The length of the solid cube of gold is [tex]$\boxed{{\text{1}}{\text{.08 in}}}$[/tex]
Further explanation:
Specific gravity:
Specific gravity is defined as the ratio of the density of the substance to the density of water. It is a unitless quantity as it includes the ratio of two similar quantities.
The formula to calculate specific gravity of gold is as follows:
[tex]{\text{Specific gravity of gold}}=\frac{{{\text{Density of gold}}}}{{{\text{Density of water}}}}[/tex] …… (1)
Density is considered as a characteristic property of the substance. It is an intensive property. It is defined as the mass per unit volume. It is generally represented by [tex]$\rho$[/tex] .
The formula to calculate the density of gold is,
[tex]\text{Density of gold}(\rho)=\dfrac{\text{Mass of gold(M)}}{\text{Volume of gold(V)}}[/tex] …… (2)
Rearrange equation (1) to calculate the density of gold.
[tex]${\text{Density of gold}}=\left({{\text{Specific gravity of gold}}}\right)\left({{\text{Density of water}}}\right)$[/tex] …… (3)
The specific gravity of gold is 19.3.
The density of water is 1 g/mL.
Substitute these values in equation (2).
[tex]$\align{{\text{Density of gold}}&=\left({{\text{19}}{\text{.3}}}\right)\left({\frac{{{\text{1 g}}}}{{{\text{1 mL}}}}}\right)\cr&=19.3\;{\text{g/mL}}\cr}$[/tex]
Rearrange equation (2) to calculate the volume of gold.
[tex]{\text{Volume of gold}}=\frac{{{\text{Mass of gold}}}}{{{\text{Density of gold}}}}[/tex] …… (4)
The mass of gold is 0.4 kg. Firstly, we have to convert the mass of gold from kg to g. The conversion factor for this is as follows:
[tex]${\text{1 kg}}= \text{1}}{{\text{0}}^{\text{3}}}{\text{ g}}$[/tex]
So the mass of gold is calculated as follows:
[tex]$\align{{\text{Mass of gold}}&=\left({{\text{0}}{\text{.4 kg}}}\right)\left({\frac{{{\text{1}}{{\text{0}}^{\text{3}}}{\text{ g}}}}{{{\text{1 kg}}}}}\right)\cr&=400\;{\text{g}}\cr}$[/tex]
Mass of gold is 400 g.
Density of gold is 19.3 g/mL.
Substitute these values in equation (4).
[tex]$\align{{\text{Volume of gold}}&=\left({{\text{400 g}}}\right)\left({\frac{{1\;{\text{mL}}}}{{{\text{19}}{\text{.3 g}}}}}\right)\cr&={\text{20}}{\text{.725 mL}}\cr}$[/tex]
The formula to calculate the volume of a gold cube is as follows:
[tex]${\text{Volume of gold cube}}={\left({{\text{Length of gold cube}}}\right)^3}$[/tex] …… (5)
Rearrange equation (5) to calculate the length of the gold cube.
[tex]${\text{Length of gold cube}}=\root3\of{{\text{Volume of gold cube}}}$[/tex] …… (6)
The volume of gold is 20.725 mL or [tex]${\text{20}}{\text{.725 c}}{{\text{m}}^{\text{3}}}$[/tex] .
Substitute this value in equation (6).
[tex]$\align{{\text{Length of gold cube}}&=\root3\of{{\text{20}}{\text{.725 c}}{{\text{m}}^3}}\cr&=2.7468\;{\text{cm}}\cr&\approx{\text{2}}{\text{.75 cm}}\cr}$[/tex]
The conversion factor required to convert the length of the gold cube from cm to inches is as follows:
[tex]${\text{1 cm}}=0.3937\;{\text{in}}$[/tex]
So the length of the gold cube is calculated as follows:
[tex]$\align{{\text{Length of gold cube}}&=\left({{\text{2}}{\text{.75 cm}}}\right)\left({\frac{{{\text{0}}{\text{.3937 in}}}}{{{\text{1 cm}}}}}\right)\cr&={\text{1}}{\text{.082675 in}}\cr&\approx{\bf{1}}{\bf{.08 in}}\cr}$[/tex]
Learn more:
1. Calculate the molar mass of phenol: https://brainly.com/question/5671083
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Answer details:
Grade: High School
Subject: Chemistry
Chapter: Density
Keywords: specific gravity, density, volume of gold cube, length of gold cube, inches, cm, 1.08 in, 2.75 cm, 19.3, 19.3 g/mL, density of gold, density of water, 1 g/mL, conversion factor.
