 11.13 (b) What is the speed of an electron with kinetic energy of 120\hspace{1mm}eV .

Answer:

The relation between speed and kinetic energy of a particle is

\\v=\sqrt{\frac{2K}{m}}\\ v=\sqrt{\frac{2\times 120\times 1.6\times 10^{-19}}{9.1\times 10^{-31}}}\\ v=6.495\times 10^{6}\ m\ s^{-1}

Q: 11.13 (c) What is the de Broglie wavelength of an electron with kinetic energy of 120\hspace{1mm}eV

Answer:

De Broglie wavelength is given by

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{5.911\times 10^{-24}}\\ \lambda =1.12\times 10^{-10}\ m\\

The de Broglie wavelength associated with the electron is 0.112 nm

Q: 11.14 (a) The wavelength of light from the spectral emission line of sodium is 589\hspace{1mm}nm . Find the kinetic energy at which an electron would have the same de Broglie wavelength.

Answer:

The momentum of a particle with de Broglie wavelength of 589 nm is

\\p=\frac{h}{\lambda }\\ p=\frac{6.62\times 10^{-34}}{589\times 10^{-9}}\\ p=1.12\times 10^{-27}\ kg\ m\ s^{-1}

The Kinetic Energy of an electron moving with above-mentioned momentum is

\\K=\frac{p^{2}}{2m_{e}}\\ K=\frac{(1.12\times 10^{-27})^{2}}{2\times 9.1\times 10^{-31}}\\ K=6.89\times 10^{-25}\ J

Q: 11.14 (b) The wavelength of light from the spectral emission line of sodium is 589\hspace{1mm}nm . Find the kinetic energy at which a neutron would have the same de Broglie wavelength.

Answer:

The momentum of the neutron would be the same as that of the electron.

The kinetic energy of neutron would be

\\K=\frac{p^{2}}{2m_{n}}\\ K=\frac{(1.12\times 10^{-27})^{2}}{2\times 1.675\times 10^{-27}}\\ K=3.74\times 10^{-28}\ J

Q: 11.15 (a) What is the de Broglie wavelength of a bullet of mass 0.040\hspace{1mm}kg travelling at the speed of 1.0\hspace{1mm}km/s .

Answer:

The momentum of the bullet is

\\p=mv\\ p=0.04\times 10^{3}\\ p=40\ kg\ m\ s^{-1}

De Broglie wavelength is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{40}\\ \lambda =1.655\times 10^{-35}\ m

Q: 11.15 (b) What is the de Broglie wavelength of a ball of mass 0.060\hspace{1mm}kg moving at a speed of 1.0\hspace{1mm}m/s .

Answer:

The momentum of the ball is

\\p=mv\\ p=0.06\ kg\ m\ s^{-1}

De Broglie wavelength is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{0.06}\\ \lambda =1.1\times 10^{-32}\ m

Q: 11.15 (c) What is the de Broglie wavelength of a dust particle of mass 1.0\times 10^-^9\hspace{1mm}kg drifting with a speed of 2.2\hspace{1mm}m/s ?

Answer:

The momentum of the dust particle is

\\p=mv\\ p=10^{-9}\times 2.2\\ p=2.2\times 10^{-9} kg\ m\ s^{-1}

De Broglie wavelength is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{2.2\times 10^{-9} }\\ \lambda =3.01\times 10^{-25}\ m

Q: 11.16 (a) An electron and a photon each have a wavelength of 1.00\hspace{1mm}nm . Find their momenta.

Answer:

Their momenta depend only on the de Broglie wavelength, therefore, it will be the same for both the electron and the photon

\\p=\frac{h}{\lambda }\\ p=\frac{6.62\times 10^{-34}}{10^{-9}}\\ p=6.62\times 10^{-25}kg\ m\ s^{-1}

Q: 11.16 (b) An electron and a photon each have a wavelength of 1.00\hspace{1mm}nm . Find the energy of the photon.

Answer:

The energy of the photon is given by

\\E=\frac{hc}{\lambda }\\

h is the Planks constant, c is the speed of the light and lambda is the wavelength

E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{10^{-9}}\\

E=1.86\times 10^{-16}\ J

Q: 11.16 (c) An electron and a photon each have a wavelength of 1.00\hspace{1mm}nm . Find the kinetic energy of electron.

