Problem 23 - University Physics with Modern Physics

Question

The density of water is 999.73 \(\mathrm{kg} / \mathrm{m}^{3}\) at a temperature of \(10^{\circ} \mathrm{C}\) and 958.38 \(\mathrm{kg} / \mathrm{m}^{3}\) at a temperature of \(100^{\circ} \mathrm{C} .\) Calculate the average coefficient of volume expansion for water in that range of temperature.

Step-by-Step Solution

Answer

The average coefficient of volume expansion for water is approximately \( 2.07 \times 10^{-4} \, \mathrm{°C}^{-1} \).

Step 1: Understand the Formula for Volume Expansion

The formula for average volume expansion is given by \( \beta = \frac{\Delta V}{V_0 \times \Delta T} \), where \( \beta \) is the average coefficient of volume expansion, \( \Delta V \) is the change in volume, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature.

Step 2: Understand the Relationship Between Volume and Density

The volume \( V \) is inversely proportional to the density \( \rho \), i.e., \( V = \frac{m}{\rho} \), where \( m \) is the mass of water. For unit mass, the volume is simply the reciprocal of the density \( V = \frac{1}{\rho} \).

Step 3: Calculate Initial and Final Volume

For the initial temperature \( 10^{\circ} \mathrm{C} \), \( V_0 = \frac{1}{999.73} \, \mathrm{m}^3/\mathrm{kg} \). For \( 100^{\circ} \mathrm{C} \), \( V_f = \frac{1}{958.38} \, \mathrm{m}^3/\mathrm{kg} \).

Step 4: Calculate Change in Volume

Find \( \Delta V = V_f - V_0 = \frac{1}{958.38} - \frac{1}{999.73} \). Calculate this value to find \( \Delta V \).

Step 5: Calculate Change in Temperature

The change in temperature is given by \( \Delta T = 100^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C} = 90^{\circ} \mathrm{C} \).

Step 6: Substitute Values into Volume Expansion Formula

Substitute the values from steps 3, 4, and 5 into the formula: \( \beta = \frac{\Delta V}{V_0 \times \Delta T} = \frac{\left(\frac{1}{958.38} - \frac{1}{999.73}\right)}{\frac{1}{999.73} \times 90} \).

Step 7: Compute the Coefficient

Perform the arithmetic computations to find \( \beta \). The final value will be the average coefficient of volume expansion over the given temperature range.

Key Concepts

Density of Water

Density is a property that defines the mass per unit volume of a substance. For water, this density can change with temperature. At 10°C, water has a density of 999.73 kg/m³, and at 100°C, it becomes 958.38 kg/m³. As a liquid heats up, it usually expands, causing its density to decrease if the mass remains constant. Understanding the density changes with temperature helps us in calculating the thermal expansion, which is crucial for accurate thermal expansion calculations. This density-temperature relationship is particularly important in natural and engineering systems where precise measurements are vital. In systems where fluid density plays a critical role, such as climate modeling or engineering applications, acknowledging and accounting for these shifts ensures accuracy.

Volume-Temperature Relationship

The relationship between volume and temperature is fundamental to understanding thermal expansion. As temperature increases, most substances expand and their volume increases. For water, the volume at any given temperature can be calculated using the formula:

• \( V = \frac{1}{\rho} \)

This equation shows that volume is inversely proportional to density. So, as the density of water decreases with an increase in temperature, it results in an increase in volume.This principle helps us calculate changes in volume when the temperature changes, which is essential for determining the coefficient of volume expansion. Knowing how volume changes with temperature allows us to anticipate and design for expansion in various scientific and industrial contexts.

Thermal Expansion Calculations

To calculate how much a substance's volume will change with temperature, we use the thermal expansion coefficient. This is where the formula comes into play:

• \( \beta = \frac{\Delta V}{V_0 \times \Delta T} \)

Here, \( \Delta V \) is the change in volume, \( V_0 \) is the initial volume, and \( \Delta T \) is the change in temperature. Using this formula, we can derive \( \beta \), which represents the average coefficient of volume expansion. For the water example, \( \beta \) is calculated over a temperature change from 10°C to 100°C.Performing these calculations allows us to predict how much water will expand when heated. Such predictions are crucial for designing containers and systems that need to handle temperature changes without failure or leakage. Understanding thermal expansion calculations ensures safety and functionality in various applications.