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Question: The relative density of mercury is \(13.6\) . Its density in \(C.G.S\) unit is \(X\,gc{m^{ - 3}}\) ....

The relative density of mercury is 13.613.6 . Its density in C.G.SC.G.S unit is X gcm−3X\,gc{m^{ - 3}} . Find XX
(A) 1313
(B) 1414
(C) 13.613.6
(D) 14.614.6

Explanation

Solution

The relative density is also specified by the specific gravity. It is the unit less term and its formula is given below. Use the formula, substitute the known values, to find the C.G.SC.G.S unit of the density of the mercury, by considering the density of mercury in SISI units.

Useful formula:
The formula of the relative density of the mercury is given by

RD=ρρwRD = \dfrac{\rho }{{{\rho _w}}}

Where RDRD is the relative density of the mercury, ρ\rho is the density of the mercury and ρw{\rho _w} is the density of the water.

Complete step by step solution:
It is given that the
Relative density of the mercury, ρ=13.6\rho = 13.6
Use the formula of the relative density given above,

RD=ρρwRD = \dfrac{\rho }{{{\rho _w}}}

Substituting the value of the relative density and also the density of the water as

1 gcm−31\,gc{m^{ - 3}} (in the C.G.S.C.G.S. system of units)
13.6=ρ113.6 = \dfrac{\rho }{1}
By performing the basic arithmetic operation, we get

ρ=13.6 gcm−3\rho = 13.6\,gc{m^{ - 3}}

By converting the C.G.S.C.G.S. units into the SISI system of units. The gram is converted into kilograms and the centimeter is converted into meters.

ρ=13.6×10−3×1(10−2)3  Kgm−3\rho = 13.6 \times {10^{ - 3}} \times \dfrac{1}{{{{\left( {{{10}^{ - 2}}} \right)}^3}}}\,\,Kg{m^{ - 3}}

By performing the simplification,

ρ=13.6×10−310−6 Kgm−3\rho = \dfrac{{13.6 \times {{10}^{ - 3}}}}{{{{10}^{ - 6}}}}\,Kg{m^{ - 3}}

By cancelling the similar terms in the right hand side of the equation, we get

ρ=13.6×103 Kgm−3\rho = 13.6 \times {10^3}\,Kg{m^{ - 3}} ( SISI )
ρ=13.6 gcm−3\rho = 13.6\,gc{m^{ - 3}} ( C.G.S.C.G.S. )

Hence the density of the mercury in the C.G.S.C.G.S. is obtained as 13.613.6 .

Thus the option (C) is correct.

Note: It is known that the value of the density of the mercury in SISI unit is 1360013600 , by converting the kilogram into the gram and the meter into the centimeter, 13600×10001003=13.613600 \times \dfrac{{1000}}{{{{100}^3}}} = 13.6 . Hence the answer can be checked in this way of a simple solution.