Problem 80 - Chemistry The Science in Context

Question

A sealed, flexible foil bag of potato chips, containing \(0.500 \mathrm{L}\) of air, is carried from Boston \(\left(P_{\mathrm{atm}}=1.0 \mathrm{atm}\right)\) to Denver \(\left(P_{\text {atm }}=0.83 \mathrm{atm}\right)\) a. What would the volume of the bag be upon arrival in Denver? b. If the structural limitations of the bag allow for only a \(10 \%\) expansion in the volume of the bag, what is the pressure in the bag? c. If the bag is placed in checked luggage, the pressure and temperature in the hold will decrease considerably during flight to \(T=210 \mathrm{K}\) and \(P=126\) torr. Calculate the pressure in the bag during flight.

Step-by-Step Solution

Answer

Answer: The volume of the bag in Denver is approximately 0.602 L, the pressure after expanding by 10% is approximately 0.909 atm, and the pressure during flight in the luggage hold is approximately 434.67 torr.

Step 1: (a) Volume of the bag in Denver

To find the volume in Denver, we can use Boyle's Law: \(P_1 V_1 = P_2 V_2\), where \(P_1\) and \(V_1\) are the initial pressure and volume, and \(P_2\) and \(V_2\) are the final pressure and volume. In this case, we are given that \(P_1 = 1.0 \mathrm{atm}\) and \(V_1 = 0.500 \mathrm{L}\). The pressure in Denver is given as \(P_2 = 0.83 \mathrm{atm}\). To find \(V_2\), we can rearrange the formula as \(V_2 = P_1 V_1 / P_2\) Plugging in the given values: \(V_2 = \frac{\left(1.0 \mathrm{atm}\right)\left(0.500 \mathrm{L}\right)}{0.83 \mathrm{atm}} = 0.602 \mathrm{L}\) So the volume of the bag upon arrival in Denver is approximately \(0.602 \mathrm{L}\).

Step 2: (b) Pressure in the bag with 10% volume expansion

We are now asked to find the pressure in the bag if the volume expands by 10%. First, let's find the new volume (which is still in Boston) after a 10% expansion: \(V_3 = V_1(1 + 10\%) = 0.500 \mathrm{L} \cdot (1 + 0.1) = 0.550 \mathrm{L}\) Now, we will use Boyle's Law again to find the pressure with the new volume: \(P_1 V_1 = P_3 V_3\). To find \(P_3\), we can rearrange the formula as \(P_3 = P_1 V_1 / V_3\) Plugging in the given values: \(P_3 = \frac{\left(1.0 \mathrm{atm}\right)\left(0.500 \mathrm{L}\right)}{0.550 \mathrm{L}} = 0.909 \mathrm{atm}\) So the pressure in the bag after a 10% volume expansion is approximately \(0.909 \mathrm{atm}\).

Step 3: (c) Pressure in the bag during flight

We are given the new temperature and pressure during the flight in the luggage hold: \(T_4 = 210 \mathrm{K}\) and \(P_4 = 126 \mathrm{torr}\). We are not given the initial temperature, but assuming it is room temperature, we can estimate it to be \(T_1 = 298 \mathrm{K}\). We will use the Ideal Gas Law to find the pressure in the bag during the flight. To do this, we first need to find the number of moles and the gas constant \(R\). The number of moles can be found using \(n = P_1 V_1 / R T_1\). Converting the initial pressure to torr to match the given pressure during flight: \(P_1 = 1.0 \mathrm{atm} \cdot \frac{760 \mathrm{torr}}{1 \mathrm{atm}} = 760 \mathrm{torr}\) Now, using \(R = 62.36 \mathrm{L \cdot torr \cdot K^{-1} \cdot mol^{-1}}\) and rearranging for \(n\), we have: \(n = \frac{(760 \mathrm{torr})(0.500 \mathrm{L})}{(62.36 \mathrm{L \cdot torr \cdot K^{-1} \cdot mol^{-1}})(298\mathrm{K})} = 0.0206 \mathrm{mol}\) Now, we can use the Ideal Gas Law with the new temperature and pressure: \(P_5 V_3 = nR T_4\) (where \(P_5\) is the pressure in the bag during the flight). To find \(P_5\), rearrange the formula as \(P_5 = \frac{nRT_4}{V_3}\): \(P_5 = \frac{(0.0206 \mathrm{mol})(62.36 \mathrm{L \cdot torr \cdot K^{-1} \cdot mol^{-1}})(210 \mathrm{K})}{0.550 \mathrm{L}} = 434.67 \mathrm{torr}\) So the pressure in the bag during the flight is approximately \(434.67 \mathrm{torr}\).

Key Concepts

ideal gas law

The Ideal Gas Law is a fundamental principle in chemistry and physics that describes the behavior of ideal gases. It combines several individual gas laws including Boyle's Law, Charles's Law, and Avogadro's Law into one comprehensive equation: \[ PV = nRT \]where:

• \( P \) represents the pressure of the gas,


• \( V \) represents the volume,


• \( n \) is the number of moles of the gas,


• \( R \) is the ideal gas constant, and


• \( T \) is the temperature in Kelvin.

It is used to predict how a gas will behave under certain conditions. The Ideal Gas Law reveals that the pressure, volume, and temperature of a gas are interrelated. If one condition changes, the others will adjust to maintain the equality of the equation. This is helpful in scenarios like calculating how a change in altitude might affect a sealed bag of air, as it directly influences pressure and volume, a situation where the Ideal Gas Law becomes instrumental.

pressure-volume relationship

The pressure-volume relationship of a gas is fundamentally described by Boyle's Law, one of the cornerstones of gas behavior studies. Boyle's Law states that the volume of a given mass of gas is inversely proportional to its pressure at constant temperature. Mathematically, it is expressed as:\[ P_1 V_1 = P_2 V_2 \]This equation indicates that when the volume of a gas increases, its pressure decreases given the temperature remains constant, and vice versa.

This relationship is crucial in understanding how gases will react when compressed or expanded. For example, in the exercise with the potato chip bag traveling from Boston to Denver, we see Boyle's Law in action. As the atmospheric pressure in Denver is lower than in Boston, the volume of the bag expands to balance the pressure difference.

altitude effects on gases

Altitude has a significant impact on gases, primarily affecting their pressure and volume. At higher altitudes, atmospheric pressure decreases because the air is less dense compared to lower altitudes. This can be predicted using the concepts of the ideal gas law and Boyle's Law combined.

• At higher altitudes, the pressure exerted on a container's exterior is less, which typically allows gases inside to expand if the temperature is constant.


• This is why an airtight bag, like the potato chip bag in the exercise, expands upon arriving at higher altitudes like Denver.


• Fluctuations in temperature and pressure due to altitude changes should be carefully managed, especially in aviation where cabin and luggage hold environments vary significantly. Managing these changes is crucial to ensure the integrity of packaged products during transportation.

These concepts help explain why increasing altitude can lead to challenges in maintaining constant conditions for gases, making it vital to understand and predict these changes using gas laws.