Imagine removing one electron from
Question: Imagine removing one electron from He4and He3. Their energy levels, as worked out on the basis of Bohr model will be very close. Explain why. Solution: The energies of He4 and He3 will be very close according to the Bohr model because both have 1 electron as in case of the hydrogen atom and the nucleus is 4 times heavier than the H-atoms. Therefore, the stability of these atoms remains close to the H atom and so is their energy levels....
Read More →The mass of an H-atom is less than the sum
Question: The mass of an H-atom is less than the sum of the masses of a proton and electron. Why is this? Solution: We know that, E = ∆mc2 The mass of the hydrogen atom is mp + mp BE/c2 Where BE = 13.6 eV....
Read More →The simple Bohr model is not applicable to He4 atom because
Question: The simple Bohr model is not applicable to He4atom because (a) He4is an inert gas (b) He4has neutrons in the nucleus (c) He4has one more electron (d) electrons are not subject to central forces Solution: (c) He4has one more electron (d) electrons are not subject to central forces...
Read More →Prove the following
Question: Let $E_{n}=\frac{-1}{8 \epsilon_{0}^{2}} \frac{m e^{4}}{n^{2} h^{2}}$be the energy of the nth level of H-atom. If all the H-atoms are in the ground state and radiation of frequency (E2-E1)/h falls on it (a) it will not be absorbed at all (b) some of the atoms will move to the first excited state (c) all atoms will be excited to the n = 2 state (d) no atoms will make a transition to the n = 3 state Solution: (b) some of the atoms will move to the first excited state (d) no atoms will ma...
Read More →The Balmer series for the H-atom can be observed
Question: The Balmer series for the H-atom can be observed (a) if we measure the frequencies of light emitted when an excited atom falls to the ground state (b) if we measure the frequencies of light emitted due to transitions between excited states and the first excited state (c) in any transition in a H-atom (d) as a sequence of frequencies with the higher frequencies getting closely packed Solution: (b) if we measure the frequencies of light emitted due to transitions between excited states a...
Read More →What is the probability that in a group of two people, both
Question: What is the probability that in a group of two people, both will have the same birthday, assuming that there are 365 days in a year and no one has his/her birthday on 29th February? Solution: We know that, Probability of occurring $=1$ - the probability of not occurring Let's calculate for the probability of not occurring, i.e. probability such that both of them don't have a birthday on the same day. For suppose the first person has a birthday on a particular day then the other person ...
Read More →The Bohr model for the spectra of an H-atom
Question: The Bohr model for the spectra of an H-atom (a) will not be applicable to hydrogen in the molecular from (b) will not be applicable as it is for a He-atom (c) is valid only at room temperature (d) predicts continuous as well as discrete spectral lines Solution: (a) will not be applicable to hydrogen in the molecular from (b) will not be applicable as it is for a He-atom...
Read More →Consider aiming a beam of free electrons towards free protons.
Question: Consider aiming a beam of free electrons towards free protons. When they scatter, an electron and a proton cannot combine to produce an H-atom, (a) because of energy conservation (b) without simultaneously releasing energy in the form of radiation (c) because of momentum conservation (d) because of angular momentum conservation Solution: (a) because of energy conservation (b) without simultaneously releasing energy in the form of radiation...
Read More →An ionised H-molecule consists of an electron and two protons.
Question: An ionised H-molecule consists of an electron and two protons. The protons are separated by a small distance of the order of angstrom. In the ground state, (a) the electron would not move in circular orbits (b) the energy would be (2)4 times that of an H-atom (c) the electrons, the orbit would go around the protons (d) the molecule will soon decay in a proton and an H-atom Solution: (a) the electron would not move in circular orbits (c) the electrons, the orbit would go around the prot...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2}}{x^{2}+6 x+12} d x$ Solution: Given $I=\int \frac{x^{2}}{x^{2}+6 x+12} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ $\Rightarrow \int \frac{\mathrm{x}^{2}}{\mathrm{x}^{2}+6 \mathrm{x}+12} \mathrm{dx}=\int\left(\frac{-6 \mathrm{x}-12}{\mat...
Read More →A set of atoms in an excited state decays.
Question: A set of atoms in an excited state decays. (a) in general to any of the states with lower energy (b) into a lower state only when excited by an external electric field (c) all together simultaneously into a lower state (d) to emit photons only when they collide Solution: (a) in general to any of the states with lower energy...
Read More →What is the probability that a leap year has 53 Sundays?
Question: What is the probability that a leap year has 53 Sundays? Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ A leap has 366 days i.e. 52 weeks $+2$ days. So, there will be 52 Sundays for sure (because every week has one Sunday) So, we want another Sunday from the remaining two days. The two days may be Sunday, Monday or Monday, Tuesday or Tuesday, Wednesday or Wednesday, Thursday or Thursday, Fri...
Read More →Two H atoms in the ground state collide inelastically.
Question: Two H atoms in the ground state collide inelastically. The maximum amount by which their combined kinetic energy is reduced is (a) 10.20 eV (b) 20.40 eV (c) 13.6 eV (d) 27.2 eV Solution: (a) 10.20 eV...
Read More →O2 molecule consists of two oxygen atoms.
