Density, which is the mass per unit volume, is a constant for a material, and does not depend on the size of the sample tested
[tex]Density = \mathbf{\dfrac{m}{v}} = \rho = \mathbf{Constant \ for \ the \ material}[/tex]
The steps taken to arrive at the above statement are as follows:
The density of a substance is its mass per unit volume. Mathematically, density can be expressed as follows;
[tex]Density = \mathbf{\dfrac{Mass}{Volume}}[/tex]
Let m₁ represent the mass and let v₁ represent the volume, of an object. Let m₂ represent the mass of double the the object, so that we have;
m₂ = 2 × m₁
The volume of m₂, v₂ = 2 × v₁
The ratio of the difference in the change in mass to the change in volume between the samples, m₁ and m₂, is given as follows;
[tex]\dfrac{\Delta m}{\Delta v} = \dfrac{m_2 - m_1}{v_2 - v_1} = \dfrac{2 \times m_1 - m_1}{2 \times v_1 - v_1} = \dfrac{m_1}{v_1} = \rho_1[/tex]
Therefore as Δv = v₂ - v₁ approaches zero, that is as the mass m₁ approaches m₂, we have;
[tex]\lim \limits_{\Delta v \to 0} \dfrac{\Delta m}{\Delta v} =\mathbf{ \dfrac{dm}{dv}}[/tex]
Which gives;
[tex]\mathbf{\lim \limits_{\Delta v \to 0} \dfrac{\Delta m}{\Delta v} } = \dfrac{dm}{dv} = \mathbf{\dfrac{Mass \ of \, sample}{Volume \ of \, sample}} = \mathbf{Density, \ \rho}[/tex]
Therefore, the rate of change of mass to volume of a sample to the volume of the sample is constant which gives that the mass of a sample is directly proportional to the volume of the sample and the ratio of the mass of the sample to the volume of the sample is a constant equal to the density of the substance
Which gives that the density of a substance which is the mass/volume is constant for all masses of the substance and does not depend on the size of the sample tested
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