Answer:

The kinetic energy of the electron is. In the below equation p is the momentum

\\K=\frac{p^{2}}{2m_{e}}\\ K=\frac{(6.62\times 10^{-25})^{2}}{2\times 9.1\times 10^{-31}}\\ K=2.41\times 10^{-19}\ J

Q: 11.17 (a) For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.4*10-10 m?

Answer:

For the given wavelength momentum of the neutron will be p given by

\\p=\frac{h}{\lambda }\\ p=\frac{6.62\times 10^{-34}}{1.4\times 10^{-10}}\\ p=4.728\times 10^{-24}kg\ m\ s^{-1}

The kinetic energy K would therefore be

\\K=\frac{p^{2}}{2m}\\ K=\frac{(4.728\times 10^{-24})^{2}}{2\times 1.675\times 10^{-27}}\\ K=6.67\times 10^{-21}J

Q : 11.17 (b) Also find the de Broglie wavelength of a neutron, in thermal equilibrium with matter, having an average kinetic energy of (3/2)\hspace{1mm}kT at 300\hspace{1mm}K .

Answer:

The kinetic energy of the neutron is

\\K=\frac{3}{2}kT\\ K=\frac{3}{2}\times 1.38\times 10^{-23}\times 300\\ K=6.21\times 10^{-21} J

Where k Boltzmann's Constant is 1.38 \times 10 -23 J/K

The momentum of the neutron will be p

\\p=\sqrt{2m_{N}K}\\ p=\sqrt{2\times 1.675\times 10^{-27}\times 6.21\times 10^{-21}}\\ p=4.56\times 10^{-24}kg\ m\ s^{-1}

Associated De Broglie wavelength is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{4.56\times 10^{-24}}\\ \lambda =1.45\times 10^{-10} m

De Broglie wavelength of the neutron is 0.145 nm.

Q: 11.18 Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).

Answer:

For a photon we know that it's momentum (p) and Energy (E) are related by following equation

E=pc

we also know

E=h\nu

Therefore the De Broglie wavelength is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{h}{E/c}\\ \lambda =\frac{hc}{h\nu }\\ \lambda =\frac{c}{\nu }

The above de Broglie wavelength is equal to the wavelength of electromagnetic radiation.

Q: 11.19 What is the de Broglie wavelength of a nitrogen molecule in air at 300\hspace{1mm}K ? Assume that the molecule is moving with the root-mean-square speed of molecules at this temperature. (Atomic mass of nitrogen =14.0076\hspace{1mm} \mu )

Answer:

Since the molecule is moving with the root-mean-square speed the kinetic energy K will be given by

K=3/2 kT where k is the Boltzmann's constant and T is the absolute Temperature

In the given case Kinetic Energy of a Nitrogen molecule will be

\\K=\frac{3}{2}\times 1.38\times 10^{-23}\times 300\\ K=6.21\times 10^{-21}J

Mass of Nitrogen molecule = 2 \times 14.0076 \times 1.66 \times 10 -27 =4.65 \times 10 -26 kg

The momentum of the molecule is

\\p=\sqrt{2mK}\\ p=\sqrt{2\times 4.65\times 10^{-26}\times 6.21\times 10^{-21}}\\ p=2.4\times 10^{-23}kg\ m\ s^{-1}

Associated De Broglie wavelength is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{6.62\times 10^{-34}}{2.4\times 10^{-23}}\\ \lambda= 2.75\times 10^{-11}\ m

The nitrogen molecule will have a De Broglie wavelength of 0.0275 nm.

NCERT solutions for class 12 physics chapter 11 dual nature of radiation and matter additional exercise

Q: 11.20 (a) Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of 500\hspace{1mm}V with respect to the emitter. Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/m is given to be 1.76\times 10^1^1\hspace{1mm}kg^-^1 .

Answer:

The kinetic energy of an electron accelerated through Potential Difference V is K=eV where e the electronic charge.

Speed of the electrons after being accelerated through a potential difference of 500 V will be

\\v=\sqrt{\frac{2K}{m_{e}}}\\ v=\sqrt{\frac{2eV}{m_{e}}}\\ v=\sqrt{2\times 1.76\times 10^{11}\times 500}\\ v=1.366\times 10^{7}ms^{-1} Specific charge is e/m e =1.366 \times 10 11 C/kg

Q: 11.20 (b) Use the same formula you employ in (a) to obtain electron speed for an collector potential of 10 MV. Do you see what is wrong? In what way is the formula to be modified?

