We know that the change in density (∆ρ) of a material depends on its coefficient of volumetric thermal expansion (β) and the change in temperature (∆t).
The relationship between these variables can be expressed as:
∆V = Vβ∆t
where ∆V is the change in volume of the material, V is its original volume, β is the coefficient of volumetric thermal expansion, and ∆t is the change in temperature.
We also know that density (ρ) is the mass (m) of a material divided by its volume:
ρ = m/V
Taking the derivative of this equation with respect to time gives:
∆ρ/∆t = (dm/dt)/V - m/(V^2) * (dV/dt)
Since we are assuming that the mass of the material is constant, dm/dt = 0, and the first term on the right-hand side of the equation is zero.
Putting in the expression for ∆V and simplifying, we get:
∆ρ/∆t = -ρβ∆t
Therefore,
∆ρ = -ρβ∆t
which shows that the change in density (∆ρ) is proportional to the negative product of the original density (ρ), coefficient of volumetric thermal expansion (β), and change in temperature (∆t).
Show that ∆ρ = -ρβ∆t
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