Question

Cyclopropane, \(\mathrm{C}_{3} \mathrm{H}_{6}\), is a general anesthetic. A 5.0-L sample has a pressure of \(5.0 \mathrm{~atm}\). What is the volume, in liters, of the anesthetic given to a patient at a pressure of \(1.0 \mathrm{~atm}\) with no change in temperature and amount of gas?

Short answer

The volume of the anesthetic at 1.0 atm is 25.0 L.

Step-by-step solution

- Understand Boyle's Law

Boyle's Law states that the pressure of a given mass of gas is inversely proportional to its volume when the temperature is kept constant. Mathematically, it is expressed as: \[ P_1 V_1 = P_2 V_2 \]

- Identify and Assign Values

The initial pressure, \(P_1\), is given as 5.0 atm and the initial volume, \(V_1\), is 5.0 L. The final pressure, \(P_2\), is given as 1.0 atm. We need to find the final volume, \(V_2\).

- Rearrange Boyle's Law

Rearrange the formula to solve for \(V_2\): \[ V_2 = \frac{P_1 V_1}{P_2} \]

- Substitute Known Values

Substitute the values into the rearranged formula: \[ V_2 = \frac{5.0 \mathrm{~atm} \times 5.0 \mathrm{~L}}{1.0 \mathrm{~atm}} \]

- Calculate the Final Volume

Perform the calculation: \[ V_2 = 25.0 \mathrm{~L} \]

Key concepts

Gas Laws

Gas laws are a set of rules describing the behavior of gases. They help us understand how different factors affect the state of a gas. Factors such as pressure, volume, and temperature all play a crucial role in determining how a gas behaves.

Some important gas laws include:

• Boyle's Law: Deals with pressure and volume.
• Charles's Law: Focuses on temperature and volume.
• Avogadro's Law: Looks at volume and the amount of gas.

Gas laws are essential for calculations involving gases in many scientific fields. They can be used to predict the behavior of gases under various conditions, aiding in fields such as chemistry, physics, and engineering.

Pressure-Volume Relationship

One of the central concepts in gas laws is the pressure-volume relationship. Specifically discussed by Boyle's Law, this principle states that the volume of a given amount of gas is inversely proportional to its pressure, provided the temperature remains constant.

This can be expressed mathematically using the formula:
\text{ Boyle's Law Equation:} \[ P_1 V_1 = P_2 V_2 \]

Here,\( P_1\): Initial pressure, \( V_1\): Initial volume, \( P_2\): Final pressure,\( V_2\): Final volume.

In the exercise, we see an initial pressure of 5.0 atm in a 5.0-L sample. When the pressure is reduced to 1.0 atm, the volume increases to keep the product \(P \times V\) consistent. By rearranging the equation to solve for the final volume \( V_2\), we find that when \(P_2 = 1.0~ \text{atm}\), the corresponding \(V_2\) becomes 25.0 L. This direct computation underlines how the pressure-volume relationship works.

Inverse Proportionality

Inverse proportionality is a key concept when studying Boyle's Law. It means that as one variable increases, the other decreases proportionally. In the context of gas laws, this applies to the relationship between pressure and volume.

If the temperature and amount of gas remain constant, then an increase in pressure will result in a decrease in volume, and vice versa. This explains why, in the exercise, when the pressure decreased from 5.0 atm to 1.0 atm, the volume increased from 5.0 L to 25.0 L.

Understanding inverse proportionality helps to predict how gases will behave under different conditions. This is especially useful in real-world applications such as:

• Anaesthesia administration where controlling the volume of gases is vital.
• The function of syringes and pistons, where pressure changes affect volume movements.
• Scuba diving, where divers must manage how pressure changes influence gas volume in tanks and lungs.

Overall, inverse proportionality makes it clear why changes in pressure directly impact gas volume in gas law scenarios.