Answer:

Using the same formula we get the speed of electrons to be 1.88 \times 10 9 m/s. This is wrong because the speed of the electron is coming out to be more than the speed of light. This discrepancy is occurring because the electron will be travelling at very large speed and in such cases(relativistic) the mass of the object cannot be taken to be the same as the rest mass.

In such a case

m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}

where m is the relativistic mass, m 0 is the rest mass of the body, v is the very high speed at which the body is travelling and c is the speed of light.

Q : 11.21 (a) A monoenergetic electron beam with electron speed of 5.20\times 10^6\hspace{1mm}ms^-^1 is subject to a magnetic field of \dpi{100} 1.30\times 10^{-4}\hspace{1mm}T normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals \dpi{100} 1.76\times 10^1^1\hspace{1mm}Ckg^-^1 .

Answer:

The force due to the magnetic field on the electron will be F b =evB (since the angle between the velocity and magnetic field is 90 o )

This F b acts as the centripetal force required for circular motion. Therefore

\\F_{b}=\frac{mv^{2}}{r}\\ evB=\frac{mv^{2}}{r}\\ r=\frac{mv}{eB}\\ r=\frac{5.2\times 10^{6}}{1.76\times 10^{11}\times 1.3\times 10^{-4}}\\ r=0.227 m

Q: 11.21 (b) Is the formula you employ in (a) valid for calculating radius of the path of a 20\hspace{1mm}MeV electron beam? If not, in what way is it modified?

Answer:

The formula used in (a) can not be used. As the electron would be travelling at a very high speed we can not take its mass to be equal to its rest mass as its motion won't be within the non-relativistic limits.

The value for the mass of the electron would get modified to

m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}

where m is the relativistic mass, m 0 is the rest mass of the body, v is the very high speed at which the body is travelling and c is the speed of light.

The radius of the circular path would be

r=\frac{m_{e}v}{eB\sqrt{1-\frac{v^{2}}{c^{2}}}}

Q: 11.22 An electron gun with its collector at a potential of 100\hspace{1mm}V fires out electrons in a spherical bulb containing hydrogen gas at low pressure ( \sim 10^-^2\hspace{1mm}mm of Hg). A magnetic field of 2.83\times 10^4\hspace{1mm}T curves the path of the electrons in a circular orbit of radius 12.0\hspace{1mm}cm (The path can be viewed because the gas ions in the path focus the beam by attracting electrons, and emitting light by electron capture; this method is known as the ‘fine beam tube’ method.) . Determine e/m from the data.

Answer:

The kinetic energy of an electron after being accelerated through a potential difference of V volts is eV where e is the electronic charge.

The speed of the electron will become

v=\sqrt{\frac{2eV}{m_{e}}}

Since the magnetic field curves, the path of the electron in circular orbit the electron's velocity must be perpendicular to the magnetic field.

The force due to the magnetic field is therefore F b =evB

This magnetic force acts as a centripetal force. Therefore

\\\frac{m_{e}v^{2}}{r}=evB\\ \\\frac{m_{e}v}{r}=eB\\ \frac{m_{e}}{r}\times \sqrt{\frac{2eV}{m_{e}}}=eB\\ \sqrt{\frac{e}{m_{e}}}=\frac{\sqrt{2V}}{Br}\\ \frac{e}{m_{e}}=\frac{2V}{r^{2}B^{2}}\\ \frac{e}{m_{e}}=\frac{2\times 100}{(2.83\times 10^{-4})^{2}\times (0.12)^{2}}\\ \frac{e}{m_{e}}=1.73\times 10^{11}C\ kg^{-1}

Q: 11.23 (b) From your answer to (a), guess what order of accelerating voltage (for electrons) is required in such a tube?

Answer:

In such a tube where X-ray of energy 27.6 keV is to be produced the electrons should be having an energy about the same value and therefore accelerating voltage should be of order 30 KeV.

Q: 11.25 (b) Estimating the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about photons! The second number tells you why our eye can never ‘count photons’, even in barely detectable light.The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive (\sim10^-^1^0\hspace{1mm}Wm^2). Take the area of the pupil to be about 0.4 cm2, and the average frequency of white light to be about 6\times10^1^4\ Hz.

