Water’s coefficient of volume expansion in the temperature range from 0°C to about 20°C is given approximately by β = α + bT + cT² , with α = - 6.43 x 10⁻⁵ (C°)⁻¹ , b = 1.70 x 10⁻⁵ (C°)⁻² , and c = -2.02 x 10⁻⁷ ((C°)⁻³. Using the formula for density from Problem 22, show that water has its greatest density at approximately 4.0°C.

Verified step by step guidance

1. Start by recalling the relationship between the coefficient of volume expansion (β) and density (ρ). The density of a substance is inversely proportional to its volume, so changes in volume due to temperature will affect the density. The formula for density as a function of temperature can be written as ρ(T) = ρ₀ / (1 + βΔT), where ρ₀ is the reference density and ΔT is the temperature change.
2. Substitute the given expression for β into the formula. The coefficient of volume expansion is given as β = α + bT + cT². Replace β in the density formula to get ρ(T) = ρ₀ / (1 + (α + bT + cT²)ΔT).
3. To find the temperature at which water has its greatest density, note that the density is maximized when the rate of change of density with respect to temperature (dρ/dT) is zero. Use the chain rule to differentiate ρ(T) with respect to T, and set dρ/dT = 0.
4. Simplify the derivative. Since ρ(T) = ρ₀ / (1 + βΔT), the derivative involves the term β = α + bT + cT². Differentiate β with respect to T to get dβ/dT = b + 2cT. Substitute this into the derivative of ρ(T) and solve for T when dρ/dT = 0.
5. Solve the resulting equation for T. This will involve setting b + 2cT = 0 (since the maximum density occurs when the expansion coefficient β is zero). Solve for T to find the temperature at which water has its greatest density. Substitute the given values of b and c to calculate the approximate value of T, which should be around 4.0°C.

Key Concepts

Coefficient of Volume Expansion

The coefficient of volume expansion (β) quantifies how the volume of a substance changes with temperature. For water, this coefficient varies with temperature, indicating that as water heats up, its volume increases, which in turn affects its density. The formula provided incorporates linear and quadratic temperature terms, reflecting the non-linear behavior of water's expansion in the specified temperature range.

Density and Temperature Relationship

Density is defined as mass per unit volume and is influenced by temperature changes. As water warms, it expands, leading to a decrease in density. However, water exhibits unique behavior where its density reaches a maximum at around 4.0°C, after which further heating causes it to expand and decrease in density, a property critical for understanding aquatic ecosystems and climate effects.

Mathematical Modeling of Density

The mathematical modeling of density involves using equations that relate temperature to volume expansion and, consequently, density. By substituting the coefficients into the density formula, one can derive the temperature at which water's density is maximized. This approach highlights the importance of mathematical relationships in predicting physical properties and behaviors of substances under varying conditions.