Question

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

Short answer

Given a 15.0 L helium tank with an initial pressure of \(1.00 \times 10^{2}\) atm, we can inflate 742 balloons of 2.00 L each to a pressure of 1.00 atm, assuming the temperature remains constant and the tank cannot be emptied below 1.00 atm.

Step-by-step solution

Write down the initial conditions

The initial volume of the helium gas tank, \(V_1\) is 15.0 L. The initial pressure of the helium gas tank, \(P_1\) is \(1.00 \times 10^{2}\) atm. The minimum allowable pressure in the helium tank, \(P_{min}\), is 1.00 atm.

Apply the Ideal Gas Law

Since the temperature remains constant, we can use the Ideal Gas Law to determine the relationship between pressure and volume for a constant temperature process: \[\frac{P_1}{V_1} = \frac{P_2}{V_2}\] where \(P_1\) and \(V_1\) are the initial pressure and volume of the helium gas and \(P_2\) and \(V_2\) are the final pressure and volume of the helium gas.

Calculate the final volume

We are given the final pressure of the helium gas tank, \(P_2\), which is the minimum allowable pressure, 1.00 atm. So, we can solve for the final volume, \(V_2\), using the pressure-volume relationship: \[V_2 = \frac{P_1}{P_2} \times V_1\] \[V_2 = \frac{1.00 \times 10^{2}}{1.00} \times 15.0\] \[V_2 = 1500\ L\]

Calculate the helium gas used for inflating balloons

Now we can find the difference in volume before and after inflating balloons to obtain the volume of helium gas used for inflating balloons: \[V_{used} = V_2 - V_{min}\] \[V_{used} = 1500\ L - 15.0\ L\] \[V_{used} = 1485\ L\]

Calculate the number of balloons filled with helium gas

Each balloon has a volume of 2.00 L and is inflated at a pressure of 1.00 atm. Now we can determine the number of balloons, \(n\), that can be filled using the helium gas: \[n = \frac{V_{used}}{V_{balloon}}\] \[n = \frac{1485\ L}{2.00\ L}\] \[n = 742.5\] Since we cannot inflate half a balloon, the number of balloons that can be inflated is 742 balloons.

Key concepts

pressure-volume relationship

The pressure-volume relationship, often referred to as Boyle's Law, is one of the primary principles within the study of gases. It states that, for a given mass of gas at constant temperature, the volume of the gas is inversely proportional to its pressure. This means that if you increase the pressure on the gas, its volume will decrease, and if you reduce the pressure, the volume will increase, as long as the temperature doesn't change.

If we imagine pressing down on a syringe filled with air, as the pressure increases because the space is getting smaller, the air is compressed and the volume decreases. Similarly, if we pull the plunger back, the pressure decreases and the volume of air expands. In mathematical terms, this relationship can be represented by the formula \(P_1V_1 = P_2V_2\), where \(P_1\) and \(P_2\) are the initial and final pressures, and \(V_1\) and \(V_2\) are the initial and final volumes respectively.

When considering our textbook exercise, this pressure-volume relationship allows us to calculate the volume of gas needed to fill balloons from a high-pressure tank. By rearranging the formula, we can determine how much the volume of the gas will expand when the pressure is decreased to fill the balloons.

gas laws

The Ideal Gas Law is a crucial equation in understanding the behavior of gases under various conditions, combining several simpler gas laws that describe how temperature, volume, and pressure are interrelated. It is typically expressed as \(PV = nRT\), where \(P\) stands for the pressure of the gas, \(V\) represents the volume, \(n\) is the amount of substance of gas (in moles), \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Ideal Gas Law and Constant Temperature

When temperature remains constant (isothermal process), as we see in the exercise, the Ideal Gas Law simplifies to Boyle's Law for pressure-volume relationship. This is because the number of moles of gas and the temperature are constant, so the product of pressure and volume at one state (initial condition) will equal the pressure and volume product at another state (final condition).

This law is not only theoretical but practical. For instance, it helps us understand why a sealed bag of chips might appear puffed up at higher elevations—it is because the external pressure is lower, giving rise to a greater volume of the air within the bag (given the internal temperature and number of gas molecules remains unchanged).

molar volume

The concept of molar volume is related to the volume that one mole of a gas occupies under certain conditions. Typically, at standard temperature and pressure (STP, which is 0 degrees Celsius and 1 atm), one mole of any ideal gas occupies 22.4 liters. This can be a powerful tool for predicting the behavior of gases during chemical reactions or physical transformations, like inflating balloons.

In the textbook exercise, although we don’t directly calculate the moles of helium, understanding the molar volume allows us to appreciate the vast amount of space that gases can occupy. For example, if we were to inflate balloons with one mole of helium at STP, we would need a 22.4-liter container to hold it all, or conversely, it would fill up to 11 balloons that each have a volume of 2.00 liters.

This insight also emphasizes the practical utilization in industries. For instance, knowing the molar volume helps scuba divers understand how much air can be compressed into their tanks and how long they can breathe underwater at different depths (pressures).