Problem 5 - Astronomy Journey to the Cosmic Frontier

Question

Suppose the temperature of a normal gas doubles while its density remains the same. What happens to the pressure of the gas?

Step-by-Step Solution

Answer

The pressure of the gas doubles.

Step 1: Identify the Ideal Gas Law

The ideal gas law, given by the equation \( PV = nRT \), relates the pressure \( P \), volume \( V \), number of moles \( n \), gas constant \( R \), and temperature \( T \) of a gas.

Step 2: Understand the Implications

Since the density remains the same and density \( \rho = \frac{m}{V} \) (where \( m \) is mass and \( V \) is volume), we can infer that the volume does not change. Thus, we can focus on the equation \( P = \frac{nRT}{V} \) under constant density.

Step 3: Analyze the Temperature Change

The problem states that the temperature of the gas doubles. This means \( T \to 2T \). Substitute this into the equation: \( P = \frac{nR(2T)}{V} \).

Step 4: Calculate the Change in Pressure

Since \( n \), \( R \), and \( V \) are constant, the pressure \( P \) doubles as the temperature \( 2T \) appears in the numerator: \( P \to 2P \). Thus, pressure doubles.

Key Concepts

Gas Pressure

Gas pressure is a fundamental concept in understanding how gases behave. It is defined as the force exerted by gas particles against the walls of a container. This force results from the collision of gas molecules with the container boundaries. The more frequent and vigorous the collisions, the higher the pressure.

Pressure can be influenced by several factors:

• **Number of Molecules**: More molecules mean more collisions, leading to higher pressure.
• **Temperature**: Higher temperatures provide energy to molecules, increasing their speed and collision force, thus increasing pressure.
• **Volume**: In a larger container, particles have more space to move, which may reduce pressure if the number of particles remains constant.

The ideal gas law describes how pressure is related to other gas properties. According to this law, for a given amount of gas, pressure is directly proportional to temperature (when volume and number of moles are constant). Thus, understanding gas pressure is key to predicting how gases react to changes in temperature or volume.

Gas Temperature

Gas temperature is a critical property that determines the energy level of gas molecules. Temperature is a measure of the average kinetic energy of the gas particles. When you increase the temperature of a gas, its molecules move faster, which leads to more frequent and energetic collisions with the container walls, affecting the gas' pressure.

In practical terms, increasing the gas temperature can result in increased pressure if the volume and the number of gas molecules remain constant, as seen in the ideal gas law. This relationship shows why a balloon might expand in warm weather – the increased temperature raises the pressure inside.

In the scenario where the temperature doubles, such as given in the exercise, this means the gas molecules have twice the kinetic energy, assuming no change in the density of the gas. As a result, this doubling of temperature directly leads to a doubling of pressure according to the ideal gas law.

Gas Density

Gas density is the measure of how much mass a gas has in a given volume, formulated as density (\( \rho \) = \( \frac{m}{V} \)). It is a valuable property that gives insights into how gas molecules are distributed in space.

Unlike liquids and solids, gas density is highly variable and can significantly change with variations in pressure and temperature. For gases, density is closely tied to other gas properties through the ideal gas law:

• **Pressure**: Increasing pressure, while keeping temperature constant, typically increases density since the gas molecules are pushed closer together.
• **Temperature**: Raising the temperature at constant pressure tends to decrease density as the gas expands and its volume increases.

In the provided exercise, when the temperature of a gas doubles but the density stays the same, the volume must remain constant to balance the mass over the volume (since density is unchanged). This context helps us relate gas density to practical applications, illustrating how changes in temperature can affect gas properties while maintaining a steady density.