Problem 59 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Expansion and Contraction of Gases \(\quad\) In \(1787,\) Jacques Charles noticed that gases expand when heated and contract when cooled. A particular gas follows the model2\(y=\frac{5}{3} x+455\) where \(x\) is the temperature in Celsius and \(y\) is the volume in cubic centimeters. (a) What is the volume when the temperature is \(27^{\circ} \mathrm{C} ?\) (b) What is the temperature when the volume is 605 cubic centimeters? (c) Determine what temperature gives a volume of 0 cubic centimeters.

Step-by-Step Solution

Answer

(a) 500 cc, (b) 90°C, (c) -273°C.

Step 1: Determine the Volume at 27°C

Use the formula \( y = \frac{5}{3}x + 455 \) to find the volume when \( x = 27 \). Substitute \( x = 27 \) into the equation: \[y = \frac{5}{3} \times 27 + 455\]Calculate:\[\begin{align*}y &= \frac{135}{3} + 455 \y &= 45 + 455 \y &= 500.\end{align*}\]The volume is 500 cubic centimeters.

Step 2: Determine the Temperature for Volume of 605 cc

To find the temperature when the volume is 605 cubic centimeters, set \( y = 605 \) in the equation:\[605 = \frac{5}{3}x + 455\]First, subtract 455 from both sides:\[605 - 455 = \frac{5}{3}x\]This simplifies to:\[150 = \frac{5}{3}x\]Multiply both sides by \( \frac{3}{5} \) to solve for \( x \):\[x = 150 \times \frac{3}{5} = 90\]The temperature is 90°C.

Step 3: Find Temperature for Volume of 0 cc

Set \( y = 0 \) in the formula to find the temperature:\[0 = \frac{5}{3}x + 455\]Subtract 455 from both sides:\[-455 = \frac{5}{3}x\]Multiply both sides by \( \frac{3}{5} \) to solve for \( x \):\[x = -455 \times \frac{3}{5} = -273\]The temperature is -273°C.

Key Concepts

Charles's Law

Charles's Law is a fundamental principle in chemistry that describes the relationship between the volume and temperature of a gas. It specifically states that for a given mass of gas at constant pressure, the volume is directly proportional to its temperature in Kelvin. This means that when the temperature of a gas increases, its volume also increases, and vice versa. This law is crucial for understanding how gases behave under different thermal conditions. When applying Charles's Law, it's important to always use temperature measurements in Kelvin. The Kelvin scale is the standard for scientific calculations involving temperature because it starts at absolute zero, where molecular motion theoretically stops. Charles's Law helps us to predict and explain numerous phenomena related to gases, including the rising of a hot air balloon and the fact that car tires might become over-inflated on a hot day.

Linear Equations

Linear equations are mathematical expressions that describe a straight line when plotted on a graph. These equations take the form of \(y = mx + b\), where \(m\) represents the slope of the line, \(x\) is the independent variable, and \(b\) is the y-intercept. In the context of Charles's Law, the linear equation \(y = \frac{5}{3}x + 455\) represents the relationship between temperature (\(x\)) and volume (\(y\)).
The slope \(\frac{5}{3}\) indicates how much the volume of the gas changes for every unit change in temperature. The y-intercept 455 gives us the volume of the gas at 0°C. Linear equations are powerful tools because they allow us to calculate unknown values when given some other known values. Such equations are essential in various scientific and engineering applications where predicting or finding specific variables is necessary.

Temperature-Volume Relationship

The temperature-volume relationship is at the heart of understanding how gases expand and contract. In simple terms, as the temperature of a gas increases, its volume also increases, and as the temperature decreases, its volume decreases. This relationship is crucial in practical applications such as meteorology, automotive engineering, and even in culinary arts when dealing with gas appliances.

• When a gas is heated, the particles gain kinetic energy and move more vigorously, causing the gas to expand.
• Conversely, when a gas is cooled, its particles lose kinetic energy, and the gas contracts.

This principle helps explain everyday occurrences such as the inflation of a balloon in the sun or the shrinking of an air mattress in cooler temperatures. The consistent proportional change in volume with temperature is mathematically described by Charles's Law.

Solving Equations

Solving equations involves finding the value of a variable that satisfies a given mathematical expression. In our exercise, we solve the equation \(y = \frac{5}{3}x + 455\) to find specific values for either temperature or volume. Here are the steps:

In Step 1, to find the volume at a temperature of 27°C, substitute \(x = 27\) into the equation and calculate to find \(y = 500\) cm³.
In Step 2, to find the temperature for a specified volume (605 cm³), set \(y = 605\), then solve for \(x\) by manipulating the equation to find that \(x = 90\)°C.
In Step 3, to determine when the volume is zero, set \(y = 0\) and solve for \(x\), resulting in \(x = -273\)°C. These solutions involve algebraic manipulation by isolating the variable of interest, often through steps like addition, subtraction, multiplication, or division.