Answer:

The minimum perceivable intensity of white light(I)=10 -10 Wm -2

Area of the pupil(A)=0.4 cm 2 =4 \times 10 -5 m 2

Power of light falling on our eyes at minimum perceivable intensity is P

P=IA

P=10 -10 \times 4 \times 10 -5

P=4 \times 10 -15 W

The average frequency of white light( \nu )=6 \times 10 14 Hz

The average energy of a photon in white light is

\\E=h\nu \\ E=6.62\times 10^{-34}\times 6\times 10^{14}\\ E=3.972\times 10^{-19} J

Number of photons reaching our eyes is n

\\n=\frac{P}{E}\\ n=\frac{4\times 10^{-15}}{3.972\times 10^{-19}}\\ n=1.008\times 10^{4}s^{-1}

Q: 11.26 Ultraviolet light of wavelength 2271\hspace{1mm}\dot{A} from a 100\hspace{1mm}W mercury source irradiates a photo-cell made of molybdenum metal. If the stopping potential is -1.3\hspace{1mm}V , estimate the work function of the metal. How would the photo-cell respond to a high intensity (\sim 10^5\hspace{1mm}Wm^2) red light of wavelength 6382\hspace{1mm}\dot{A} produced by a He-Ne laser?

Answer:

The energy of the incident photons is E given by

\\E=\frac{hc}{\lambda }\\ E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{2271\times 10^{-10}\times 1.6\times 10^{-19}}\\ E=5.465\ eV

Since stopping potential is -1.3 V work function is

\\\phi _{0}=5.465-1.3\\ \phi _{0}=4.165 eV

The energy of photons which red light consists of is E R

\\E_{R}=\frac{hc}{\lambda _{R}}\\ E_{R}=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{6382\times 10^{-10}\times 1.6\times 10^{-19}}\\ E_{R}=1.945eV

Since the energy of the photons which red light consists of have less energy than the work function, there will be no photoelectric emission when they are incident.

Q: 11.27 Monochromatic radiation of wavelength 640.2\hspace{1mm}nm (1\hspace{1mm}nm=10^-^9\hspace{1mm}m) from a neon lamp irradiates photosensitive material made of caesium on tungsten. The stopping voltage is measured to be 0.54\hspace{1mm}V . The source is replaced by an iron source and its 427.2\hspace{1mm}nm line irradiates the same photo-cell. Predict the new stopping voltage.

Answer:

The wavelength of photons emitted by the neon lamp=640.2 nm

The energy of photons emitted by the neon lamp is E given by

\\E_{1}=\frac{hc}{\lambda }\\ E_{1}=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{640.2\times 10^{-9}\times 1.6\times 10^{-19}}\\ E_{1}=1.939eV

Stopping potential is 0.54 V

Work function is therefore

\\\phi _{0}=1.939-0.54\\\phi _{0}=1.399 eV

The wavelength of photons emitted by the iron source=427.2 nm

The energy of photons emitted by the ion source is

\\E_{2}=\frac{hc}{\lambda }\\ E_{2}=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{427.2\times 10^{-9}\times 1.6\times 10^{-19}}\\ E_{2}=2.905eV

New stopping voltage is

\\E_{2}-\phi _{0}=2.905-1.399=1.506V

Q: 11.28 A mercury lamp is a convenient source for studying frequency dependence of photoelectric emission since it gives a number of spectral lines ranging from the UV to the red end of the visible spectrum. In our experiment with rubidium photo-cell, the following lines from a mercury source were used:

\lambda _1=3650\hspace{1mm}\dot{A},\lambda _2=4047\hspace{1mm}\dot{A}, \lambda _3=4358\hspace{1mm}\dot{A}, \lambda _4=5461\hspace{1mm}\dot{A},\lambda _5=6907\hspace{1mm}\dot{A}.

The stopping voltages, respectively, were measured to be

V _0_1=1.28\hspace{1mm}V,V _0_2=0.95\hspace{1mm}V,V _0_3=0.74\hspace{1mm}V, V_0_4=0.16\hspace{1mm}V,V _0_5=0\hspace{1mm}V.

Determine the value of Planck’s constant h , the threshold frequency and work function for the material.

Answer:

\\h\nu =\phi _{0}+eV\\ V=(\frac{h}{e})\nu -\phi _{0}\\

where V is stopping potential, h is planks constant, e is electronic charge, \nu is frequency of incident photons and \phi _{0} is work function of metal in electron Volts.

