A particular balloon is designed by its manufacturer to be in- flated to a volume of no more than 2.5 L. If the balloon is filled with 2.0 L helium at sea level, is released, and rises to an alti- tude at which the atmospheric pressure is only 500. mm Hg, will the balloon burst? (Assume temperature is constant.)
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.
help with this problem? explain why the ballon will burst.
![### Problem Statement:
A particular balloon is designed by its manufacturer to be inflated to a volume of no more than 2.5 L. If the balloon is filled with 2.0 L helium at sea level, is released, and rises to an altitude at which the atmospheric pressure is only 500. mm Hg, will the balloon burst? (Assume temperature is constant.)
### Detailed Explanation:
This problem involves understanding the relationship between pressure and volume for a gas, where temperature is assumed to be constant. We use Boyle's Law for this purpose. Boyle's Law states that for a fixed amount of gas at a constant temperature, the volume (V) and pressure (P) of the gas are inversely proportional, which can be mathematically expressed as:
\[ P_1 V_1 = P_2 V_2 \]
#### Given Data:
- The balloon's maximum volume: \( V_{max} = 2.5 \, \text{L} \)
- Initial volume of helium at sea level: \( V_1 = 2.0 \, \text{L} \)
- Pressure at sea level: \( P_1 = 760 \, \text{mm Hg} \) (standard atmospheric pressure)
- Pressure at altitude: \( P_2 = 500 \, \text{mm Hg} \)
#### Problem Solving:
We need to determine the final volume of the helium in the balloon at the lower pressure of 500 mm Hg and check if it exceeds the maximum allowed volume.
Using Boyle's Law:
\[ P_1 V_1 = P_2 V_2 \]
We can rearrange this equation to solve for \( V_2 \):
\[ V_2 = \frac{P_1 V_1}{P_2} \]
Substitute the known values into the equation:
\[ V_2 = \frac{760 \, \text{mm Hg} \times 2.0 \, \text{L}}{500 \, \text{mm Hg}} \]
\[ V_2 = \frac{1520 \, \text{L mm Hg}}{500 \, \text{mm Hg}} \]
\[ V_2 = 3.04 \, \text{L} \]
#### Conclusion:
Since the final volume of the helium when the balloon rises to an altitude with pressure of 500 mm Hg is 3.04](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5222c9e-84ca-49ca-a26b-1a6cb57ce931%2F21c04954-8c7e-4598-a0f9-2a4d306cbc88%2Fihe7ap5_processed.png&w=3840&q=75)
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