29. Solve problems using Boyle's, Charles's, or the Combined gas law, such as those below. a. A sample of gas is contained in a flexible container at 2.51 atm. The container expands to 3.5 L as the pressure decreases to 1.89 atm. What was the original volume? b. Originally, a balloon has a volume of 0.302 L at 1.53 atm and 25.7°C. The balloon expands until the final volume is 0.594 L at a pressure of 0.892 atm. What is the final temperature of the balloon?
Ideal and Real Gases
Ideal gases obey conditions of the general gas laws under all states of pressure and temperature. Ideal gases are also named perfect gases. The attributes of ideal gases are as follows,
Gas Laws
Gas laws describe the ways in which volume, temperature, pressure, and other conditions correlate when matter is in a gaseous state. The very first observations about the physical properties of gases was made by Robert Boyle in 1662. Later discoveries were made by Charles, Gay-Lussac, Avogadro, and others. Eventually, these observations were combined to produce the ideal gas law.
Gaseous State
It is well known that matter exists in different forms in our surroundings. There are five known states of matter, such as solids, gases, liquids, plasma and Bose-Einstein condensate. The last two are known newly in the recent days. Thus, the detailed forms of matter studied are solids, gases and liquids. The best example of a substance that is present in different states is water. It is solid ice, gaseous vapor or steam and liquid water depending on the temperature and pressure conditions. This is due to the difference in the intermolecular forces and distances. The occurrence of three different phases is due to the difference in the two major forces, the force which tends to tightly hold molecules i.e., forces of attraction and the disruptive forces obtained from the thermal energy of molecules.
![### Solving Gas Law Problems: Boyle's, Charles's, and Combined Gas Law
#### Problem 29
**Solve problems using Boyle's, Charles's, or the Combined gas law, such as those below.**
**a. Sample Gas Problem (Boyle's Law)**
A sample of gas is contained in a flexible container at \(2.51 \, \text{atm}\). The container expands to \(3.5 \, \text{L}\) as the pressure decreases to \(1.89 \, \text{atm}\). What was the original volume?
**b. Balloon Expansion Problem (Combined Gas Law)**
Originally, a balloon has a volume of \(0.302 \, \text{L}\) at \(1.53 \, \text{atm}\) and \(25.7^\circ \text{C}\). The balloon expands until the final volume is \(0.594 \, \text{L}\) at a pressure of \(0.892 \, \text{atm}\). What is the final temperature of the balloon?
### Explanation of Relevant Gas Laws
**Boyle's Law:**
\[ P_1V_1 = P_2V_2 \]
(At constant temperature, the pressure of a gas is inversely proportional to its volume.)
**Charles's Law:**
\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]
(At constant pressure, the volume of a gas is directly proportional to its temperature in Kelvin.)
**Combined Gas Law:**
\[ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} \]
(The relationship between pressure, volume, and temperature of a fixed amount of gas.)
### Detailed Solution Approach
**a. For Boyle's Law Problem:**
Given:
- \( P_1 = 2.51 \, \text{atm} \)
- \( P_2 = 1.89 \, \text{atm} \)
- \( V_2 = 3.5 \, \text{L} \)
Need to find \( V_1 \):
Using Boyle's Law:
\[ P_1V_1 = P_2V_2 \]
\[ V_1 = \frac{P_2V_2}{P_1} \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b52cd77-b3d2-4b5d-8903-b23362bef087%2F9d30a25f-c0bd-4d76-a6f0-b3513eda5001%2F379bjob_processed.png&w=3840&q=75)
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