To calculate the planks constant from the above date we plot the stopping potential vs frequency graph

\nu_{1}=\frac{c}{\lambda_{1} }=\frac{3\times 10^{8}}{3650\times 10^{-10}}=8.219\times 10^{14}\ Hz

\nu_{2}=\frac{c}{\lambda_{2} }=\frac{3\times 10^{8}}{4047\times 10^{-10}}=7.412\times 10^{14}\ Hz

\nu_{3}=\frac{c}{\lambda_{3} }=\frac{3\times 10^{8}}{4358\times 10^{-10}}=6.884\times 10^{14}\ Hz

\nu_{4}=\frac{c}{\lambda_{4} }=\frac{3\times 10^{8}}{5461\times 10^{-10}}=5.493\times 10^{14}\ Hz

\nu_{5}=\frac{c}{\lambda_{5} }=\frac{3\times 10^{8}}{6907\times 10^{-10}}=4.343\times 10^{14}\ Hz

V _0_1=1.28\hspace{1mm}V,V _0_2=0.95\hspace{1mm}V,V _0_3=0.74\hspace{1mm}V, V_0_4=0.16\hspace{1mm}V,V _0_5=0\hspace{1mm}V.

The plot we get is

1646200511071

From the above figure, we can see that the curve is almost a straight line.


The slope of the above graph will give the Plank's constant divided by the electronic charge. Planks constant calculated from the above chart is

\\h=\frac{\left ( 1.28-0.16 \right )\times 1.6\times 10^{-19}}{(8.214-5.493)\times 10^{14}}\\ h=6.573\times 10^{-34}\ Js

Planks constant calculated from the above chart is therefore 6.573\times 10^{-34}\ Js

Q: 11.29 The work function for the following metals is given: Na:2.75\hspace{1mm}eV,K:2.30\hspace{1mm}eV,Mo:4.17\hspace{1mm}eV,Ni:5.15\hspace{1mm}eV.

Which of these metals will not give photoelectric emission for a radiation of wavelength 3300\hspace{1mm}\dot{A} from a He-Cd laser placed 1\hspace{1mm}m away from the photocell? What happens if the laser is brought nearer and placed 50\hspace{1mm}cm away?

Answer:

The wavelength of the incident photons= 3300\dot{A}

The energy of the incident photons is

\\E=\frac{hc}{\lambda }\\ E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{3300\times 10^{-10}\times 1.6\times 10^{-19}}\\ E=3.16 eV

Mo and Ni will not give photoelectric emission for radiation of wavelength 3300\hspace{1mm}\dot{A} from a He-Cd .

If the laser is brought nearer no change will be there in case of Mo and Ni although there will be more photoelectrons in case of Na and K.

Q: 11.30 Light of intensity 10^-^5\hspace{1mm}Wm^-^2 falls on a sodium photo-cell of surface area 2\hspace{1mm}cm^2 . Assuming that the top 5 layers of sodium absorb the incident energy, estimate time required for photoelectric emission in the wave-picture of radiation. The work function for the metal is given to be about 2\hspace{1mm}eV . What is the implication of your answer? (Effective atomic area of a sodium atom = 10 -20 m 2 )

Answer:

Intensity of Incident light(I) = 10^-^5\hspace{1mm}Wm^-^2

The surface area of the sodium photocell (A)=2 cm 2 = 2 \times 10 -4 m 2

The rate at which energy falls on the photo cell=IA=2 \times 10 - 9 W

The rate at which each of the 5 surfaces absorbs energy= IA/5=4 \times 10 -10 W

Effective atomic area of a sodium atom (A')= 10 -20 m 2

The rate at which each sodium atom absorbs energy is R given by

\\R=\frac{IA}{5}\times \frac{A'}{A}\\ R=\frac{10^{-5}\times 10^{-20}}{5}\\ R=2\times 10^{-26}J/s

The time required for photoelectric emission is

\\t=\frac{\phi _{0}}{R}\\ t=\frac{2\times 1.6\times 10^{-19}}{2\times 10^{-26}}\\ t=1.6\times 10^{7}s\\ t\approx 0.507 \ years

Q: 11.31 Crystal diffraction experiments can be performed using X-rays, or electrons accelerated through appropriate voltage. Which probe has greater energy? (For quantitative comparison, take the wavelength of the probe equal to 1\hspace{1mm}\dot{A}, which is of the order of inter-atomic spacing in the lattice) (m_e=9.11\times 10^-^3^1\hspace{1mm}kg).

