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Question
The coefficient of volume expansion of glycerin is 49 × 10–5 K–1. What is the fractional change in its density for a 30 °C rise in temperature?
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Solution 1
Coefficient of volume expansion of glycerin, αV = 49 × 10–5 K–1
Rise in temperature, ΔT = 30°C
Fractional change in its volume =`(triangle V)/V`
This change is related with the change in temperature as:
`(triangle V)/V = alpha_V triangle T`
`V_(T_2) - V_(T_1) = V_(T_1) alpha_V triangle T`
`m/rho_(T_2) - m/rho_(T_1) = m/rho_(T_1) alpha_1 triangleT`
Where,
m = Mass of glycerine
`rho_(T_1)` = Initial density at `T_1`
`rho_(T_2)` = Final density at `T_2`
`(rho_(T_1) - rho_(T_2))/rho_(T_2) = alpha_1 triangle T`
Where
`(rho_(T_1) - rho_(T_2_))/(rho_T_2)` = = Fractional change in density
∴Fractional change in the density of glycerin = 49 ×10–5 × 30 = 1.47 × 10–2
Solution 2
Here `gamma= 49 xx 10^(-5) ""^@C^(-1)`, `triangle T = 30 ""^@C`
As` V = V + triangle V = V(1+gamma triangle T)`
`V' = V(1+49xx10^(-5)xx30) = 1.0147 V`
Since `rho = m/V, rho^n = m/(V') = m/1.0147V = 0.9855 rho`
Fractional change in density = `(rho - rho')/rho`
`= (rho - 0.9855 rho)/rho = 0.0145`
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