Problem 110 - Chemistry: The Central Science

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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)

Step 2: 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: \(P_{f} =\frac{695~\text{torr}}{760~\text{torr/atm}} = 0.914~\text{atm}\)

Step 3: 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: \[P_1(V_1 + V_2) = P_f(V_1 + V_2)\]

Step 4: 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 \(V_1\): \(V_1 = 1.60~\text{L}\)

Step 5: Write the final answer

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

Key Concepts

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.