Answer:

According to De Broglie's equation

p=\frac{h}{\lambda }

The kinetic energy of an electron with De Broglie wavelength 1\hspace{1mm}\dot{A} is given by

\\K=\frac{p^{2}}{2m_{e}}\\ K=\frac{h^{2}}{\lambda ^{2}2m_{e}} \\K=\frac{(6.62\times 10^{-34})^{2}}{2\times 10^{-20}\times 9.11\times 10^{-31}\times 1.6\times 10^{-19}}\\ K=149.375\ eV

The kinetic energy of photon having wavelength 1\hspace{1mm}\dot{A} is

\\E=\frac{hc}{\lambda }\\ E=\frac{6.62\times 10^{-34}\times 3\times 10^{8}}{10^{-10}\times 1.6\times 10^{-19}}\\ E=12.375keV

Therefore for the given wavelength, a photon has much higher energy than an electron.

Q:11.32 (a) Obtain the de Broglie wavelength of a neutron of kinetic energy 150\hspace{1mm}eV . As you have seen in Exercise 11.31, an electron beam of this energy is suitable for crystal diffraction experiments. Would a neutron beam of the same energy be equally suitable? Explain.(m_e=9.11\times 10^-^3^1\hspace{1mm}kg).

Answer:

Kinetic energy of the neutron(K)=150eV

De Broglie wavelength associated with the neutron is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{h}{\sqrt{2m_{N}K }}\\ \lambda =\frac{6.62\times 10^{-34}}{\sqrt{2\times 1.675\times 10^{-27}\times 150\times 1.6\times 10^{-19}}}\\ \lambda =2.327\times 10^{-12}m

Since an electron beam with the same energy has a wavelength much larger than the above-calculated wavelength of the neutron, a neutron beam of this energy is not suitable for crystal diffraction as the wavelength of the neutron is not of the order of the dimension of interatomic spacing.

Q: 11.32 (b) Obtain the de Broglie wavelength associated with thermal neutrons at room temperature (27\hspace{1mm}^{\circ}C ) . Hence explain why a fast neutron beam needs to be thermalised with the environment before it can be used for neutron diffraction experiments.

Answer:

Absolute temperature = 273+27=300K

Boltzmann's Constant=1.38 \times 10 -23 J/mol/K

The de Broglie wavelength associated with the neutron is

\\\lambda =\frac{h}{p}\\ \lambda =\frac{h}{\sqrt{2m_{N}K}} \\\lambda =\frac{h}{\sqrt{3kT}}\\ \lambda =\frac{6.62\times 10^{-34}}{\sqrt{3\times 1.38\times 10^{-23}\times 300}}\\ \lambda =1.446 \dot{A}

Since this wavelength is comparable to the order of interatomic spacing of a crystal it can be used for diffraction experiments. The neutron beam is to be thermalised so that its de Broglie wavelength attains a value such that it becomes suitable for the crystal diffraction experiments.

Q:11.33 An electron microscope uses electrons accelerated by a voltage of 50\hspace{1mm}kV . Determine the de Broglie wavelength associated with the electrons. If other factors (such as numerical aperture, etc.) are taken to be roughly the same, how does the resolving power of an electron microscope compare with that of an optical microscope which uses yellow light?

Answer:

The potential difference through which electrons are accelerated(V)=50kV.

Kinetic energy(K) of the electrons would be eV where e is the electronic charge

The De Broglie wavelength associated with the electrons is

\\\lambda =\frac{h}{\sqrt{2m_{e}K}}\\ \lambda =\frac{6.62\times 10^{-34}}{\sqrt{2\times 9.11\times 10^{-31}\times 1.6\times 10^{-19}\times 50000}}\\ \lambda =5.467\times 10^{-12} m

The wavelength of yellow light = 5.9 \times 10 -7 m

The calculated De Broglie wavelength of the electron microscope is about 10 5 more than that of yellow light and since resolving power is inversely proportional to the wavelength the resolving power of electron microscope is roughly 10 5 times than that of an optical microscope.

Q: 11.35 Find the typical de Broglie wavelength associated with a He atom in helium gas at room temperature ( 27\hspace{1mm}^{\circ}C ) and 1 atm pressure, and compare it with the mean separation between two atoms under these conditions.

