Problem 94 - A Graphical Approach to Precalculus with Limits

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 kelvins (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 volume is approximately 1.105 liters.

Step 1: Understand the Relationship

The volume of a gas varies inversely with pressure and directly with temperature. This can be expressed as \( V \propto \frac{T}{P} \) where \( V \) is the volume, \( T \) is the temperature, and \( P \) is the pressure.

Step 2: Write the Proportionality as an Equation

Convert the proportionality to an equation using a constant of proportionality \( k \). The equation is \( V = k \frac{T}{P} \).

Step 3: Find the Constant of Proportionality

Use the initial conditions (\( V = 1.3 \) L, \( T = 300 \) K, \( P = 18 \) N/cm²) to find \( k \):\[ 1.3 = k \frac{300}{18} \]Solving for \( k \), we get:\[ k = \frac{1.3 imes 18}{300} \approx 0.078 \]

Step 4: Calculate the New Volume

Using the new temperature and pressure conditions (\( T = 340 \) K, \( P = 24 \) N/cm²), the volume \( V' \) can be calculated as:\[ V' = k \frac{T'}{P'} = 0.078 \frac{340}{24} \]Calculate \( V' \):\[ V' = 0.078 \times 14.1667 \approx 1.105 \]

Step 5: Present the Solution

The volume of the gas at 340 K and 24 N/cm² is approximately 1.105 liters.

Key Concepts

Volume of a Gas

The volume of a gas is an important concept in both chemistry and physics. It's the space that the gas molecules occupy. According to the principle of inverse variation, the volume of a gas varies inversely with pressure and directly with temperature. This means that as pressure increases, volume decreases if temperature remains constant. On the other hand, if temperature increases, the volume will also increase when pressure is held constant. When these principles are combined, it allows us to calculate the behavior of gases under different conditions. This relationship helps scientists in real-world applications such as in the design of engines and even in predicting weather patterns.

A simple equation that expresses this relationship is:

• \( V \propto \frac{T}{P} \)

Pressure

Pressure is the force that a gas exerts on the walls of its container. It is vital to understand this to predict and manipulate gas behavior. Pressure is usually measured in newtons per square centimeter (N/cm²) or other units like atmospheres (atm) and Pascals (Pa).

This force comes from gas molecules repeatedly hitting the walls of a container. The more these molecules move and collide, the higher the pressure. If we increase the volume without adding more gas, fewer collisions occur, which reduces pressure. Conversely, compressing the gas into a smaller volume increases these collisions, hence the pressure goes up. This behavior follows the principle of inverse variation in the context of gases, where lower volume results in higher pressure, assuming unchanged temperature.

Temperature

Temperature is a measure of how hot or cold something is, but in the context of gases, it indicates the average kinetic energy of the molecules. In scientific measurements, we use the Kelvin scale because it starts at absolute zero, the theoretical point where molecular activity ceases. This scale allows for more precise calculations involving gas laws.

In gas behavior, temperature directly affects how fast molecules move. Higher temperatures result in faster movement and, therefore, create more pressure if the volume is constant. Alternatively, to keep pressure constant when temperature increases, the volume must increase as well, showcasing direct variation. For our example, a temperature increase from 300 K to 340 K leads to the expectation of increased gas volume, assuming that pressure has not disproportionately increased.

Proportionality Constant

The proportionality constant, \( k \), is a pivotal component in the mathematical relationship of gas volume, temperature, and pressure. This constant helps translate the proportional relationship into a precise equation that can be used for calculations.

• Using the formula \( V = k \frac{T}{P} \), \( k \) represents the specific characteristics of a gas under a set of standardized conditions.

It's determined using initial conditions, such as temperature, pressure, and volume, allowing the equation to be applied universally for that particular gas scenario.

In the given exercise, we defined \( k \) as approximately 0.078 based on initial conditions of a 1.3-liter volume at 300 Kelvin and pressure of 18 N/cm². With this constant, we can now calculate volume changes as pressure and temperature vary, always considering the same type or condition of gases.