Question: O2molecule consists of two oxygen atoms. In the molecule, nuclear force between the nuclei of the two atoms (a) is not important because nuclear forces are short-ranged (b) is as important as an electrostatic force for binding the two atoms (c) cancels the repulsive electrostatic force between the nuclei (d) is not important because the oxygen nucleus have an equal number of neutrons and protons Solution: (a) is not important because nuclear forces are short-ranged...
Read More →For the ground state,
Question: For the ground state, the electron in the H-atom has an angular momentum = ℏ, according to the simple Bohr model. Angular momentum is a vector and hence there will be infinitely many orbits with the vector pointing in all possible directions. In actuality, this is not true, (a) because Bohr model gives incorrect values of angular momentum (b) because only one of these would have a minimum energy (c) angular momentum must be in the direction of spin of electron (d) because electrons go ...
Read More →The simple Bohr model cannot be directly applied
Question: The simple Bohr model cannot be directly applied to calculate the energy levels of an atom with many electrons. This is because (a) of the electrons not being subject to a central force (b) of the electrons colliding with each other (c) of screening effects (d) the force between the nucleus and an electron will no longer be given by Coulombs law Solution: (a) of the electrons not being subject to a central force...
Read More →What is the probability that an ordinary year has 53 Tuesdays?
Question: What is the probability that an ordinary year has 53 Tuesdays? Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no. of outcomes }}$ An ordinary year has 365 days i.e. it has 52 weeks $+1$ day. So, there will be 52 Tuesdays for sure(because every week has 1 Tuesday) So, we want another Tuesday that to from that 1 day left(as there is only one Tuesday left after 52 weeks) This one day can be, Monday, Tuesday, Wedn...
Read More →The binding energy of an H-atom,
Question: The binding energy of an H-atom, considering an electron moving around a fixed nucleus (proton), is $B=\frac{m e^{4}}{8 n^{2} \epsilon_{0}^{2} h^{2}} \quad(m=$ electron mass $)$. If one decides to work in a frame of reference where the electron is at rest, the proton would be moving around it. By similar arguments, the binding energy would be $B=\frac{M e^{4}}{8 n^{2} \epsilon_{0}^{2} h^{2}} \quad(\mathrm{M}=$ proton mass $)$ This last expression is not correct because (a) n would not ...
Read More →Tickets numbered from 1 to 12 are mixed up together, and then a ticket is withdrawn at random.
Question: Tickets numbered from 1 to 12 are mixed up together, and then a ticket is withdrawn at random. Find the probability that the ticket has a number which is a multiple of 2 or 3. Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total no. of outcomes are 12 Desired output is picking a number which is multiple of 2 or 3 . So, desire outputs are 2 , $3,4,6,8,9,10,12$. Total no.of desired outputs are...
Read More →Taking the Bohr radius as a0 = 53pm,
Question: Taking the Bohr radius as a0= 53pm, the radius of Li++ion in its ground state, on the basis of Bohrs model, will be about (a) 53 pm (b) 27 pm (c) 18 pm (d) 13 pm Solution: (c) 18 pm...
Read More →A card is drawn at random from a well-shuffled pack of 52 cards.
Question: A card is drawn at random from a well-shuffled pack of 52 cards. What is the probability that the card bears a number greater than 3 and less than 10? Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no. of Desired outcomes }}{\text { Total no.of outcomes }}$ Total no. of outcomes are 52 Desired output is a number greater than 3 and less than 10 . There will be four sets of each card naming $A, 1,2,3,4,5,6,7,8,9,10, J, Q, K .$ So, there will be a tota...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2}\left(x^{4}+4\right)}{x^{2}+4} d x$ Solution: Given $I=\int \frac{x^{2}\left(x^{4}+4\right)}{x^{2}+4} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ $\Rightarrow \int \frac{x^{2}\left(x^{4}+4\right)}{x^{2}+4} d x=\int\left(-\frac{80}{x^{2}+4...
Read More →If a letter is chosen at random from the English alphabet,
Question: If a letter is chosen at random from the English alphabet, find the probability that the letter is chosen is (i) a vowel (ii) a consonant Solution: (i) We know that, Probability of occurrence of an event $=\frac{\text { Total no. of Desired outcomes }}{\text { Total no. of outcomes }}$ Total possible outcomes are alphabets from a to $z$ Desired outcomes are $a, e, i, o, u$ Total no. of outcomes are 26 and desired outputs are 5 Therefore, the probability of picking a vowel $=\frac{5}{26...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{3}+x^{2}+2 x+1}{x^{2}-x+1} d x$ Solution: Given $I=\int \frac{x^{3}+x^{2}+2 x+1}{x^{2}-x+1} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ $\Rightarrow \int \frac{x^{3}+x^{2}+2 x+1}{x^{2}-x+1} d x=\int \frac{3 x-1}{x^{2}-x+1}+x+2 d x$ $=\int \...
Read More →Consider a 20 W bulb emitting light of wavelength 5000 A °
Question: Consider a 20 W bulb emitting light of wavelength 5000 A and shining on a metal surface kept at a distance 2m. Assume that the metal surface has work function of 2 eV and that each atom on the metal surface can be treated as a circular disk of radius 1.5 A . (i) Estimate no. of photons emitted by the bulb per second. [Assume no other losses] (ii) Will there be photoelectric emission? (iii) How much time would be required by the atomic disk to receive energy equal to work function (2 eV...
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