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An ideal gas at a pressure of \(1.50 \mathrm{~atm}\) is contained in a bulb of unknown volume. A stopcock is used to connect this bulb with a previously evacuated bulb that has a volume of \(0.800 \mathrm{~L}\) as shown here. When the stopcock is opened the gas expands into the empty bulb. If the temperature is held constant during this process and the final pressure is 695 torr, what is the volume of the bulb that was originally filled with gas?

Short Answer

Expert verified
The volume of the bulb that was originally filled with gas is \(1.60~\mathrm{L}\).

Step by step solution

01

Write down the given information

Given information is: 1. Initial pressure of the gas, \(P_1\)=1.50 atm 2. Volume of the evacuated bulb, \(V_2\)=0.800 L 3. Final pressure when gas expands, \(P_f\)=695 torr (convert to atm)
02

Convert torr to atm

To work with consistent units, convert the final pressure from torr to atm using the conversion factor 1 atm = 760 torr: \(P_{f} =\frac{695~\text{torr}}{760~\text{torr/atm}} = 0.914~\text{atm}\)
03

Apply the ideal gas law

We will use the ideal gas law formula: \[\frac{n_1}{n_2} = \frac{P_1V_1}{P_2V_2}\] The fact that initial and final temperatures are constant and the number of moles of the gas is constant in this process allows us to rewrite the equation: \[\frac{n_1}{n_2} = \frac{P_1(V_1 + V_2)}{P_f(V_1 + V_2)}\] As the temperature and number of moles of the gas are the same for both cases, we can drop \(\frac{n_1}{n_2}\) term from the equation; By doing that, we will get: \[P_1(V_1 + V_2) = P_f(V_1 + V_2)\]
04

Solve for the unknown volume

We want to find the volume of the bulb that was originally filled with the gas \(V_1\). Rearrange the equation to isolate \(V_1\): \(V_1 = \frac{P_fV_2 - P_1V_2}{P_1 - P_f}\) Now, substitute the known values and calculate \(V_1\): \(V_1 = \frac{0.914~\text{atm} \cdot 0.800~\text{L} - 1.50~\text{atm} \cdot 0.800~\text{L}}{1.50~\text{atm} - 0.914~\text{atm}}\) Calculating the value of \(V_1\): \(V_1 = 1.60~\text{L}\)
05

Write the final answer

The volume of the bulb that was originally filled with gas is 1.60 L.

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Key Concepts

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

Partial Pressure
When dealing with gases, "partial pressure" is a key concept. It refers to the pressure that a single type of gas exerts within a mixture of gases. In our exercise, the final pressure measured after the gas expands is an expression of its partial pressure in the new volume.
In technical terms, when a mixture of gases is in a container, each gas exerts pressure as if it were the only gas present. This is because gases in a mixture behave independently of each other. Dalton's Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of all individual gases in the mixture.
  • The final pressure is measured in torr (695 torr) and can be converted into another unit for consistency, like atm.
  • This conversion helps maintain unit coherence across calculations, often necessary to solve ideal gas law problems correctly.
For this exercise, this conversion was key in calculating the volume of the original gas bulb. Therefore, understanding partial pressure and its unit conversions is vital in the analysis of such problems.
Gas Expansion
Gas expansion is an important process that involves a gas increasing its volume. This process follows fundamental laws of chemistry and physics, particularly the ideal gas law. In the problem at hand, we see gas move from a bulb of unknown volume into a second, known volume bulb. The stopcock allows the gas to flow freely between these two connected bulbs.
  • Since gas tends to fill any container it is in fully, once the stopcock is opened, the gas molecules spread out until they occupy both bulbs.
  • While expanding, the energy dispersed and the pressure changes, demonstrated here by the reading of 695 torr when the bulbs are fully filled by the gas.
This is a common scenario in physics and chemistry problems where understanding the relationships of volume, pressure, and temperature, thanks to the ideal gas law, helps explain how expansion impacts gas behavior.
Volume Calculation
Calculating the volume of gas is a practical application of the ideal gas law, which is often needed in science problems. In our exercise, the unknown volume of the initial bulb is determined using properties of ideal gases. The initial step involved setting up and rearranging the ideal gas law equation to find the unknown.
  • Given constants like initial and final pressures, alongside known volumes, can be plugged into the equation.
  • The conversion of the final pressure from torr to atm is crucial for consistency in the measurements used for calculations.
  • Solving the rearranged formula gives you the original volume, which was found to be 1.60 L in this example.
This example illustrates not only how mathematical manipulations of formulas can provide solutions, but also reinforces the importance of accuracy in conversions and scientific notations in day-to-day problem solving.

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Most popular questions from this chapter

Newton had an incorrect theory of gases in which he assumed that all gas molecules repel one another and the walls of their container. Thus, the molecules of a gas are statically and uniformly distributed, trying to get as far apart as possible from one another and the vessel walls. This repulsion gives rise to pressure. Explain why Charles's law argues for the kineticmolecular theory and against Newton's model.

A 6.53-g sample of a mixture of magnesium carbonate and calcium carbonate is treated with excess hydrochloric acid. The resulting reaction produces \(1.72 \mathrm{~L}\) of carbon dioxide gas at \(28^{\circ} \mathrm{C}\) and 743 torr pressure. (a) Write balanced chemical equations for the reactions that occur between hydrochloric acid and each component of the mixture. (b) Calculate the total number of moles of carbon dioxide that forms from these reactions. (c) Assuming that the reactions are complete, calculate the percentage by mass of magnesium carbonate in the mixture.

Assume that an exhaled breath of air consists of \(74.8 \% \mathrm{~N}_{2}, 15.3 \% \mathrm{O}_{2}, 3.7 \% \mathrm{CO}_{2}\), and \(6.2 \%\) water vapor. (a) If the total pressure of the gases is \(0.985 \mathrm{~atm}\), calculate the partial pressure of each component of the mixture. (b) If the volume of the exhaled gas is \(455 \mathrm{~mL}\) and its temperature is \(37^{\circ} \mathrm{C}\), calculate the number of moles of \(\mathrm{CO}_{2}\) exhaled. (c) How many grams of glucose \(\left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right)\) would need to be metabolized to produce this quantity of \(\mathrm{CO}_{2}\) ? (The chemical reaction is the same as that for combustion of \(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\). See Section \(3.2\) and Problem 10.57.)

(a) What conditions are represented by the abbreviation STP? (b) What is the molar volume of an ideal gas at STP? (c) Room temperature is often assumed to be \(25^{\circ} \mathrm{C}\). Calculate the molar volume of an ideal gas at \(25^{\circ} \mathrm{C}\) and 1 atm pressure. (d) If you measure pressure in bars instead of atmospheres, calculate the corresponding value of \(R\) in L-bar/mol-K.

Determine whether each of the following changes will increase, decrease, or not affect the rate with which gas molecules collide with the walls of their container: (a) increasing the volume of the container, (b) increasing the temperature, (c) increasing the molar mass of the gas.

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