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Chapter 2: Problem 30

In a common demonstration, a bottle is heated and stoppered with a hard-boiled egg that's a little bigger than the bottle's neck. When the bottle is cooled, the pressure difference between inside and outside forces the egg into the bottle. Suppose the bottle has a volume of \(0.500 \mathrm{~L}\) and the temperature inside it is raised to \(80.0^{\circ} \mathrm{C}\) while the pressure remains constant at 1.00 atm because the bottle is open. (a) How many moles of air are inside? (b) Now the egg is put in place, sealing the bottle. What is the gauge pressure inside after the air cools back to the ambient temperature of \(25^{\circ} \mathrm{C}\) but before the egg is forced into the bottle?

Short Answer

Expert verified
(a) The number of moles of air inside the bottle is approximately \(n \approx 0.0171 \mathrm{~mol}\). (b) The gauge pressure inside the bottle when the air cools back to the ambient temperature of 25掳颁, but before the egg is forced into the bottle, is approximately \(\text{gauge pressure} \approx -0.156\mathrm{~atm}\).

Step by step solution

01

(Step 1: Ideal Gas Law)

The Ideal Gas Law relates the pressure, volume, temperature, and amount of substance (number of moles): \(PV=nRT\) where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.
02

(Step 2: Calculate the number of moles)

For part (a), we are given the volume of the bottle (0.500 L), the pressure (1.00 atm), and the temperature (80.0掳颁). However, we need to convert the temperature to Kelvin: \(80.0 + 273.15 = 353.15\) K. Now we can solve for the number of moles using the Ideal Gas Law: \[ n = \frac{PV}{RT} \] Substitute the given values and the value of the gas constant R (0.0821 Latm/molK): \[ n = \frac{(1.00 \mathrm{~atm})(0.500 \mathrm{~L})}{(0.0821 \mathrm{~Latm/molK})(353.15 \mathrm{~K})} \] Calculate the number of moles: \[ n \approx 0.0171 \mathrm{~mol} \]
03

(Step 3: Determine the new pressure)

For part (b), we want to determine the gauge pressure when the inside temperature decreases to 25掳颁 (298.15 K). We can use the Ideal Gas Law again to find the new pressure. Since the number of moles and the volume do not change, we can write the ratio of the initial state to the final state: \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \] where \(P_1\) is the initial pressure (1.00 atm), \(T_1\) is the initial temperature (353.15 K), \(P_2\) is the new pressure, and \(T_2\) is the new temperature (298.15 K).
04

(Step 4: Calculate the new pressure)

Solve for the new pressure, \(P_2\): \[ P_2 = \frac{P_1 T_2}{T_1} \] Substitute the known values: \[ P_2 = \frac{(1.00 \mathrm{~atm})(298.15 \mathrm{~K})}{353.15 \mathrm{~K}} \] Calculate the new pressure: \[ P_2 \approx 0.844 \mathrm{~atm} \]
05

(Step 5: Calculate the gauge pressure)

The gauge pressure is the difference between the new pressure and the initial pressure: \[ \text{gauge pressure} = P_2 - P_1 \] Substitute the values of \(P_2\) and \(P_1\): \[ \text{gauge pressure} = 0.844 \mathrm{~atm} - 1.00 \mathrm{~atm} \] Calculate the difference: \[ \text{gauge pressure} \approx -0.156\mathrm{~atm} \]

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Key Concepts

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

Moles of Air Calculation
Understanding the amount of substance in a given volume requires fundamental knowledge of the Ideal Gas Law, stated as PV = nRT, where n represents the number of moles of a gas. Moles are a unit of measurement in chemistry, used to express the amount of a substance. When dealing with gases, calculating moles can provide insights into the amount of particles in a specific volume at a known pressure and temperature.

To find the moles of air in the scenario, we need to rearrange the Ideal Gas Law equation to solve for n, thus n = PV / RT. It's crucial, however, to substitute values with the correct units: pressure in atmospheres (atm), volume in liters (L), and the temperature in Kelvin (K). With the given conditions, using the known value of the gas constant R = 0.0821 Latm/molK, we can determine the moles present in the bottle before the egg is inserted.

The calculation effectively bridges quantitative chemistry and physics, offering students a clear link between abstract numbers and real-world applications, like the egg-in-the-bottle demonstration.
Temperature to Kelvin Conversion
Temperature conversion from degrees Celsius to Kelvin is a critical step in working with gases and applying the Ideal Gas Law. The Kelvin scale is the SI unit for temperature and is preferred in scientific calculations because it begins at absolute zero, the lowest possible temperature.

To convert Celsius to Kelvin, one must add 273.15 to the Celsius temperature. In our example, an ambient temperature of 25掳颁 is converted to 298.15 K, and the heated bottle temperature of 80.0掳颁 is converted to 353.15 K. This conversion aligns the temperature with the absolute scale necessary for the Ideal Gas Law calculations, ensuring accuracy when determining the behavior of the gas under various conditions. Simple yet pivotal, understanding this conversion helps students grasp why gases expand when heated and contract when cooled, rooted in the kinetic molecular theory.
Gauge Pressure
Gauge pressure represents the difference between the actual pressure inside a container and the atmospheric pressure outside. It's often what we measure in everyday life because most pressure gauges are calibrated to read zero in the earth's atmosphere, hence indicating only the excess pressure.

In our described experiment, the gauge pressure after cooling the bottle will be the new internal pressure, P2, minus the external pressure, typically 1 atm. Mathematically, it's expressed as gauge pressure = P2 - P1. If gauge pressure is negative, as in the given solution, the atmospheric pressure is higher than the internal pressure, which corresponds to our situation right before the egg is sucked into the bottle.

Understanding gauge pressure helps in comprehending phenomena like the one described in the exercise, but also has practical implications in daily life, such as checking tire pressure or the operation of HVAC systems.

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Most popular questions from this chapter

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?

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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 \(200^{\circ} \mathrm{C} ?\)

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