Problem 37 - Astronomy: At Play in the Cosmos

Question

Two regions of the ISM have identical pressure, but the temperature of region 1 is 2.5 times that of region \(2 .\) How does the density of region 1 compare with that of region \(2 ?\)

Step-by-Step Solution

Answer

The density of region 1 is 0.4 times the density of region 2.

Step 1: Understanding the Relationship Between Pressure, Temperature, and Density

Using the ideal gas law, we know that pressure \( P \), volume \( V \), density \( \rho \), temperature \( T \), and gas constant \( R \) are related by the formula: \[ P = \rho \cdot R \cdot T. \]Since both regions have identical pressure, we can set up the equation:\[ \rho_1 \cdot R \cdot T_1 = \rho_2 \cdot R \cdot T_2. \]

Step 2: Substitute the Given Condition

We are given that the temperature of region 1, \( T_1 \), is 2.5 times the temperature of region 2, \( T_2 \). Thus, we can write:\[ T_1 = 2.5 \cdot T_2. \]Substituting this into the equation from step 1, we have:\[ \rho_1 \cdot R \cdot (2.5 \cdot T_2) = \rho_2 \cdot R \cdot T_2. \]

Step 3: Simplify the Equation to Find Density Relationship

We can cancel \( R \cdot T_2 \) from both sides of the equation:\[ \rho_1 \cdot 2.5 = \rho_2. \]This simplifies to:\[ \rho_1 = \frac{\rho_2}{2.5}. \]

Step 4: Interpret the Results

The equation \( \rho_1 = \frac{\rho_2}{2.5} \) tells us that region 1 has a density that is \( \frac{1}{2.5} \) or 0.4 times the density of region 2. Thus, region 1 is less dense than region 2 by a factor of 2.5.

Key Concepts

Pressure

Pressure is a fundamental concept in physics that describes the force exerted per unit area. When dealing with gases, pressure is influenced by various factors such as volume, density, and temperature. In the context of the Ideal Gas Law, pressure can be mathematically represented as:\[ P = \rho \cdot R \cdot T \]where:

• \( P \) is the pressure of the gas.
• \( \rho \) is the density of the gas.
• \( R \) is the specific gas constant.
• \( T \) is the absolute temperature.

Knowing that the pressure is consistent across different regions of an interstellar medium (ISM) helps to understand the balance maintained in gaseous systems. If the pressure is identical, changes in temperature or density must occur to maintain this balance. Changes in density and temperature can result in the same pressure, which is an intriguing aspect of the Ideal Gas Law.

Temperature

Temperature is a measure of the average kinetic energy of particles in a substance. In gases, higher temperatures correlate with increased particle kinetic energy, which often leads to greater volume and pressure unless confined. Within the Ideal Gas Law, temperature is an absolutely crucial variable:\[ P = \rho \cdot R \cdot T \]If we consider two different regions, the temperature directly influences the other two variables—density and pressure. When it is stated that temperature in region 1 is 2.5 times that of region 2, it's a driver of significant density changes, assuming pressure remains constant.

• Increased temperature leads to potential expansiveness in gas unless confined by pressure or limited volume.
• Conversely, if pressure is maintained (as by the assignment), an increase in temperature would typically result in a decrease in density.

Recognizing the interplay between temperature and density allows one to understand how gases behave in different environmental settings, such as interstellar spaces.

Density

Density is the mass per unit volume of a substance. For gases, it's an aspect of their state of matter that directly influences both temperature and pressure under the ideal gas assumptions. The Ideal Gas Law connects density to other parameters as follows:\[ P = \rho \cdot R \cdot T \]In considering the exercise, where region 1's temperature is 2.5 times that of region 2, it's important to determine how this variance affects density while pressure remains constant.

• Given that temperature and pressure are key factors, if temperature increases and pressure does not, then density must decrease to maintain the equilibrium defined by the equation.
• In the exercise's result, the density of region 1 is found to be \( \frac{1}{2.5} \) times that of region 2, illustrating the inverse relationship between temperature and density.

This principle demonstrates how interdependently these quantities act under the assumptions of the Ideal Gas Law, offering insights into both theoretical and practical physics applications.