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Chapter 7: Problem 13

Boyle's Law says that if the temperature is held constant, then the volume, \(V_{s}\) of a fixed quantity of gas is inversely proportional to the pressure, \(P .\) That is, \(V=\frac{k}{p}\) for some constant \(k\). What happens to the volume if: a. The pressure triples? b. The pressure is multiplied by \(n\) ? c. The pressure is halved? d. The pressure is divided by \(n\) ?

Short Answer

Expert verified
a. Volume becomes one-third. b. Volume becomes one-\( n^{th} \). c. Volume becomes twice. d. Volume becomes \( n \) times.

Step by step solution

01

Understand Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. Mathematically, it is represented as: \[ V = \frac{k}{P} \] where \( V \) is volume, \( P \) is pressure, and \( k \) is a constant.
02

Determine the effect of tripling the pressure

If the pressure triples, let the new pressure be \( 3P \). Substitute \( 3P \) into the Boyle's Law equation: \[ V = \frac{k}{3P} \] The volume becomes one-third of the original volume.
03

Determine the effect of multiplying the pressure by \( n \)

If the pressure is multiplied by \( n \), let the new pressure be \( nP \). Substitute \( nP \) into the Boyle's Law equation: \[ V = \frac{k}{nP} \] The volume becomes one-\( n^{th} \) of the original volume.
04

Determine the effect of halving the pressure

If the pressure is halved, let the new pressure be \( \frac{P}{2} \). Substitute \( \frac{P}{2} \) into the Boyle's Law equation: \[ V = \frac{k}{\frac{P}{2}} = \frac{2k}{P} \] The volume becomes twice the original volume.
05

Determine the effect of dividing the pressure by \( n \)

If the pressure is divided by \( n \), let the new pressure be \( \frac{P}{n} \). Substitute \( \frac{P}{n} \) into the Boyle's Law equation: \[ V = \frac{k}{\frac{P}{n}} = \frac{nk}{P} \] The volume becomes \( n \) times the original volume.

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Key Concepts

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

inverse proportionality
Inverse proportionality means that as one value increases, the other decreases. In the context of Boyle's Law, the volume of a gas is inversely proportional to its pressure. This relationship means that if you increase the pressure on a gas, its volume will decrease in a predictable way.

To visualize this, consider a balloon. When you squeeze the balloon (increase the pressure), it gets smaller (decrease in volume). The mathematical representation of inverse proportionality in Boyle's Law is:
\[ V = \frac{k}{P} \]

Here, V is the volume, P is the pressure, and k is a constant, representing a specific scenario with constant temperature and gas amount.
pressure and volume relationship
The pressure and volume relationship described by Boyle's Law is fascinating. When the pressure on a given amount of gas increases, the volume decreases, and vice versa.

Let's clarify with the systematic examples:
  • If the pressure triples, \[V = \frac{k}{3P} \], so the volume becomes one-third.
  • If the pressure is multiplied by n, \[V = \frac{k}{nP} \], the volume becomes one-\bth\b of the original.
  • If the pressure is halved, \[V = \frac{2k}{P} \], the volume doubles.
  • If the pressure is divided by n, \[V = \frac{nk}{P} \], the volume becomes n times the original.

This relationship shows a clear and predictable pattern: more pressure means less volume, less pressure means more volume.
constant temperature effects
Boyle's Law holds true only if the temperature remains constant. This condition is crucial because temperature changes can affect the volume and pressure in different ways.

In a fixed temperature scenario, the gas molecules鈥� kinetic energy remains unchanged. The only way to alter the volume is through changing pressure. This controlled environment allows straightforward application of Boyle's Law.

For example, compressing a gas inside a cylinder without changing the temperature will reduce its volume precisely according to inverse proportions. Real-life applications are found in syringes, hydraulic systems, and even our lungs as we inhale and exhale.

Understanding the constant temperature condition helps make accurate predictions using Boyle鈥檚 Law in various practical scenarios.

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Most popular questions from this chapter

(Graphing program recommended.) If \(x\) is positive, for what values of \(x\) is \(3 \cdot 2^{x}<3 \cdot x^{2}\) ? For what values of \(x\) is \(3 \cdot 2^{x}>3 \cdot x^{2} ?\)

Begin with the function \(f(x)=x^{4}\) a. Create a new function \(g(x)\) by vertically stretching \(f(x)\) by a factor of \(6 .\) b. Create a new function \(h(x)\) by vertically compressing \(f(x)\) by a factor of \(\frac{1}{2}\). c. Create a new function \(j(x)\) by first vertically stretching \(f(x)\) by a factor of 2 and then reflecting it across the \(x\) -axis.

The data in the table satisfy the equation \(y=k x^{n}\), where \(n\) is a positive integer. Find \(k\) and \(n\). $$ \begin{array}{ccccc} \hline x & 2 & 3 & 4 & 5 \\ y & 1 & 2.25 & 4 & 6.25 \\ \hline \end{array} $$

(Graphing program recommended.) Assume a person weighing \(P\) pounds is standing at the center of a \(4^{\prime \prime} \times 12^{\prime \prime}\) fir plank that spans a distance of \(L\) feet. The downward deflection \(D_{\text {deflection }}\) (in inches) of the plank can be described by $$ D_{\text {deflection }}=\left(5.25 \cdot 10^{-7}\right) \cdot P \cdot L^{3} $$ a. Graph the deflection formula, \(D_{\text {deflection }}\), assuming \(P=200\) pounds. Put values of \(L\) (from 0 to 25 feet) on the horizontal axis and deflection (in inches) on the vertical axis. b. A rule used by architects for estimating acceptable deflection, \(D_{\text {safe }},\) in inches, of a beam \(L\) feet long bent downward as a result of carrying a load is $$ D_{\text {safe }}=0.05 L $$ Add to your graph from part (a) a plot of \(D_{\text {safe. }}\) c. Is it safe for a 200 -pound person to sit in the middle of a \(4^{\prime \prime} \times 12^{\prime \prime}\) fir plank that spans 20 feet? What maximum span is safe for a 200 -pound person on a \(4^{\prime \prime} \times 12^{\prime \prime}\) plank?

The frequency, \(F\) (the number of oscillations per unit of time), of an object of mass \(m\) attached to a spring is inversely proportional to the square root of \(m\). a. Write an equation describing the relationship. b. If a mass of \(0.25 \mathrm{~kg}\) attached to a spring makes three oscillations per second, find the constant of proportionality, c. Find the number of oscillations per second made by a mass of \(0.01 \mathrm{~kg}\) that is attached to the spring discussed in part (b).
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