Express the density of a solid substance varying with temperature

[endeavor]

I need to show that the density of a solid substance varies with temperature as d = d₀(1 - 3aT), where a is the thermal coefficient and T is the change in temperature.

I know V = V₀ (1 + 3aT) and V = m/d. Since m is constant,

d₀ = d(1 + 3aT)
d₀ - d₀3aT = d + d3aT - d₀3aT
d₀(1 - 3aT) = d +3aT(d - d₀)

Am I now supposed to assume that the d only changes a little compared to d₀, so that (d - d₀) = 0?

 

[mukundpa]

do = d(1 + 3aT)
d = do/(1+3aT) = do(1+3aT)^-1

now expanding using binomial theorem and as a is very small the higher terms can be neglected, gives

d = do(1- 3aT + ...)


MP

 

[kyle8921]

No, you cannot assume that (d - d0) = 0 as it would not accurately represent the relationship between density and temperature. The equation d = d0(1 - 3aT) shows that the density of a solid substance decreases as temperature increases, and the magnitude of this decrease is determined by the thermal coefficient (a) and the change in temperature (T). This relationship is not affected by the initial density (d0) of the substance. So, as the temperature increases, the density of the solid substance will decrease according to this equation. This is an important concept to consider when studying the properties of solid substances and their behavior at different temperatures.