A) \[\sqrt{{{\rho }_{A}}\,{{\rho }_{B}}}\]
B) \[\frac{{{\rho }_{A}}+\,{{\rho }_{B}}}{2}\]
C) \[\frac{{{\rho }_{A}}\,{{\rho }_{B}}}{{{\rho }_{A}}+\,{{\rho }_{B}}}\]
D) \[\frac{2{{\rho }_{A}}\,{{\rho }_{B}}}{{{\rho }_{A}}+\,{{\rho }_{B}}}\]
Correct Answer: D
Solution :
The density of mixture is \[\rho =\frac{Total\,\,mass}{Total\,\,volume}=\frac{{{m}_{A}}+{{m}_{B}}}{\frac{{{m}_{A}}}{{{\rho }_{A}}}+\frac{{{m}_{B}}}{{{\rho }_{B}}}}=\frac{({{m}_{A}}+{{m}_{B}}){{\rho }_{A}}\,{{\rho }_{B}}}{({{m}_{A}}\,{{\rho }_{B}}+{{m}_{B}}\,{{\rho }_{A}})}\]as \[{{m}_{A}}={{m}_{B}}=m\] \[\therefore \,\,\,\,\,\,\,\,\,\,\rho =\frac{2m\,{{\rho }_{A}}{{\rho }_{B}}}{m({{\rho }_{A}}+{{\rho }_{B}})}=\frac{2{{\rho }_{A}}\,{{\rho }_{B}}}{{{\rho }_{A}}\,+\,{{\rho }_{B}}}\]
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