Question

A 15.0-L tank is filled with helium gas at a pressure of \(1.00 \times 10^{2}\). How many balloons (each \(2.00 \mathrm{~L}\) ) can be inflated to a pressure of \(1.00 \mathrm{~atm}\), assuming that the temperature remains constant and that the tank cannot be emptied below \(1.00\) atm?

Short answer

Using the Ideal Gas Law and assuming constant temperature, 742 balloons each with a volume of 2.00 L can be inflated to a pressure of 1.00 atm using the helium from the 15.0-L tank, while keeping the tank pressure above 1.00 atm.

Step-by-step solution

Identify the given values and the goal

Given: - Initial tank volume (V1) = 15.0 L - Initial pressure (P1) = 1.00 x 10² - Balloon volume (Vb) = 2.00 L - Balloon pressure (Pb) = 1.00 atm - Tank cannot be emptied below 1.00 atm (Pt) Goal: - Find the number of balloons that can be inflated #Step 2: Set up the Ideal Gas Law equation and modify it #

Set up the Ideal Gas Law equation

We will use Boyle's Law in this problem, as the temperature remains constant. Boyle's Law states that the initial pressure times the initial volume is equal to the final pressure times the final volume. We'll treat the tank and the balloon as separate systems and set up two equations using Boyle's Law: Initial tank equation: \(P_1V_1 = P_2V_2\) Balloon equation: \(P_bV_b = P_3V_3\) Since the gas in the two systems comes from the same source and has equal pressure, we can set the final pressures of the tank and balloon equal: \(P_2 = P_3\) #Step 3: Calculate the final tank pressure#

Calculate the final tank pressure

To find the final tank pressure, we first need to impose the condition that the tank cannot be emptied below 1.00 atm. Therefore, the final tank pressure is: Final tank pressure (P2) = 1.00 atm #Step 4: Calculate the final tank volume#

Calculate the final tank volume

Using Boyle's Law for the tank, we can calculate the final tank volume (V2) as follows: \(P_1V_1 = P_2V_2\) \(V_2 = \frac{P_1V_1}{P_2}\) Now substitute the given values: \(V_2 = \frac{(1.00\times10^2)(15.0)}{1.00}\) \(V_2 = 1500.0\,\mathrm{L}\) #Step 5: Calculate the amount of gas that can be taken out from the tank #

Calculate the amount of gas that can be taken out from the tank

Now we can determine the amount of gas that can be taken out from the tank by calculating the difference between the initial tank volume and the final tank volume: Amount of gas = Initial tank volume - Final tank volume Amount of gas = 15.0 L - 1500.0 L Amount of gas = -1485.0 L Since this is negative, we know that 1485 L of helium can be taken out from the tank before it reaches 1 atm. #Step 6: Calculate the number of balloons that can be inflated#

Calculate the number of balloons that can be inflated

Now we can calculate the number of balloons that can be inflated using the 1485 L of helium: Number of balloons = Amount of gas / Balloon volume Number of balloons = (-1485.0 L) / (2.00 L) Number of balloons = 742.50 Since we cannot inflate half a balloon, we need to round down the number of balloons to the nearest whole number: Number of balloons = 742 So, 742 balloons each with a volume of 2.00 L can be inflated to a pressure of 1.00 atm using the helium from the 15.0-L tank.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental concept in chemistry that describes the relationship between pressure, volume, temperature, and the number of moles of gas. Although Boyle's Law, which focuses on the inverse relationship between pressure and volume at constant temperature, is often used for practical applications, the Ideal Gas Law combines Boyle's Law with Charles's Law and Avogadro's Law. The Ideal Gas Law is typically expressed as:
\[ PV = nRT \]

• \(P\) represents the gas pressure.
• \(V\) stands for the volume.
• \(n\) is the number of moles of gas.
• \(R\) is the ideal gas constant.
• \(T\) symbolizes the temperature in Kelvin.

Boyle's Law specifically applies when temperature and moles of gas remain constant, therefore the relationship simplifies to \(P_1V_1 = P_2V_2\). Understanding these relationships allows us to predict how a gas will behave when its environment changes.

Gas Pressure

Gas pressure is a measure of the force exerted by gas molecules as they collide with the walls of their container. This phenomenon arises because gas molecules are in constant, random motion. This pressure can be understood and measured in various units such as Pascals (Pa), atmospheres (atm), and millimeters of mercury (mmHg).
In a container such as a tank or a balloon, the pressure of the gas is directly proportional to the force the gas molecules exert per unit area. When the helium gas in the exercise is compressed into a smaller volume like in balloons, the increased concentration of molecules increases the frequency of collisions, thus increasing the pressure within the balloon.
Key factors affecting gas pressure include:

• Temperature: Raising the temperature increases the kinetic energy and velocity of gas molecules, leading to increased pressure.
• Volume: As the volume decreases (while temperature and gas quantity remain constant), pressure increases as molecules collide more frequently.

Volume Calculations

Volume calculations are critical when working with gases, as they allow us to understand how much space a gas occupies under varying conditions. In the scenario of inflating balloons, we used Boyle's Law to relate the volume of gas in different states of pressure.
The calculation is based on the formula derived from Boyle's Law:
\[ V_2 = \frac{P_1V_1}{P_2} \]This formula computes the final volume \(V_2\) after a change in pressure from \(P_1\) to \(P_2\), assuming temperature remains constant.

• The initial tank volume and pressure are given traits, used to calculate the potential volume the gas would occupy at a lower pressure.
• Determining the number of balloons that can be inflated involves dividing the extractable gas volume by the balloon’s specific volume capacity. In our example, the ability to inflate 742 balloons comes directly from understanding these calculations effectively.

Mastering these calculations ensures precise applications in practical and theoretical scenarios involving gases.