Question

Suppose a steel tank is filled with a gas at sea level. This tank keeps the volume constant. If the tank is placed in the bed of a truck and driven up a mountain road to an elevation of \(10,000 \mathrm{ft}\), what happens to the gas particles?

Short answer

As the tank is driven up the mountain to a higher altitude, the external pressure decreases. In response to this change, the gas particles inside the tank, which is of a constant volume, move faster. This increase in particle movement corresponds to an increase in the internal pressure of the gas in the tank.

Step-by-step solution

Understanding Basic Gas Laws

Boyle's Law states that for a fixed amount of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure. This implies that if the volume is kept constant (as in our case), and the external pressure decreases (as we drive up the mountain), the pressure of the gas inside the tank should increase.

Visualizing the Scenario

Conceptually, at sea level, the gas particles in the tank are at equilibrium with the external pressure. As the external environmental pressure decreases with altitude, the gas particles inside the tank find less resistance to their movement, leading to an increase in their speed and thus an increase in internal pressure.

Relating Pressure and Particle Movement

According to Gay-Lussac's law, the pressure of a gas is directly proportional to its absolute temperature if the volume is kept constant, which means higher the pressure, higher the kinetic energy of the gas particles, and therefore faster they move. So in this case as the internal pressure increases due to decrease in external pressure, the gas particles will move faster.

Key concepts

Pressure

Pressure is a fundamental concept in the study of gases and is defined as the force exerted per unit area. In the context of gas laws, pressure is essential to understand how gases behave under varying conditions.
As you drive up a mountain, atmospheric pressure decreases. This change impacts a sealed tank filled with gas. When external pressure decreases, the internal pressure in the tank increases if the volume remains constant. This is an effect aligned with Boyle's Law, which states that the pressure of a gas is inversely proportional to its volume, provided temperature remains unchanged.
Thus, as elevation increases and external pressure drops, the pressure within a fixed volume container will inevitably rise. In practical terms, this means gas molecules will exert greater force on the walls of the tank, increasing the internal pressure.

• This increase in pressure translates to molecules moving more energetically.
• Pressure is a direct consequence of particle collisions with the tank walls.

Understanding these nuances of pressure helps us predict how gases will respond to environmental changes and is vital in scenarios like transportation of pressurized containers through varying altitudes.

Volume

Volume refers to the amount of space a gas occupies. In this exercise, the tank's volume is constant, meaning it doesn't change as the truck moves up the mountain. In the realm of gas laws, volume plays a crucial role, especially in Boyle's Law.
Boyle's Law highlights that at a constant temperature, the pressure of a gas is inversely related to its volume. Hence, when the volume is unchanged and the external pressure decreases, the pressure inside the tank inevitably increases as a compensatory measure.

• Keeping the volume constant simplifies calculations as one variable, pressure, is then solely responsible for changes.
• The fixed volume ensures that any variation in the internal pressure is because of changes in external conditions, like altitude.

These insights into volume help us understand how gases behave in enclosed systems, predicting their reactions to outside environmental changes. In practical applications, it assures that the pressure inside remains stable unless external conditions prompt a shift.

Kinetic Energy

Kinetic energy in gases relates to the motion and speed of particles. According to Gay-Lussac's law, as internal pressure increases, the kinetic energy of gas particles also increases if the volume stays the same.
In our scenario, as the truck climbs a mountain, atmospheric pressure decreases. This decrease means the internal pressure has to increase (since volume stays constant), boosting the kinetic energy of the gas particles.

• Kinetic energy can be thought of as the energy of motion—faster-moving particles have more kinetic energy.
• When particles move faster, they collide more energetically with the walls of the container, increasing pressure further.

This increase in kinetic energy not only results in higher pressure but also suggests that the temperature inside the tank may increase slightly, according to the collision theory and Gay-Lussac's law. Thus, understanding kinetic energy is crucial for predicting and managing the behavior of gases in variable altitude conditions.