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Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblem 11a
Chapter 17, Problem 11a

The density of water at 4°C is 1.00 x 10³ kg / m³. What is water’s density at 94°C? Assume a constant coefficient of volume expansion.

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Understand the concept: The density of a substance changes with temperature due to thermal expansion. The relationship between the initial and final densities can be derived using the coefficient of volume expansion (β). The formula for volume expansion is ΔV = βV₀ΔT, where ΔV is the change in volume, V₀ is the initial volume, β is the coefficient of volume expansion, and ΔT is the temperature change.
Relate density to volume: Since density (ρ) is inversely proportional to volume (V), the relationship between the initial density (ρ₀) and the final density (ρ) can be expressed as ρ = ρ₀ / (1 + βΔT). This equation accounts for the change in volume due to thermal expansion.
Identify the given values: From the problem, the initial density of water (ρ₀) is 1.00 × 10³ kg/m³, the initial temperature (T₀) is 4°C, the final temperature (T) is 94°C, and the temperature change (ΔT) is T - T₀ = 94°C - 4°C = 90°C. The coefficient of volume expansion (β) for water is typically around 2.1 × 10⁻⁴ /°C.
Substitute the known values into the formula: Use the equation ρ = ρ₀ / (1 + βΔT). Substitute ρ₀ = 1.00 × 10³ kg/m³, β = 2.1 × 10⁻⁴ /°C, and ΔT = 90°C into the formula to calculate the final density.
Simplify the expression: Perform the arithmetic operations in the denominator to find the factor by which the density changes. Then divide the initial density by this factor to determine the final density of water at 94°C.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Density

Density is defined as mass per unit volume and is a fundamental property of materials. For liquids like water, density can change with temperature due to thermal expansion or contraction. Understanding how density varies with temperature is crucial for solving problems related to fluid mechanics and thermodynamics.
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Coefficient of Volume Expansion

The coefficient of volume expansion quantifies how much a material's volume changes with temperature. For liquids, this coefficient is typically positive, indicating that the volume increases as temperature rises. This concept is essential for calculating changes in density when temperature varies, particularly in fluids like water.
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Thermal Expansion

Thermal expansion refers to the increase in volume of a substance as its temperature increases. In the case of water, it expands when heated, leading to a decrease in density. This phenomenon is critical for understanding how temperature affects the physical properties of liquids and is key to solving the given problem regarding water's density at different temperatures.
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