Question

The air in a 5.00-L tank has a pressure of \(1.20\) atm. What is the new pressure, in atm, when the air is placed in tanks that have the following volumes, if there is no change in temperature and amount of gas? a. \(1.00 \mathrm{~L}\) b. \(2500 . \mathrm{mL}\) c. \(750 . \mathrm{mL}\)

Short answer

a. 6.00 atmb. 2.40 atmc. 8.00 atm

Step-by-step solution

- Identify the given data

Given: \( V_1 = 5.00 \, L \)\( P_1 = 1.20 \, atm \)

- State Boyle's Law

Boyle's Law states that for a given mass of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. The formula is given by: \( P_1 V_1 = P_2 V_2 \)

- Calculate the new pressure for part a

Given: \( V_2 = 1.00 \, L \)Using Boyle's Law:\( P_2 = \frac{P_1 V_1}{V_2} \)Substitute the known values:\( P_2 = \frac{1.20 \, atm \, \times \, 5.00 \, L}{1.00 \, L} = 6.00 \, atm \)

- Calculate the new pressure for part b

First, convert the volume to liters:\( V_2 = 2500 \, mL = 2.50 \, L \)Using Boyle's Law:\( P_2 = \frac{P_1 V_1}{V_2} \)Substitute the known values:\( P_2 = \frac{1.20 \, atm \, \times \, 5.00 \, L}{2.50 \, L} = 2.40 \, atm \)

- Calculate the new pressure for part c

First, convert the volume to liters:\( V_2 = 750 \, mL = 0.750 \, L \)Using Boyle's Law:\( P_2 = \frac{P_1 V_1}{V_2} \)Substitute the known values:\( P_2 = \frac{1.20 \, atm \, \times \, 5.00 \, L}{0.750 \, L} = 8.00 \, atm \)

Key concepts

gas laws

The behavior of gases can be described by various gas laws, which show how pressure, volume, temperature, and the number of gas particles are related to one another. The main gas laws include Boyle's Law, Charles's Law, Gay-Lussac's Law, and Avogadro's Law. These laws are crucial for understanding how gases react under different conditions. In this exercise, we focus on Boyle's Law which specifically looks at the pressure-volume relationship for a gas at constant temperature.

pressure-volume relationship

The pressure-volume relationship is key to understanding how gases behave. According to Boyle's Law, the pressure of a gas is inversely proportional to its volume when the temperature and amount of gas remain constant. This means that if the volume of a gas decreases, its pressure increases, and vice versa, as long as the temperature stays the same.
Let's look at the formula from Boyle's Law:
\( P_1 V_1 = P_2 V_2 \)
Here, \( P_1 \) and \( V_1 \) represent the initial pressure and volume, while \( P_2 \) and \( V_2 \) represent the new pressure and volume. By using this formula, you can calculate how changes in volume will affect the pressure of the gas.

ideal gas law

The Ideal Gas Law is another critical concept in understanding gas behavior. It combines several principles from the individual gas laws into one comprehensive formula: \( PV = nRT \)
Here, \( P \) represents pressure, \( V \) is volume, \( n \) is the number of moles of gas, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin. This law is useful for calculating the behavior of gases under a variety of conditions. Though we did not use the Ideal Gas Law directly in this exercise, understanding this foundational formula helps reinforce why Boyle's Law, a component of the Ideal Gas Law, holds true. The Ideal Gas Law assumes that gas particles do not attract or repel each other and that they occupy no volume themselves, which is often a good approximation for many gases under standard conditions.

inverse proportionality

Inverse proportionality is a fundamental concept in Boyle's Law. When two quantities are inversely proportional, as one increases, the other decreases at the same rate. For gases, this means that if you decrease the volume, the pressure will increase proportionally, as long as the temperature and amount of gas remain unchanged. In mathematical terms, this relationship is expressed as:
\( P \propto \frac{1}{V} \)
This equation highlights how increasing the volume will reduce the pressure, and decreasing the volume will increase the pressure. Understanding inverse proportionality helps you predict how changes to your gas's volume will affect its pressure and vice versa, allowing for accurate problem-solving in related exercises.