Experimental measurements show that volume of a nucleus is proportional to its mass number $$A.$$ If $$R$$ is the radius of the nucleus assumed to be spherical, then its volume and mass no. relation is given by
$$\eqalign{
& {\text{Volume}}\left( V \right) \propto {\text{mass}}\,{\text{no}}{\text{.}}\left( A \right) \cr
& {\text{or}}\,\,\left( {\frac{4}{3}\pi {R^3}} \right) \propto A \cr
& {\text{or}}\,\,R \propto {A^{\frac{1}{3}}} \cr
& \therefore \frac{{{R_{Al}}}}{{{R_{Te}}}} = \frac{{{{\left( {27} \right)}^{\frac{1}{3}}}}}{{{{\left( {125} \right)}^{\frac{1}{3}}}}} = \frac{3}{5} = \frac{6}{{10}} \cr} $$
4.
Calculate binding energy of $$_{92}{U^{238}}.$$
Given $$M\left( {{U^{238}}} \right) = 238.050783\,amu,{m_n} = 1.008665\,amu$$ and $${m_P} = 1.007825\,amu$$
6.
A radioactive sample $${S_1}$$ having an activity $$5\mu Ci$$ has twice the number of nuclei as another sample $${S_2}$$ which has an activity of $$10\,\mu Ci.$$ The half lives of $${S_1}$$ and $${S_2}$$ can be
Note : In a nucleus neutron converts into proton as follows
$$n \to {p^ + } + {e^{ - 1}}$$
Thus, decay of neutron is responsible for $$\beta $$-radiation is carbon atom. origination
8.
A radioactive substance with decay constant of $$0.5\,{s^{ - 1}}$$ is being produced at a constant rate of $$50$$ nuclei per second. If there are no nuclei present initially, the time (in second) after which $$25$$ nuclei will be present is
The mass of radioactive substance remained is,
$$M = {M_0}{\left( {\frac{1}{2}} \right)^n}$$
Here, final mass, $$M = 1\,g,$$ initial mass, $${M_0} = 256\,g,$$ half life period, $${T_{\frac{1}{2}}} = 12.5\,h$$
So, $$1 = 256{\left( {\frac{1}{2}} \right)^n}\,\,{\text{or}}\,\,\frac{1}{{256}} = {\left( {\frac{1}{2}} \right)^n}$$
or $${\left( {\frac{1}{2}} \right)^8} = {\left( {\frac{1}{2}} \right)^n}$$
Comparing the powers on both the sides, we get
$$n = 8 = \frac{t}{{{T_{\frac{1}{2}}}}}$$
$$\eqalign{
& \therefore t = 8{T_{\frac{1}{2}}} \cr
& = 8 \times 12.5 \cr
& = 100\,h \cr} $$