Answer:

The kinetic energy K of a He atom is given by

K=\frac{3}{2}kT

m He i.e. mass of one atom of He can be calculated as follows

\\m_{He}=\frac{4\times 10^{-3}}{N_{A}}\\ =\frac{4\times 10^{-3}}{6.023\times 10^{23}}\\ m_{He}=6.64\times 10^{-27}\ kg (N A is the Avogadro's Number)

De Broglie wavelength is given by

\\\lambda =\frac{h}{p}\\ \lambda =\frac{h}{\sqrt{2m_{He}K}}\\ \lambda =\frac{h}{\sqrt{3m_{He}kT}}\\ \lambda =\frac{6.62\times 10^{-34}}{\sqrt{3\times 6.64\times 10^{-27}\times 1.38\times 10^{-23}\times 300}}\\ \lambda =7.27\times 10^{-11}\ m

The mean separation between two atoms is given by the relation

\\d=\left ( \frac{V}{N} \right )^{\frac{1}{3}}\\

From the ideal gas equation we have

\\PV=nRT\\ PV=\frac{NRT}{N_{A}}\\ \frac{V}{N}=\frac{RT}{PN_{A}}

The mean separation is therefore

\\d=\left ( \frac{RT}{PN_{A}} \right )^{\frac{1}{3}}\\ d=\left ( \frac{kT}{P} \right )^{\frac{1}{3}}\\ d=\left ( \frac{1.38\times 10^{-23}\times 300}{1.01\times 10^{5}} \right )^{\frac{1}{3}}\\ d=3.35\times 10^{-9}\ m

The mean separation is greater than the de Broglie wavelength.

Q: 11.36 Compute the typical de Broglie wavelength of an electron in a metal at 27\hspace{1mm}^{\circ}C and compare it with the mean separation between two electrons in a metal which is given to be about 2*10-10m.

Answer:

The de Broglie wavelength associated with the electrons is

\\\lambda =\frac{h}{\sqrt{3m_{e}kT}}\\ \lambda =\frac{6.62\times 10^{-34}}{\sqrt{3\times 9.1\times 10^{-31}\times 1.38\times 10^{-23}\times 300}}\\ \lambda =6.2\times 10^{-9}\ nm

The de Broglie wavelength of the electrons is comparable to the mean separation between two electrons.

Answer the following questions:

Q: 11.37 (a ) Quarks inside protons and neutrons are thought to carry fractional charges \dpi{100} [(+2/3)e;(-1/3)e] . Why do they not show up in Millikan’s oil-drop experiment?

Answer:

Quarks are thought to be tight within a proton or neutron by forces which grow tough if one tries to pull them apart. That is event though fractional charges may exist in nature, the observable charges are still integral multiples of the charge of the electron

Answer the following questions:

Q: 11.37 (b) What is so special about the combination e/m ? Why do we not simply talk of e and m separately?

Answer:

The speed of a charged particle is given by the relations

v=\sqrt{2K\left ( \frac{e}{m} \right )}

or

v=Br\left ( \frac{e}{m} \right )

As we can see the speed depends on the ratio e/m it is of such huge importance.

Answer the following questions:

Q: 11.37 (c) Why should gases be insulators at ordinary pressures and start conducting at very low pressures?

Answer:

At ordinary pressure due to a large number of collisions among themselves, the gases have no chance of reaching the electrodes while at very low pressure these collisions decrease exponentially and the gas molecules have a chance of reaching the respective electrodes and therefore are capable of conducting electricity.

Answer the following questions:

Q: 11.37 (d) Every metal has a definite work function. Why do all photoelectrons not come out with the same energy if incident radiation is monochromatic? Why is there an energy distribution of photoelectrons?

Answer:

The work function is defined as the minimum energy below which an electron will never be ejected from the metal. But when photons with high energy are incident it is possible that electrons from different orbits get ejected and would, therefore, come out of the atom with different kinetic energies.

Answer the following questions:

Q : 11.37 (e) The energy and momentum of an electron are related to the frequency and wavelength of the associated matter wave by the relations:
E=hv,p=\frac{h}{\lambda }
But while the value of \lambda is physically significant, the value of v (and therefore, the value of the phase speed v \lambda ) has no physical significance. Why?

Answer:

The absolute energy has no significance because of the reference point being arbitrary and thus the inclusion of an arbitrary constant rendering the value of \nu\lambda and . \nu to have no physical significance as such.

The group speed is defined as

V_{G}=\frac{h}{\lambda m}

Due to the significance of the group speed the absolute value of wavelength has physical significance.

Dual Nature of Matter and Radiation Class 12 Solutions

There are 37 questions discussed in Dual Nature of Radiation and Matter NCERT solutions. The exercise questions of ch 11 Physics Class 12 is divided into two parts: exercise and additional exercise. Questions number 20 to 37 of Dual Nature of Matter and Radiation Class 12 are part of additional exercises.

