Question

You have a gas at \(25^{\circ} \mathrm{C}\) confined to a cylinder with a movable piston. Which of the following actions would double the gas pressure? (a) Lifting up on the piston to double the volume while keeping the temperature constant; (b) Heating the gas so that its temperature rises from \(25^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\), while keeping the volume constant; (c) Pushing down on the piston to halve the volume while keeping the temperature constant.

Short answer

In conclusion, pushing down on the piston to halve the volume while keeping the temperature constant (action c) would double the gas pressure.

Step-by-step solution

Action (a): Doubling the Volume

For this action, we'll lift the piston to double the volume of the gas while keeping the temperature constant. Using the Ideal Gas Law, we can write the initial and final states as follows: Initial state: \(PV = nRT\) Final state: \(P_{final} (2V) = nR(298.15)\) We want to determine the relationship between the initial and final pressures: \(\frac{P_{final}}{P} = \frac{nRT}{2nR(298.15)} = \frac{1}{2}\) The pressure in this case is halved, not doubled. So, action (a) does not double the gas pressure.

Action (b): Heating the Gas

In this action, we'll heat the gas so that its temperature rises from \(25^{\circ} C\) to \(50^{\circ} C\) while keeping the volume constant. First, convert the final temperature to Kelvin: \(50^{\circ}C + 273.15 = 323.15 K\) Now, let's use the Ideal Gas Law for the initial and final states: Initial state: \(PV = nR(298.15)\) Final state: \(P_{final}V = nR(323.15)\) Using the relationship between the initial and final pressures, we get: \(\frac{P_{final}}{P} = \frac{nR(323.15)}{nR(298.15)} = \frac{323.15}{298.15}\) The pressure does not double in this case. Therefore, action (b) does not double the gas pressure.

Action (c): Halving the Volume

For this action, we'll push down on the piston to halve the volume of the gas while keeping the temperature constant. Using the Ideal Gas Law, we can write the initial and final states as follows: Initial state: \(PV = nRT\) Final state: \(P_{final} \frac{V}{2} = nRT\) To determine the relationship between the initial and final pressures, we get: \(\frac{P_{final}}{P} = \frac{2nRT}{nRT} = 2\) The pressure doubles in this case. Thus, action (c) doubles the gas pressure. In conclusion, pushing down on the piston to halve the volume while keeping the temperature constant (action c) would double the gas pressure.

Key concepts

Gas Pressure

Gas pressure is an essential concept when studying gases and their behaviors. It represents the force exerted by gas molecules as they collide with the walls of their container. In the context of the ideal gas law, pressure (\(P\)) is one of the primary variables.

Here’s how pressure relates to gas behavior:

• Gas molecules are constantly in motion. When they hit container walls, they exert force, which we perceive as pressure.
• Temperature and volume are crucial in determining this pressure. As temperature increases, molecules move faster, and pressure tends to increase if volume is constant.
• When considering a confined space like a cylinder, any change in volume impacts pressure, especially if temperature or the number of molecules remains unchanged.

Understanding gas pressure helps explain why changing the volume or temperature of a gas affects how much pressure it exerts.

Volume and Temperature Relationship

The relationship between volume and temperature is fundamental when analyzing gases using the ideal gas law. This relationship is often described by Charles's Law, which states that at constant pressure, the volume of a gas is directly proportional to its absolute temperature.

If we consider \(V\) to be the volume and \(T\) the temperature in Kelvin, an increase in temperature typically leads to an increase in volume and vice versa:

• When the temperature of a gas increases at constant pressure, the gas molecules move more vigorously. This increases their kinetic energy and can cause the volume of the gas to expand if the container allows it.
• On the flip side, decreasing the temperature tends to reduce the volume because the gas molecules lose kinetic energy and move less.

In situations where volume is held constant and temperature changes, like in action (b) from the exercise, volume doesn't increase, but pressure does due to the heightened energy of the molecules.

Movable Piston

A movable piston is a common tool used to demonstrate the ideal gas law in action. It involves a cylinder with a piston that can slide up or down to change the gas volume within the cylinder.

Here's what happens when working with a movable piston:

• Changing the piston's position directly alters the gas volume. If the piston is raised, the volume increases; if lowered, the volume decreases.
• This change in volume directly affects gas pressure when the temperature is held constant. According to Boyle's Law, pressure and volume are inversely related. So, decreasing volume by moving the piston down increases pressure, as demonstrated in action (c) from the exercise.
• The piston adds a dynamic element to studying gas behavior. It's easier to demonstrate how gas behavior changes in response to volume alterations.

Movable pistons are vital in experiments illustrating these concepts by allowing direct observation of pressure and volume changes under controlled conditions.