Concept explainers
A liquid has a density ρ. (a) Show that the fractional change in density for a change in temperature ΔT is Δρ/ρ = −β ΔT. (b) What does the negative sign signify? (c) Fresh water has a maximum density of 1.000 0 g/cm3 at 4.0°C. At 10.0°C, its density is 0.999 7 g/cm3. What is β for water over this temperature interval? (d) At 0°C, the density of water is 0.999 9 g/cm3. What is the value for β over the temperature range 0°C to 4.00°C?
(a)
To show: The fraction change in density for a change in temperature is given by
Answer to Problem 57AP
The fraction change in density for a change in temperature is
Explanation of Solution
Given info: The density of the liquid is
The expression for the coefficient of the volume expansion is,
Here,
The formula for the density is,
Here,
Differentiate the above equation with respect to volume.
For very small change in volume and density the expression is,
Substitute
Conclusion:
Therefore, the fraction change in density for a change in temperature is
(b)
The significance of the negative sign.
Answer to Problem 57AP
the negative sign signifies that when the temperature increases, the density decreases.
Explanation of Solution
Given info: The density of the liquid is
The expression for the fraction of the density is,
The negative sign in the above expression shows that as the temperature increases, the density of the water is decreases.
Conclusion:
Therefore, the negative sign signifies that when the temperature increases, the density decreases.
(c)
The value of
Answer to Problem 57AP
The value of
Explanation of Solution
Given info: The density of the liquid is
Recall the equation (1) and rearrange for
Substitute
Conclusion:
Therefore, the value of
(d)
The value of
Answer to Problem 57AP
The value of
Explanation of Solution
Given info: The density of the liquid is
The expression for
Substitute
Conclusion:
Therefore, the value of
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Chapter 19 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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