NCERT Solutions for Class 12 Physics Chapter-wise

Below, we have provided the exercise-wise NCERT textbook solutions and questions:

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Class 12 Physics ch 11 NCERT Solutions: Important Formulas and Diagrams

  • Energy of photon(E): E=hv=hc

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Where: v= frequency of photon and = wavelength of photon

  • Momentum of photon: p= h

Where: h= plank’s constant and is wavelength of photon

  • Velocity of charged particle passing through a potential v:

v= 2eVm

Where: e= charge of electron and m is the mass of electron

  • Einstein’s Photoelectric Equation: Energy Quantum of Radiation

1694500175331

Where: Kmax is the maximum kinetic energy of the emitted electron and is the work function of metal

Dual Nature of Matter and Radiation Class 12-Topics

The topic discussed in the Class 12 NCERT Physics chapter Dual Nature of Matter and Radiation are Electron Emission, Photoelectric Effect and It's Experimental Study, Wave Theory of Light and Photoelectric Effect, Einsteins Photo Electric Equation, Particle Nature of Light, Wave Nature of Matter and Davisson and Germer Experiment.

Importance of NCERT Solutions for Class 12 Physics Chapter 11

  • The chapter Dual Nature of Radiation and Matter comes under modern Physics part of Class 12 NCERT book. NCERT Class 12 Physics Solutions is important for competitive exams like NEET and JEE Mains.
  • As far as CBSE 12th board exam is considered, 3 to 5 marks questions are expected from this chapter. In CBSE board exam 2019, three marks questions were asked from this chapter.
  • The CBSE NCERT solutions for Class 12 Physics chapter 11 Dual Nature of Radiation and Matter will help to score well in the board and competitive exams.

Key Feathers of Class 12 Physics ch 11 NCERT Solutions

  1. Comprehensive Coverage: These class 12 physics chapter 11 exercise solutions encompass all the topics and questions presented in Chapter 11, ensuring a thorough understanding of chapter 11.

  2. Detailed Explanations: Each chapter 11 physics class 12 ncert solutions offers detailed step-by-step explanations, simplifying complex concepts for students.

  3. Clarity and Simplicity: The dual nature of radiation and matter ncert solutions are presented in clear and simple language, ensuring ease of understanding.

  4. Practice Questions: Exercise questions are included for practice and self-assessment.

  5. Exam Preparation: These class 12 chapter 11 physics ncert solutions are essential for board exam preparation and provide valuable support for competitive exams.

  6. Foundation for Advanced Study: Concepts covered in this chapter serve as the foundation for more advanced studies in Radiation and Matter physics and related fields.

  7. Free Access: These class 12 physics chapter 11 ncert solutions are available for free, ensuring accessibility to all students.

Also Check NCERT Books and NCERT Syllabus here:

NCERT solutions subject - wise

Also, check NCERT Exemplar Class 12 Solutions

Frequently Asked Question (FAQs)

1. What is the weightage of chapter Dual Nature of Radiation and Matter for CBSE board exam?

Around 3 to 5 marks questions can be expected from the NCERT Class 12 chapter 11 for CBSE board exam according to the previous year papers. Students can also use the help of NCERT Exemplar Problems for Physics for more problems.

2. Which topics are important from NCERT Class 12 Physics chapter 11 solutions?

All the topics of the chapter are important. Understand all the graphs mentioned in the NCERT book. Also understand all the formulas listed in the chapter. The CBSE papers are based on the NCERT Syllabus. So all the concepts listed in the chapter are important.

3. What is wave-particle duality in dual nature of matter and radiation class 12?

Wave-particle duality refers to the ability of particles of matter, such as electrons and photons, to exhibit both wave-like and particle-like behaviour. This means that they can display properties of both waves, such as diffraction and interference, and particles, such as quantization and the photoelectric effect.

4. How dual nature of light class 12 is important for NEET?

dual nature ncert solutions physics is important for NEET as it helps students understand the behaviour of particles and light on a microscopic level, including the properties of atoms and molecules, and how light interacts with different materials. This understanding is crucial for understanding key topics in NEET physics such as the behaviour of electrons in solids and the behaviour of light in different materials.

5. What do you mean by photoelectric effect in class 12 physics chapter 11 ncert solutions?

The photoelectric effect is a phenomenon in which electrons are emitted from a metal surface when light shines on it. The effect occurs because the light is made up of individual packets of energy, called photons, which transfer energy to electrons in the metal surface, causing them to be emitted. 



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hello,

Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.

I hope this was helpful!

Good Luck

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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