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Chapter 17: Problem 13

Boyle's law states that for a confined gas at a constant temperature, the product of the pressure and the volume is a constant. Another way of stating this law is that the pressure is inversely proportional to the volume, or that the volume is inversely proportional to the pressure. Assume a constant temperature in the following problems. The air in a cylinder is at a pressure of \(14.7 \mathrm{lb} / \mathrm{in.}^{2}\) and occupies a volume of 175 in. \(^{3}\) Find the pressure when it is compressed to 25.0 in. \(^{3} .\)

Short Answer

Expert verified
The pressure when the air is compressed to 25.0 in.cubed is 41.1 lb/in.^2.

Step by step solution

01

Write Down Boyle's Law Formula

Boyle's Law can be expressed mathematically as: \( P_1V_1 = P_2V_2 \) where \( P_1 \) and \( V_1 \) are the initial pressure and volume, and \( P_2 \) and \( V_2 \) are the final pressure and volume.
02

Insert Known Values

Insert the known initial pressure and volume values, and the final volume into the formula: \( 14.7 \, \mathrm{lb/in}^2 \times 175 \, \mathrm{in}^3 = P_2 \times 25.0 \, \mathrm{in}^3 \).
03

Solve for the Final Pressure, \( P_2 \)

Rearrange the formula to find \( P_2 \) and then solve: \( P_2 = \frac{14.7 \, \mathrm{lb/in}^2 \times 175 \, \mathrm{in}^3}{25.0 \, \mathrm{in}^3} \). Calculate the value of \( P_2 \) to find the final pressure.
04

Calculate the Final Pressure

After performing the calculation, \( P_2 = \frac{1027.5 \, \mathrm{lb/in}^2}{25.0 \, \mathrm{in}^3} = 41.1 \, \mathrm{lb/in}^2 \) which is the pressure of the air when compressed to 25.0 in. \( ^3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Boyle's Law
Boyle's Law, named after physicist Robert Boyle, is central to understanding the behavior of gases under pressure. This physical law demonstrates the pressure volume relationship, illustrating that for a fixed mass of gas, at constant temperature, the pressure and volume of a gas are inversely proportional.

This means that when the volume of a gas increases, the pressure decreases, and vice versa, assuming the temperature doesn't change. The elegance of this relationship is critical for various fields, including chemistry, physics, and even medicine, where it helps predict how gases will behave under different pressures.
Inverse Proportionality in Boyle's Law
The term inverse proportionality, as it applies to Boyle's Law, is what allows us to predict how a change in volume will affect pressure, or vice versa, keeping temperature constant. It expresses a specific mathematical relationship, where the product of two variables is constant if all the other conditions remain the same.

This proportional relationship is visually represented by a hyperbolic curve on a graph, where one variable increases as the other decreases. Understanding this relationship is key to solving problems that involve changes in volume and pressure, as demonstrated in the textbook exercise.
The Role of Gas Laws
The concept of gas laws encompasses a series of fundamental laws in physics and chemistry that describe the behavior of gases. Boyle's Law is one of these foundational principles, alongside Charles's Law, Gay-Lussac's Law, and Avogadro's Law.

When combined, these laws form the ideal gas law, a critical equation that relates the pressure, volume, temperature, and number of moles of a gas. Understanding each law individually enables students to grasp how gases will respond to different situations, which is essential for various scientific calculations and real-world applications.
Constant Temperature Conditions
Maintaining constant temperature conditions is a key aspect when applying Boyle's Law. Temperature is a measure of the average kinetic energy of particles in a substance, and when it remains unchanged, it ensures that the only factors affecting the gas are pressure and volume.

This is an example of an isothermal process, where the temperature is constant, and the energy of the system is conserved. In practical scenarios, such as the compression of gas in an engine cylinder or the expansion of lungs during breathing, Boyle's Law provides insight only when the assumption of constant temperature holds true.

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Most popular questions from this chapter

The intensity of illumination at a given point is directly proportional to the intensity of the light source and inversely proportional to the square of the distance from the light source. If a desk is properly illuminated by a 75.0 - \(\mathrm{W}\) lamp 8.00 ft from the desk, what size lamp will be needed to provide the same lighting at a distance of \(12.0 \mathrm{ft} ?\)

Newton's law of gravitation states that any two bodies attract each other with a force that is inversely proportional to the square of the distance between them. The force of attraction between the earth and some object is called the weight of that object. The law of gravitation states, then, that the weight of an object is inversely proportional to the square of its distance from the center of the earth. If a person weighs 150 lb on the surface of the earth (assume this to be 3960 mi from the center), how much will he weigh 1500 mi above the surface of the earth?

If \(y\) varies directly as \(x,\) and \(y\) is 56 when \(x\) is \(21,\) find \(y\) when \(x\) is 74.

The inverse square law states that for a surface illuminated by a light source, the intensity of illumination on the surface is inversely proportional to the square of the distance between the source and the surface. A light source located \(2.75 \mathrm{m}\) from a surface produces an illumination of \(528 \mathrm{lux}\) on that surface. Find the illumination if the distance is changed to \(1.55 \mathrm{m}\).

If \(y\) is directly proportional to the five-halves power of \(x,\) and \(y\) has the value 55.3 when \(x\) is 17.3 (a) Find the constant of proportionality. (b) Write the equation \(y=f(x)\) (c) Find \(y\) when \(x=27.4\) (d) Find \(x\) when \(y=83.6\)
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