Problem 24 - Precalculus: Functions and Graphs

Question

The ideal gas law The ideal gas law states that the volume \(V\) that a gas occupies is directly proportional to the product of the number \(n\) of moles of gas and the temperature \(T\) (in K) and is inversely proportional to the pressure \(P\) (in atmospheres). (a) Express \(V\) in terms of \(n, T, P,\) and a constant of proportionality \(k\) What is the effect on the volume if the number of moles is doubled and both the temperature and the pressure are reduced by a factor of one-half?

Step-by-Step Solution

Answer

The volume remains unchanged when the number of moles is doubled and both temperature and pressure are halved.

Step 1: Understand the Ideal Gas Law

The ideal gas law can be expressed in the equation form with a constant of proportionality. This equation is usually written as \( PV = nRT \), where \( R \) is a constant. For our purposes, we can express it as \( V = \frac{k \cdot n \cdot T}{P} \).

Step 2: Analyze the Given Conditions

The problem asks us to determine what happens to \( V \) when the number of moles \( n \) is doubled, and both temperature \( T \) and pressure \( P \) are reduced by a factor of 1/2.

Step 3: Substitute and Solve for Conditions

Substituting the new conditions into the equation yields: \[ V' = \frac{k \cdot (2n) \cdot (\frac{T}{2})}{\frac{P}{2}} \].

Step 4: Simplify the Equation

Simplify \( V' = \frac{k \cdot 2n \cdot \frac{T}{2}}{\frac{P}{2}} \) by canceling and rearranging, which simplifies as follows: \[ V' = \frac{k \cdot n \cdot T}{P} \] or \( V' = V \).

Step 5: Conclusion

Since the resulting equation remains \( V' = V \), the volume \( V \) remains unchanged despite the changes in conditions.

Key Concepts

Volume of Gas

Understanding the volume of a gas is crucial when applying the ideal gas law. In essence, the volume represents the space occupied by a gas under specific conditions of pressure and temperature. According to the ideal gas law, this volume is directly proportional to the number of moles of gas and the temperature measured in Kelvin. It is inversely proportional to the pressure exerted on the gas. The key equation representing this relationship is written as:

• \( V = \frac{k \cdot n \cdot T}{P} \)

This equation tells us that when either the number of moles or the temperature increases, the volume will also increase if all other factors remain constant. Alternatively, if pressure increases, the volume will decrease, indicating an inverse relationship.

Moles of Gas

The moles of gas, denoted as \( n \), represent the quantity of gas present in a system. They serve as a measure of the amount of substance, and in the context of the ideal gas law, they directly affect the volume the gas occupies. The more moles present, the larger the volume, assuming temperature and pressure remain constant. This relationship is essential when predicting how changes in the gas's quantity will affect its behavior. When we talk about doubling the moles, as in the exercise, we're doubling the amount of gas available to occupy a space. In the ideal gas law equation, this change directly impacts the volume, expressed as \( V = \frac{k \cdot (2n) \cdot T}{P} \), leading to a larger volume unless countered by changes in temperature or pressure.

Temperature in Kelvin

Temperature plays a crucial role in dictating the behavior of gases, as it influences the kinetic energy of gas molecules. In the ideal gas law, temperature must always be measured in Kelvin to ensure a direct proportional relationship with the volume. Here, Kelvin provides an absolute scale where zero Kelvin signifies no molecular motion.

• When temperature increases, the kinetic energy of molecules rises. This expansion increases the volume assuming constant pressure and moles.
• Conversely, reducing temperature diminishes molecular movement, effectively reducing the volume.

In the given exercise, halving the temperature (\( T \rightarrow \frac{T}{2} \)) would cause a proportional decrease in volume if pressure and moles remained unchanged.

Pressure in Atmospheres

Pressure, measured in atmospheres, indicates how much force the gas molecules exert on their container's walls. According to Boyle’s law, and subsequently the ideal gas law, pressure and volume have an inverse relationship when moles and temperature stay constant.

• An increase in pressure will reduce the volume, assuming moles and temperature don't change.
• A decrease in pressure allows the volume to increase.

In our exercise, reducing the pressure by half \( (P \rightarrow \frac{P}{2}) \) while doubling the temperature effectively counters the volume effects, resulting in no net change in volume. This highlights the interplay of different variables and how they balance each other in the context of the ideal gas law.