Page 1 Modern Physics â€“ II Important Formulae 1. Radioactivity (i) ?-emission emission A A 4 Z2 Z XY ?? ? ? ? ? ? ? ? (ii) ? - emission n P e v ? ? ? ? ? ? emission AA Z1 Z XY ?? ? ? ? ? ? ? (iii) Rutherford and Soddy law dN dN N or N dt dt ? ? ? ? ? (iv) N = N 0 e -lt Here, N are the number of remaining nuclei. (v) N d =N 0 (1 â€“ e - ?t ) Here, N d axe the number of decayed nuclei (vi) ? = decay constant 1 ? = mean life or average life = t av 1/ 2 1 0.693 t (ln 2) ?? ?? ?? ?? ?? (vii) t av > t 1/2 1/ 2 av ln 2 0.693 t 0.693t ? ? ? ?? av 1/ 2 1 t 1.44t ?? ? (viii) Probability of survival of a nucleus upto time t t s Pe ?? ? Page 2 Modern Physics â€“ II Important Formulae 1. Radioactivity (i) ?-emission emission A A 4 Z2 Z XY ?? ? ? ? ? ? ? ? (ii) ? - emission n P e v ? ? ? ? ? ? emission AA Z1 Z XY ?? ? ? ? ? ? ? (iii) Rutherford and Soddy law dN dN N or N dt dt ? ? ? ? ? (iv) N = N 0 e -lt Here, N are the number of remaining nuclei. (v) N d =N 0 (1 â€“ e - ?t ) Here, N d axe the number of decayed nuclei (vi) ? = decay constant 1 ? = mean life or average life = t av 1/ 2 1 0.693 t (ln 2) ?? ?? ?? ?? ?? (vii) t av > t 1/2 1/ 2 av ln 2 0.693 t 0.693t ? ? ? ?? av 1/ 2 1 t 1.44t ?? ? (viii) Probability of survival of a nucleus upto time t t s Pe ?? ? (ix) Probability of decay of a nucleus in time t t d P 1 e ?? ?? (x) Activity of radioactive substance, tt 00 dN R N N e R e dt ? ? ? ? ? ? ? ? ? ? ? Here, R 0 = ?N 0 = initial activity (xi) Units of activity (a)1 Curie = 1Ci = 3.7 × 10 10 dPS (b)1 Rutherford = 1 rd = 10 6 dPS (c) 1 Becquerel = 1 Bq = 1 dPS (xii) If a nucleus decays in two modes 12 12 12 TT and T TT ? ? ? ? ? ? ? Here, T is representing the half-life. (xiii) 1/ 2 1/ 2 n tt 00 00 NN 1 N .......... N 2 4 2 ?? ? ? ? ? ? ? ?? ?? n 1 1 1 1 .......... 2 4 2 ?? ? ? ? ? ? ? ?? ?? n 1 100% 50% 25%..........100 2 ?? ? ? ? ? ? ? ?? ?? Here, n = number of half-lives 1/ 2 t t ? (xiv) Successive radioactivity. Page 3 Modern Physics â€“ II Important Formulae 1. Radioactivity (i) ?-emission emission A A 4 Z2 Z XY ?? ? ? ? ? ? ? ? (ii) ? - emission n P e v ? ? ? ? ? ? emission AA Z1 Z XY ?? ? ? ? ? ? ? (iii) Rutherford and Soddy law dN dN N or N dt dt ? ? ? ? ? (iv) N = N 0 e -lt Here, N are the number of remaining nuclei. (v) N d =N 0 (1 â€“ e - ?t ) Here, N d axe the number of decayed nuclei (vi) ? = decay constant 1 ? = mean life or average life = t av 1/ 2 1 0.693 t (ln 2) ?? ?? ?? ?? ?? (vii) t av > t 1/2 1/ 2 av ln 2 0.693 t 0.693t ? ? ? ?? av 1/ 2 1 t 1.44t ?? ? (viii) Probability of survival of a nucleus upto time t t s Pe ?? ? (ix) Probability of decay of a nucleus in time t t d P 1 e ?? ?? (x) Activity of radioactive substance, tt 00 dN R N N e R e dt ? ? ? ? ? ? ? ? ? ? ? Here, R 0 = ?N 0 = initial activity (xi) Units of activity (a)1 Curie = 1Ci = 3.7 × 10 10 dPS (b)1 Rutherford = 1 rd = 10 6 dPS (c) 1 Becquerel = 1 Bq = 1 dPS (xii) If a nucleus decays in two modes 12 12 12 TT and T TT ? ? ? ? ? ? ? Here, T is representing the half-life. (xiii) 1/ 2 1/ 2 n tt 00 00 NN 1 N .......... N 2 4 2 ?? ? ? ? ? ? ? ?? ?? n 1 1 1 1 .......... 2 4 2 ?? ? ? ? ? ? ? ?? ?? n 1 100% 50% 25%..........100 2 ?? ? ? ? ? ? ? ?? ?? Here, n = number of half-lives 1/ 2 t t ? (xiv) Successive radioactivity. Suppose A and B are radioactive and C is stable. Let us further assume that initially there are only nuclei of A Then, number of nuclei of A, B and C vary with time as shown below N B are maximum when ? A N A = ? B N B 2. Fusion and Fission (i) E = mc 2 (ii) 1 amu = 931.48 2 MeV c (iii) Mass defect = [Zm p + (A - Z)m N - m X ] Here m X is mass of nucleus. (iv) Binding energy E 1 =( ?m) c 2 If ?m is represented in amu, then E 1 can be obtained in MeV by multiplying it with 931.48. (v) Binding energy per nucleon 1 2 E E Total number of nucleons ? (vi) For stability of a nucleus binding energy per nucleon is more important rather than the total binding energy. (vii) Binding energy per nucleon is of the order of few MeV (2 MeV-10 MeV). (viii) During formation of a nucleus some mass is lost. This is called mass defect. Equivalent to that mass some energy is liberated. This is called binding energy of the nucleus. (ix) In any nuclear process energy is released if total binding energy of the daughter nuclei is more than the total binding energy of the parent nuclei. (x) Binding energy per nucleon (E 2 ) versus number of nucleon (A) graph From the graph it is clear that binding energy per nucleon is maximum near iron nucleus. Page 4 Modern Physics â€“ II Important Formulae 1. Radioactivity (i) ?