Question

The volume \(V\) of a gas held at a constant temperature in a closed container varies inversely with its pressure \(P .\) If the volume of a gas is 600 cubic centimeters \(\left(\mathrm{cm}^{3}\right)\) when the pressure is 150 millimeters of mercury (mm \(\mathrm{Hg}\) ), find the volume when the pressure is \(200 \mathrm{~mm} \mathrm{Hg}\).

Short answer

The volume is 450 cubic centimeters.

Step-by-step solution

Understand the problem

The problem states that the volume of a gas varies inversely with its pressure. This means we can write the relationship as: \[V \propto \frac{1}{P}\] Alternatively, we can express this using a constant of proportionality, \( k \): \[V = \frac{k}{P}\]

Determine the constant of proportionality

We know the volume \( V \) and pressure \( P \) at one set of conditions: \[ V = 600 \, \text{cm}^3 \] \[ P = 150 \, \text{mm Hg} \] Using the formula \( V = \frac{k}{P} \), solve for \( k \): \[ 600 = \frac{k}{150} \] Multiply both sides by 150 to find \( k \): \[ k = 600 \times 150 = 90000 \, \text{cm}^3 \cdot \text{mm Hg} \]

Find the new volume

Now use the constant \( k \) to find the volume when the pressure is 200 mm Hg. Substitute \( k \) and the new \( P \) into the relationship: \[ V = \frac{k}{P} = \frac{90000}{200} \] Perform the division to find \( V \): \[ V = 450 \, \text{cm}^3 \]

Key concepts

inverse proportionality

In mathematics and science, when two quantities vary in such a way that one quantity increases as the other decreases, they are said to be inversely proportional. This concept is expressed as follows:

• If quantity one (let's call it A) increases, the other (B) decreases
• Formally, we write this as: \[ A \times B = k \]

Here, \( k \) is a constant. For example, if we talk about gas, the pressure (\( P \)) and volume (\( V \)) of a gas can be inversely proportional - as pressure increases, volume decreases and vice versa.

pressure and volume relationship

The relationship between pressure and volume of a gas is a classic example of inverse proportionality, specifically described by Boyle's Law. Boyle's Law states:

• For a fixed amount of gas at a constant temperature, the volume of the gas is inversely proportional to its pressure
• Mathematically, this is represented as: \[ V \times P = k \] where \( k \) is a constant value when temperature and amount of gas are constant

To put it simply, if you compress a gas (increase its pressure), its volume decreases. Conversely, if you reduce the pressure, the gas expands, and its volume increases. This principle helps us understand behaviors of gases in different conditions.

gas laws

Gas laws describe how gases behave under different conditions of pressure, volume, and temperature. The primary gas laws include:

• Boyle's Law: As explained, relates pressure and volume at constant temperature \( V \propto \frac{1}{P} \)
• Charles's Law: States that the volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is constant: \( V \propto T \)
• Avogadro's Law: States that the volume of a gas is directly proportional to the number of moles of gas when pressure and temperature are constant: \( V \propto n \)

Together, these relationships can be combined into the Ideal Gas Law, often written as \( PV = nRT \), where \( R \) is the gas constant. Understanding these laws allows for predicting how a gas will respond under varying conditions. This is crucial in fields like chemistry, physics, and engineering.