Problem 78 - Precalculus: Functions and Graphs

Question

Boyle's law Boyle's law for a certain gas states that if the temperature is constant, then \(p v=200,\) where \(p\) is the pressure (in \(\mathrm{Ib} / \mathrm{in}^{2}\) ) and \(v\) is the volume (in in \(^{3}\) ). If \(25 \leq v \leq 50\) what is the corresponding range for \(p ?\)

Step-by-Step Solution

Answer

The range for \( p \) is \( 4 \leq p \leq 8 \).

Step 1: Understand Boyle’s Law

Boyle's law states that the product of pressure and volume is constant for a given amount of gas, as long as the temperature remains constant. Mathematically, this is expressed as \( pv = 200 \). This means the pressure \( p \) and volume \( v \) are inversely proportional.

Step 2: Write the Equation for Pressure

From the equation \( pv = 200 \), we can express the pressure \( p \) in terms of volume \( v \):\[ p = \frac{200}{v} \].

Step 3: Calculate Minimum Pressure

To find the minimum pressure, we substitute the maximum value of \( v \) into the equation for \( p \), as pressure is inversely proportional to volume. Substitute \( v = 50 \):\[ p = \frac{200}{50} = 4 \].

Step 4: Calculate Maximum Pressure

To find the maximum pressure, we substitute the minimum value of \( v \) into the equation for \( p \). Substitute \( v = 25 \):\[ p = \frac{200}{25} = 8 \].

Step 5: Determine the Range for Pressure

Based on the calculations of minimum and maximum pressure, the range for \( p \) when \( 25 \leq v \leq 50 \) is \[ 4 \leq p \leq 8 \].

Key Concepts

Inverse Proportionality

Inverse proportionality is a mathematical relationship where one variable increases as another decreases. In the context of Boyle's Law, pressure (\( p \)) and volume (\( v \)) are inversely proportional. This means if the volume of a gas increases, the pressure decreases, provided the temperature remains constant. Conversely, if the volume decreases, the pressure increases. This relationship can be laid out simply: the product of the two variables, pressure and volume in this case, remains constant. Hence, we say:

• When volume goes up, pressure comes down.
• When volume goes down, pressure goes up.

Understanding inverse proportionality helps in visualizing how changes in volume affect pressure and vice versa, a key aspect of Boyle's Law.

Pressure-Volume Relationship

The pressure-volume relationship in gases is a foundational concept of Boyle's Law. Specifically, it is expressed as\[ p \times v = k \], where \( p \) is pressure, \( v \) is volume, and \( k \) is a constant. In our given problem, \( k \) is 200, as: \( p \times v = 200 \). This equation tells us that as we alter the volume of the gas, the pressure compensates change to keep the product constant.
This relationship also outlines the range of possible pressures. For volumes ranging from 25 to 50 cubic inches, the pressure will vary inversely:

• At maximum volume (50 in³) the pressure is minimum (4 Ib/in²).
• At minimum volume (25 in³) the pressure is maximum (8 Ib/in²).

This demonstrates the consistent behavior of gases under constant temperature conditions.

Constants in Gas Laws

Constants, such as the one in Boyle's Law, are crucial as they provide the necessary framework to understand how gases behave under different conditions. In Boyle's Law, the constant \( k \) ensures that no matter how volume and pressure change, their product remains the same under a constant temperature.
In our specific example, the constant is 200. This means the product of pressure and volume of our gas will always equal 200. It's an essential feature of gas laws as it provides predictability and stability in calculations, helping us accurately determine other variables once we know one part of the system.

Calculus in Gas Laws

Calculus often plays an underlying role in understanding gas laws, especially with concepts like differentiating equations to understand rates of change. However, in simpler terms as used in Boyle’s Law, calculus helps us identify how variables like volume and pressure interact beyond static calculation.
In calculus, when we say something is inversely proportional, we might explore how the derivative of one variable is related to another. For practical purposes and beginning stages, though, recognizing that changing one variable systematically affects another is in itself a calculus insight.
Through understanding these changes and relationships, such as with the derivative link between volume and pressure in Boyle’s law, we can better predict and understand the behavior of gases under different conditions and transitions.