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Chapter 7: Problem 1077

The variation of density of water with temperature is represented by the

Short Answer

Expert verified
The variation of density of water with temperature is represented by a quadratic function in the form: \[ \rho(T) = A + B \cdot T + C \cdot T^2 \] where ρ(T) is the density of water at temperature T, and A, B, and C are constants determined through experimental data and fitting methods. This equation allows us to analyze and predict the behavior of water at various temperatures, which is crucial in many fields including engineering, chemistry, and environmental sciences.

Step by step solution

01

Understand the Basic Concepts

The density of a substance is defined as its mass per unit volume. In the case of water, the density (ρ) varies with temperature (T) and is typically given in kilograms per cubic meter (kg/m³). The density of water decreases as the temperature increases, which means that warmer water has a lower density than colder water. The relationship between density and temperature is not a simple linear equation and requires more advanced analysis for accurate description.
02

Establish the Density and Temperature Relationship

The relationship between the density of water and temperature is often represented using a polynomial equation of a certain degree, or an empirical model based on experimental data. For the sake of simplicity, let's assume that we have the equation in the form of a quadratic function: \[ \rho(T) = A + B \cdot T + C \cdot T^2 \] Where A, B, and C are constants that have been determined through experimental data and fitting methods, and T represents temperature.
03

Analyze the Given Equation

Now that we have established the general representation of the density-temperature relationship for water, it is important to analyze the given equation (provided in the exercise) to extract the constants A, B, and C. By doing so, we can obtain the specific density-temperature equation for water to use in further analysis or applications.
04

Example Application of the Density-Temperature Relationship

Let's assume that we have found the constants A, B, and C for the given equation, and our specific density-temperature relationship is: \[ \rho(T) = A' + B' \cdot T + C' \cdot T^2 \] Where A', B', and C' are the specific constants for water. Now, we want to find the density of water at a given temperature, say 20°C, using the aforementioned equation. To find the density at this temperature, simply replace T with 20 and compute the result: \[ \rho(20) = A' + B' \cdot 20 + C' \cdot 20^2 \] Calculating the density allows us to use this information in various applications, such as designing water storage systems, analyzing buoyant forces, and understanding the behavior of water at various temperatures.
05

Reflect on the Importance of the Density-Temperature Relationship

Understanding the variation of density of water with temperature is crucial in many fields, including engineering, chemistry, and environmental sciences, as it plays a significant role in understanding and predicting the behavior of water under different conditions. By knowing this relationship, we can design systems and structures that account for the variations in the density of water, allowing for more efficient and effective solutions.

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