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6. (a) Show that the change in the density `ρ` of a substance , when the temperature changes by `∆T`, is given by `∆ρ = -betaρ_0∆T.`
(b) What is the fractional change in density of a lead sphere whose temperature decreases from `25°C` to `-40°C`?
6. (a) Show that the change in the density ρ of a substance , when the temperature changes by ∆T, is given by ∆ρ = -betaρ_0∆T.
(b) What is the fractional change in density of a lead sphere whose temperature decreases from 25°C to -40°C? Equations:
`ρ = m/V`
`∆V = βV_0∆T`
Solution:
(a)
`∆ρ =ρ -ρ_0 = m/V -m/V_0 = m/(V_0+∆V) - m/V_0`
`∆ρ = m/(V_0+βV_0∆T) - m/V_0`
`∆ρ = m/V_0(1/(1+β∆T) - 1)`
`∆ρ = ρ_0(1/(1+β∆T) - 1) = ρ_0(-β∆T/(1+β∆T))`
When `β∆T -> 0 => ∆ρ =-ρ_0β∆T`
(b)
Given `β_(l e ad) = 87times10^(-6) //°C`
`(∆ρ)/ρ_0 = (-ρ_0β∆T)/(ρ_0 ) = -β∆T`
`(∆ρ)/ρ_0 = -(87times10^(-6) //°C)(-40°C - 25°C)`
`(∆ρ)/ρ_0 = 5.7times 10^(-3)` |