Problem 54 - College Algebra

Question

For the following exercises, use the given information to answer the questions. The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

Step-by-Step Solution

Answer

The volume is 800 cubic centimeters.

Step 1: Identify the relationship

We are given that the volume of a gas varies indirectly as the pressure. This means that when one increases, the other decreases, and vice versa. This can be written as the equation: \[ V = \frac{k}{P} \]where \( V \) is the volume, \( P \) is the pressure, and \( k \) is a constant.

Step 2: Find the constant of variation

We need to determine the constant \( k \) using the initial conditions provided: the volume \( V = 1200 \) cc and pressure \( P = 200 \) mmHg. Substitute these values into the equation to find \( k \): \[ 1200 = \frac{k}{200} \]Solving for \( k \), we get\[ k = 1200 \times 200 = 240,000 \]

Step 3: Calculate the new volume

Using the value of \( k = 240,000 \) from the previous step, we can find the new volume \( V \) when the pressure \( P \) is 300 mmHg. Substitute \( k \) and the new \( P \) into the equation:\[ V = \frac{240,000}{300} \]Simplifying this gives us:\[ V = 800 \]

Step 4: Interpret the result

The final answer means that when the pressure of the gas increases to 300 mmHg, the volume decreases to 800 cubic centimeters, demonstrating the indirect relationship between pressure and volume.

Key Concepts

Volume and Pressure Relationship

In the world of gases, the relationship between volume and pressure is a crucial concept. Understanding this link helps us grasp how gases behave under different conditions. The relationship can be described as an indirect or inverse variation. This means that as the pressure on a gas increases, its volume decreases, given that the temperature remains constant.
This counterbalancing act allows us to predict how one of these metrics will change when the other shifts.

The formula that expresses this relationship is:

• \[ V = \frac{k}{P} \]

Where:

• \( V \) is the volume, representing how much space the gas occupies.
• \( P \) is the pressure, which is the force exerted by the gas.
• \( k \) is the constant of variation.

A classic example of this is found in the behavior of gases when contained in a cylinder with a piston. If you push the piston down (increasing pressure), the gas compresses (decreasing volume). Conversely, if you pull the piston up (decreasing pressure), the gas expands (increasing volume). This inverse relationship is not just theoretical but is observed in practical situations involving gases.

Constant of Variation

To solve problems involving the indirect variation of gas volume and pressure, it's crucial to know about the 'constant of variation'. This constant, denoted as \( k \) in the equation \( V = \frac{k}{P} \), holds the key to resolving such questions.
Essentially \( k \) is what binds the indirect relationship between volume and pressure.

When you know the specific values of volume and pressure for a given condition, you can determine \( k \) by rearranging the formula:

• \[ k = V \times P \]

With this equation, once you have the volume and pressure from an initial state, you can calculate \( k \), and use it to find unknown variables when conditions change. In our exercise, when the volume was 1200 cubic centimeters and the pressure was 200 mmHg, substituting into our formula gave us \( k = 240,000 \).This constant helps solve for the new volume when the pressure changes, demonstrating the consistent nature of the gas's behavior under different pressures.

Gas Laws

Gas laws are a set of rules explaining how gases interact in our universe. These laws form the basis for understanding the behavior of gases under varying conditions like temperature and pressure. Specifically relevant here is Boyle’s Law, which highlights the indirect relationship between pressure and volume while keeping temperature steady.

Boyle’s Law is captured mathematically as:

• \[ P_1V_1 = P_2V_2 \]

In this equation:

• \( P_1 \) and \( V_1 \) stand for the initial pressure and volume.
• \( P_2 \) and \( V_2 \) signify the new conditions.

Using this law, if you know three of these quantities, you can derive the unknown one.
In our example, with an initial pressure \( P_1 \) of 200 mmHg and volume \( V_1 \) of 1200 cc, increasing the pressure \( P_2 \) to 300 mmHg led us to find a new volume \( V_2 \) of 800 cc.

This demonstration of Boyle’s Law allows students to see the practical application of theoretical physics concepts. Understanding these interactions is foundational for any study related to gas dynamics or chemistry, as it provides insight into how gases will behave in numerous real-world contexts.