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{"ops":[{"insert":"A given radioactive parent has decay constant \ud835\udf06"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":" and at time t = 0 there are \ud835\udc41"},{"attributes":{"script":"super"},"insert":"1"},{"attributes":{"script":"sub"},"insert":"0"},{"insert":" parent nuclei and the number of radioactive daughter nuclei is \ud835\udc41"},{"attributes":{"script":"super"},"insert":"2"},{"attributes":{"script":"sub"},"insert":"0"},{"insert":" = 0. The daughter has decay constant \ud835\udf06"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":". Show;\n(i) that the number of daughter nuclei \ud835\udc41"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":" at time t > 0 is N"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":" = \ud835\udc41"},{"attributes":{"script":"super"},"insert":"2"},{"attributes":{"script":"sub"},"insert":"0"},{"insert":" (\ud835\udf06"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":"\/\ud835\udf06"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":" \u2212 \ud835\udf06"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":") (\ud835\udc52"},{"attributes":{"script":"super"},"insert":"\ud835\udf061\ud835\udc61"},{"insert":"\u2212 \ud835\udc52"},{"attributes":{"script":"super"},"insert":"\ud835\udf062\ud835\udc61"},{"insert":").\n(ii) The value of N"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":" is zero at t = 0 and at t = \u221e . It therefore passes through a maximum at some time tm. Show that tm is given by t"},{"attributes":{"script":"sub"},"insert":"m"},{"insert":" = \ud835\udf0f"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":"(\ud835\udc47"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":"\/\ud835\udc47"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":" \u2212 \ud835\udc47"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":")"},{"attributes":{"script":"sub"},"insert":" "},{"insert":"In[\ud835\udc47"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":"\/\ud835\udc47"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":"] where T"},{"attributes":{"script":"sub"},"insert":"1"},{"insert":" and T"},{"attributes":{"script":"sub"},"insert":"2"},{"insert":" are the half \u2013 lives of the parent and the daughter respectively, and \ud835\udf0f2 is the mean life of the daughter\n"}]}

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