/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 People buying food in sealed bag... [FREE SOLUTION] | 91Ó°ÊÓ

91Ó°ÊÓ

People buying food in sealed bags at high elevations often notice that the bags are puffed up because the air inside has expanded. A bag of pretzels was packed at a pressure of 1.00 atm and a temperature of \(22.0^{\circ} \mathrm{C}\). When opened at a summer picnic in Santa Fe, New Mexico, at a temperature of \(32.0^{\circ} \mathrm{C},\) the volume of the air in the bag is 1.38 times its original volume. What is the pressure of the air?

Short Answer

Expert verified
The pressure of the air inside the bag when opened at a summer picnic in Santa Fe, New Mexico, is approximately \(0.764 \,\text{atm}\).

Step by step solution

01

Combined gas law formula

The combined gas law formula is: \[\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\] Step 2: Convert temperatures to Kelvin
02

Convert temperatures to Kelvin

To work with the gas law, we need to convert temperatures from Celsius to Kelvin using the formula: \(K = ^{\circ}\mathrm{C} + 273.15\) Initial temperature: \(T_1 = 22.0^{\circ} \mathrm{C} + 273.15 = 295.15 K\) Final temperature: \(T_2 = 32.0^{\circ} \mathrm{C} + 273.15 = 305.15 K\) Step 3: Find the final volume
03

Finding the final volume

Since the final volume is 1.38 times the initial volume, we have: \(V_2 = 1.38 V_1\) Step 4: Solve the combined gas law formula for the final pressure
04

Solve for final pressure

We want to find the final pressure \(P_2\). Rearrange the combined gas law formula to solve for \(P_2\): \[P_2 = \frac{P_1 V_1 T_2}{T_1 V_2}\] Step 5: Plug in the values into the formula and solve
05

Calculate final pressure

Plug in the values we know (\(P_1\), \(T_1\), \(T_2\), and \(V_2 = 1.38 V_1\)). Then solve for the final pressure \(P_2\): \[P_2 = \frac{(1.00 \,\text{atm})(V_1)(305.15 \,\text{K})}{(295.15 \,\text{K})(1.38 V_1)}\] The \(V_1\) terms cancel out, so we end up with: \[P_2 = \frac{1.00 \,\text{atm} \cdot 305.15 \,\text{K}}{295.15 \,\text{K} \cdot 1.38}\] Calculate the final pressure: \[P_2 \approx 0.764 \,\text{atm}\] Thus, when opened at a summer picnic in Santa Fe, New Mexico, the pressure of the air inside the bag is approximately 0.764 atm.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gas Pressure
Understanding gas pressure is crucial when studying the behavior of gases. Gas pressure is essentially a measure of the force exerted by gas particles as they collide with the walls of their container. It's the result of billions of particles hitting the container's surface, and it can be affected by various factors, including the temperature of the gas, the volume of the container, and the amount of gas present.

For instance, when a bag of pretzels is packed at sea level, it has a certain amount of air inside at a pressure of 1.00 atm, which is standard atmospheric pressure. However, when taken to a higher elevation, such as Santa Fe, the external pressure decreases because there's less atmosphere pushing down. This difference in pressure causes the bag to puff up. The combined gas law can help predict how much the pressure of the gas will change with volume and temperature under these new conditions.
Temperature Conversion
Temperature conversion is an important step in working with gas laws since they require absolute temperature units. The Kelvin scale is used because it starts at absolute zero, the theoretical point where particles have minimum thermal motion.

To convert degrees Celsius to Kelvin, add 273.15 to the Celsius temperature. This step is vital because Celsius and Fahrenheit are relative scales with arbitrary zero points that don't reflect the true absence of heat energy. For example, the conversion from an initial temperature of 22.0°C for our pretzels to Kelvin is as follows: \(T_1 = 22.0^\circ \mathrm{C} + 273.15 = 295.15 K\). Making sure to use Kelvin ensures that proportional relationships between pressure, volume, and temperature remain consistent when using gas laws.
Gas Volume Expansion
Gas volume expansion can be observed when a gas’s temperature increases or its pressure decreases while other factors remain constant. The particles in the gas move more energetically as the temperature rises, colliding with the container's walls with greater force and more frequently, causing the volume to expand if the container is flexible or the pressure to increase if the container is rigid.

Using the pretzel bag example, the combined gas law helps us understand that at a higher elevation (lower pressure) and increased temperature, the volume of the air inside the bag expands. This is because with more kinetic energy (due to higher temperature), the gas particles push outward more vigorously. Hence, the bag puffs up to a volume that is 1.38 times its original size at Santa Fe. The combined gas law shows us that if the temperature goes up and the pressure goes down, the volume must increase for the relationship to remain balanced.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Calculate the depth to which Avogadro's number of table tennis balls would cover Earth. Each ball has a diameter of \(3.75 \mathrm{cm} .\) Assume the space between balls adds an extra \(25.0 \%\) to their volume and assume they are not crushed by their own weight.

The product of the pressure and volume of a sample of hydrogen gas at \(0.00^{\circ} \mathrm{C}\) is \(80.0 \mathrm{J}\). (a) How many moles of hydrogen are present? (b) What is the average translational kinetic energy of the hydrogen molecules? (c) What is the value of the product of pressure and volume at \(200^{\circ} \mathrm{C} ?\)

You mix 5 moles of \(\mathrm{H}_{2}\) at \(300 \mathrm{K}\) with 5 moles of He at \(360 \mathrm{K}\) in a perfectly insulated calorimeter. Is the final temperature higher or lower than \(330 \mathrm{K}\) ?

Find the number of moles in \(2.00 \mathrm{L}\) of gas at \(35.0^{\circ} \mathrm{C}\) and under \(7.41 \times 10^{7} \mathrm{N} / \mathrm{m}^{2}\) of pressure.

The gauge pressure in your car tires is 18. The gase \(2.50 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) at a temperature of \(35.0^{\circ} \mathrm{C}\) when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to \(-40.0^{\circ} \mathrm{C}\) ? Assume the tires have not gained or lost any air. 18. The gauge pressure in your car tires is 18. The gase \(2.50 \times 10^{5} \mathrm{N} / \mathrm{m}^{2}\) at a temperature of \(35.0^{\circ} \mathrm{C}\) when you drive it onto a ship in Los Angeles to be sent to Alaska. What is their gauge pressure on a night in Alaska when their temperature has dropped to \(-40.0^{\circ} \mathrm{C}\) ? Assume the tires have not gained or lost any air.

See all solutions

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.