(i) The number of nuclei of a given radioactive sample at time t = 0 and t =T are \[{{N}_{0}}\] and \[{{N}_{0}}/n,\] respectively. Obtain an expression for the half-life \[({{T}_{1/2}})\] of the nucleus in terms of n and T. |
(ii) Write the basic nuclear process underlying \[{{\beta }^{-}}\text{-}decay\]of a given radioactive nucleus. |
(iii) Differentiate between \[{{\beta }^{+}}\] and \[{{\beta }^{-}}\]-decay. |
OR |
Define the terms: |
(i) Mass defect. |
(ii) Binding energy for a nucleus and state the relation between the two, for a given nuclear reaction the binding energy per nucleon of the product nucleus is more than that for the original nucleus, is this nuclear reaction exothermic or endothermic in nature? |
(iii) What is the role of control rods in a nuclear reactor? Why are they made of cadmium? |
Answer:
(i) Given, \[N=\frac{{{N}_{0}}}{n},\] t = T where \[{{N}_{0}}\] = number of nuclei of a radio-active sample at t = 0. According to law of radioactive decay, \[N={{N}_{0}}{{e}^{-\lambda t}}\] \[\therefore \] \[\frac{{{N}_{0}}}{n}={{N}_{0}}{{e}^{-\lambda T}}\Rightarrow n={{e}^{\lambda T}}\Rightarrow \lambda =\frac{\log (n)}{T}\] Therefore, half-life, \[{{T}_{1/2}}=\frac{0.6931}{\lambda }=\frac{0.693T}{\log (n)}\] |
(ii) In \[{{\beta }^{-}}-decay\] process, a nuclei emits a negative charge from the nucleus. A neutron is convened to a proton, causing the nuclides atomic number to increase by one, but the atomic mass remains the same. |
(iii) In \[{{\beta }^{+}}\] decay, a proton converts into a neutron, emits a positron \[({{e}^{+}})\] and a neutrino (v) \[v\to n+{{e}^{+}}+v\] In \[{{\beta }^{-}}\] -decay, a proton converts into a proton, emits an electron \[({{e}^{-}})\] and an ant-neutrino (v) \[v\to p+{{e}^{-}}+v\] |
OR |
(i) Mass defect \[(\Delta \,M)\] of any nucleus \[(_{Z}^{A}X)\] is the difference between the actual mass of nucleus (M) and the sum of the masses of its constituent nucleons. \[\Delta \,M=[Z{{M}_{p}}+(A-Z){{M}_{n}}]-M\] where, \[{{M}_{p}}\] = mass of proton and \[{{M}_{n}}\] = mass of the neutron |
(ii) Binding energy is the energy required to separate a nucleus from its constituent nucleons. \[BE=(\Delta M){{c}^{2}}\] where, c = speed of light Reasons Increase in bind1ng energy per nucleon implies that more mass has been converted into energy, this would result in release of energy. Nuclear reaction is called exothermic, as energy is released. |
(iii) For a controlled chain reaction, the average number of available neutrons should never exceed one per fission. Any excess neutrons over this critical unit should be absorbed. This is what the control rods do. They are made of cadmium because cadmium has a high cross-section for neutron absorption. |
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