-emission emission A A 4 Z2 Z XY ?? ? ? ? ? ? ? ? (ii) ? - emission n P e v ? ? ? ? ? ? emission AA Z1 Z XY ?? ? ? ? ? ? ? (iii) Rutherford and Soddy law dN dN N or N dt dt ? ? ? ? ? (iv) N = N 0 e -lt Here, N are the number of remaining nuclei. (v) N d =N 0 (1 â€“ e - ?t ) Here, N d axe the number of decayed nuclei (vi) ? = decay constant 1 ? = mean life or average life = t av 1/ 2 1 0.693 t (ln 2) ?? ?? ?? ?? ?? (vii) t av > t 1/2 1/ 2 av ln 2 0.693 t 0.693t ? ? ? ?? av 1/ 2 1 t 1.44t ?? ? (viii) Probability of survival of a nucleus upto time t t s Pe ?? ? (ix) Probability of decay of a nucleus in time t t d P 1 e ?? ?? (x) Activity of radioactive substance, tt 00 dN R N N e R e dt ? ? ? ? ? ? ? ? ? ? ? Here, R 0 = ?N 0 = initial activity (xi) Units of activity (a)1 Curie = 1Ci = 3.7 × 10 10 dPS (b)1 Rutherford = 1 rd = 10 6 dPS (c) 1 Becquerel = 1 Bq = 1 dPS (xii) If a nucleus decays in two modes 12 12 12 TT and T TT ? ? ? ? ? ? ? Here, T is representing the half-life. (xiii) 1/ 2 1/ 2 n tt 00 00 NN 1 N .......... N 2 4 2 ?? ? ? ? ? ? ? ?? ?? n 1 1 1 1 .......... 2 4 2 ?? ? ? ? ? ? ? ?? ?? n 1 100% 50% 25%..........100 2 ?? ? ? ? ? ? ? ?? ?? Here, n = number of half-lives 1/ 2 t t ? (xiv) Successive radioactivity. Suppose A and B are radioactive and C is stable. Let us further assume that initially there are only nuclei of A Then, number of nuclei of A, B and C vary with time as shown below N B are maximum when ? A N A = ? B N B 2. Fusion and Fission (i) E = mc 2 (ii) 1 amu = 931.48 2 MeV c (iii) Mass defect = [Zm p + (A - Z)m N - m X ] Here m X is mass of nucleus. (iv) Binding energy E 1 =( ?m) c 2 If ?m is represented in amu, then E 1 can be obtained in MeV by multiplying it with 931.48. (v) Binding energy per nucleon 1 2 E E Total number of nucleons ? (vi) For stability of a nucleus binding energy per nucleon is more important rather than the total binding energy. (vii) Binding energy per nucleon is of the order of few MeV (2 MeV-10 MeV). (viii) During formation of a nucleus some mass is lost. This is called mass defect. Equivalent to that mass some energy is liberated. This is called binding energy of the nucleus. (ix) In any nuclear process energy is released if total binding energy of the daughter nuclei is more than the total binding energy of the parent nuclei. (x) Binding energy per nucleon (E 2 ) versus number of nucleon (A) graph From the graph it is clear that binding energy per nucleon is maximum near iron nucleus. In any nuclear process, if the products are towards the peak of this graph then binding energy per nucleon, hence total binding energy will increase. Therefore, energy will be released. In fusion reaction two or more lighter nuclei combine to make a relatively heavier nucleus. This heavy nucleus is towards peak of this graph. Therefore, energy will be released. In fission reaction a heavy nucleus breaks into two or more lighter nuclei. These lighter nuclei again lie towards peak of this graph. Therefore, energy will be released. (xi) Normally, fusion reaction is more difficult compared to fission reaction. Because to combine two or more positively charged nuclei is difficult compared to break it. (xii) Radius of a nucleus R = R 0 A 1/3 Here, R 0 =1.3 fm = 1.3 × 10 -15 m and A = mass number Thus, R ? A 1/3 (xiii) The density of any nucleus is independent of A and is of the order of 10 17 kgm -3 .Read More

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