Problem 94 - A Graphical Approach to College Algebra

Question

Solve each problem. Volume of a Gas Natural gas provides \(25 \%\) of U.S. energy. The volume of a gas varies inversely with the pressure and directly with the temperature. (Temperature must be measured in degrees Kelvin (K), a unit of measurement used in physics.) If a certain gas occupies a volume of 1.3 liters at \(300 \mathrm{K}\) and a pressure of 18 newtons per square centimeter, find the volume at \(340 \mathrm{K}\) and a pressure of 24 newtons per square centimeter.

Step-by-Step Solution

Answer

The new volume is approximately 1.105 liters.

Step 1: Understand Inverse and Direct Variation

The problem states that the volume varies inversely with pressure and directly with temperature. Thus, we can express the relationship as \( V \propto \frac{T}{P} \), where \( V \) is the volume, \( T \) is the temperature in Kelvin, and \( P \) is the pressure.

Step 2: Establish the Constant of Variation

Since \( V \propto \frac{T}{P} \), there exists a constant \( k \) such that \( V = k \frac{T}{P} \). For the given conditions, we use \( V = 1.3 \) L, \( T = 300 \) K, and \( P = 18 \) N/cm². Substitute these into the equation to find \( k \): \[ 1.3 = k \frac{300}{18} \] Therefore, \( k = 1.3 \times \frac{18}{300} = 0.078. \)

Step 3: Use the Constant to Find New Volume

Now that we have \( k = 0.078 \), calculate the new volume \( V' \) when \( T' = 340 \) K and \( P' = 24 \) N/cm². Plug these values into the equation: \[ V' = k \frac{T'}{P'} = 0.078 \times \frac{340}{24}. \] Solve for \( V' \): \[ V' = 0.078 \times \frac{340}{24} \approx 1.105. \] Therefore, the new volume is approximately 1.105 liters.

Key Concepts

Volume of a Gas

In the fascinating world of gases, understanding how volume changes with other factors is crucial. Volume, in the context of gases, refers to the amount of space that a gas occupies. It can change when conditions like pressure and temperature vary.
When we say that the volume of a gas varies inversely with pressure and directly with temperature, we are talking about a relationship that can be mathematically represented. In simple terms:

• As pressure increases, the volume decreases if the temperature remains constant.
• As temperature increases, the volume increases if the pressure is constant.

This dual dependency on pressure and temperature is expressed as: \[ V \propto \frac{T}{P} \]This tells us that if you know the temperature, pressure, and volume in one state, you can find the missing volume in a new state, assuming you know the new conditions.

Temperature and Pressure Relationship

The relationship between temperature and pressure is known as Gay-Lussac's Law in gas laws. This principle states that the pressure of a gas is directly proportional to its temperature when the volume is held constant.
The Kelvin scale is typically used for this kind of calculation because it starts at absolute zero—where molecular motion ceases—making it ideal for these physical calculations. When you alter the temperature, either increasing or decreasing it, the pressure will follow suit proportionally if the volume remains constant.
In many practical scenarios, including our problem, we see how these changes affect volume. Remember:

• Higher temperature means higher pressure, assuming volume doesn't change.
• Lower temperature means lower pressure under the same conditions.

This is crucial when predicting how gases behave in different environments.

Constant of Variation

The constant of variation, often denoted with the letter \( k \), is a key piece of the inverse and direct variation relationship. It links the directly proportional components of temperature and pressure to volume.
In the given problem:- The volume \( V \) is proportional to \( \frac{T}{P} \), thus expanding this gives us \( V = k \cdot \frac{T}{P} \).Determining \( k \) is essential, as it provides the baseline from which you calculate any unknown variables when conditions change.
In our exercise step by step solution, we calculated \( k = 0.078 \) using known values. This constant allows us to predict the new volume of the gas when conditions such as temperature (\( T \)) and pressure (\( P \)) change.
With an understanding of \( k \), we can easily navigate through different states of gas, making these calculations dependable across varying scenarios.