Problem 119 - Introductory Chemistry Atoms First

Question

What happens to the volume of a gas in a cylinder with a movable piston if: (a) The pressure is doubled while the temperature is held constant? (b) The temperature is doubled while the pressure is held constant? (c) The pressure and the temperature are both doubled? (The movable piston means the volume of the cylinder, and therefore of the gas, can change.)

Step-by-Step Solution

Answer

(a) When the pressure is doubled while the temperature is held constant, the volume of the gas in the cylinder with a movable piston will be reduced to half of its initial volume: \(V_{new} = \frac{1}{2}V\). (b) When the temperature is doubled while the pressure is held constant, the volume of the gas in the cylinder with a movable piston will be doubled compared to its initial volume: \(V_{new} = 2V\). (c) When both the pressure and the temperature are doubled, the volume of the gas in the cylinder with a movable piston remains the same as its initial volume: \(V_{new} = V\).

Step 1: (a) Pressure is doubled, temperature constant

We are given that the pressure is doubled (\(P_{new} = 2P\)) while the temperature remains constant (\(T_{new} = T\)). We will use the initial and final conditions in the ideal gas equation to find the effect on volume. Initial Condition: \(PV = nRT\) Final Condition: \((2P)V_{new} = nR(T)\) Now, divide the final condition by the initial condition: \(\frac{(2P)V_{new}}{PV} = \frac{nRT}{nRT}\) Solving for \(V_{new}\): \(V_{new} = \frac{1}{2}V\) Thus, when the pressure is doubled while the temperature is held constant, the volume of the gas in the cylinder with a movable piston will be reduced to half of its initial volume.

Step 2: (b) Temperature is doubled, pressure constant

Now we are given that the temperature is doubled (\(T_{new} = 2T\)) while the pressure remains constant (\(P_{new} = P\)). Again, use initial and final conditions in the ideal gas equation. Initial Condition: \(PV = nRT\) Final Condition: \(P(V_{new}) = nR(2T)\) Now, divide the final condition by the initial condition: \(\frac{P(V_{new})}{PV} = \frac{2nRT}{nRT}\) Solving for \(V_{new}\): \(V_{new} = 2V\) Thus, when the temperature is doubled while the pressure is held constant, the volume of the gas in the cylinder with a movable piston will be doubled compared to its initial volume.

Step 3: (c) Pressure and temperature are doubled

Here, both pressure and temperature are doubled (\(P_{new} = 2P\) and \(T_{new} = 2T\)). Use the initial and final conditions in the ideal gas equation. Initial Condition: \(PV = nRT\) Final Condition: \((2P)V_{new} = nR(2T)\) Now, divide the final condition by the initial condition: \(\frac{(2P)V_{new}}{PV} = \frac{2nRT}{nRT}\) Solving for \(V_{new}\): \(V_{new} = V\) Thus, when both the pressure and the temperature are doubled, the volume of the gas in the cylinder with a movable piston remains the same as its initial volume.

Key Concepts

Pressure-Volume Relationship

When studying gases, understanding the pressure-volume relationship is key. This relationship is governed by Boyle's Law, which states that for a given amount of gas at constant temperature, the pressure of a gas is inversely proportional to its volume. In simpler terms, if you increase the pressure, the volume decreases as long as the temperature doesn't change.

For instance, in a cylinder with a movable piston, if the pressure is doubled, the volume of the gas is halved. This happens because the gas particles are pushed closer together by the increased pressure. The relationship is expressed in the equation:\[ PV = k \]where \(P\) is pressure, \(V\) is volume, and \(k\) is a constant.

• When \(P\) is doubled, \(V\) becomes \(\frac{1}{2}V\).
• When \(V\) is halved, \(P\) becomes \(2P\).

By keeping the temperature constant, we can observe this clear inverse relationship, ensuring the total energy remains unchanged.

Temperature-Volume Relationship

Another fundamental concept in gas behavior is the temperature-volume relationship. This is explained by Charles's Law, which states that for a fixed amount of gas at constant pressure, the volume of a gas is directly proportional to its temperature. To put it simply, if you increase the temperature, the volume increases, given that the pressure is constant.

Consider again our gas within a movable cylinder. If the temperature is doubled, the volume will also double. This occurs because the gas particles gain energy and move more vigorously, pushing the piston outward to accommodate the increased kinetic energy. The relationship is described by the formula:\[ \frac{V}{T} = \text{constant} \]where \(V\) is volume and \(T\) is temperature in Kelvin.

• When \(T\) is doubled, \(V\) becomes \(2V\).
• This direct relationship allows the volume to change identically with temperature variations, provided pressure is steady.

This fundamental principle helps explain why a balloon shrinks in cold environments and expands when heated.

Gas Laws in Chemistry

Gas laws are essential tools in chemistry that describe how gases behave under various conditions. The Ideal Gas Law is a synthesis of several simpler gas laws and is typically expressed as:\[ PV = nRT \]where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin.

These gas laws encompass several specific relationships:

• Boyle's Law: Pressure and volume are inversely proportional at constant temperature.
• Charles's Law: Volume and temperature are directly proportional at constant pressure.
• Avogadro's Law: Volume and the amount of gas are directly proportional at constant pressure and temperature.

When both pressure and temperature change, as in some of our example cases, the Ideal Gas Law helps us predict the outcome:

- Doubling both the pressure and temperature keeps volume unchanged because the effects cancel each other out.- Understanding these interrelationships is crucial for solving complex chemistry problems and predicting gas behavior in real-life scenarios.