Problem 6 - University Physics with Modern Physics

Question

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds 0.900 L. The pressure of the gas inside the balloon equals air pressure 1.00 atm). (a) If the air inside the balloon is at a constant temperature of \(22.0^{\circ} \mathrm{C}\) and behaves as an ideal gas, what mass of air can you blow into one of the balloons before it bursts? (b) Repeat part (a) if the gas is helium rather than air.

Step-by-Step Solution

Answer

(a) Mass of air: 1.076 g. (b) Mass of helium: 0.148 g.

Step 1: Convert Temperature

First, we need to convert the given temperature from Celsius to Kelvin since we are working with an ideal gas. The formula to convert Celsius to Kelvin is: \[ T(\text{K}) = T(\degree C) + 273.15 \]Plugging in the given temperature:\[ T = 22.0 + 273.15 = 295.15\, \text{K} \]

Step 2: Collect Ideal Gas Law Variables

We have the following values:- Volume, \(V = 0.900\, \text{L} = 0.900 \times 10^{-3}\, \text{m}^3\) (convert liters to cubic meters)- Pressure, \(P = 1.00\, \text{atm} = 101325\, \text{Pa}\) (convert atm to pascals)- Temperature, \(T = 295.15\, \text{K}\)We will use the ideal gas law: \[ PV = nRT \]Where \(R = 8.314\, \text{J}\, \text{mol}^{-1}\, \text{K}^{-1}\) is the ideal gas constant.

Step 3: Solve for Moles of Gas

Rearrange the ideal gas law to solve for the moles of gas, \( n \):\[ n = \frac{PV}{RT} \]Substitute the known values:\[ n = \frac{101325 \times 0.900 \times 10^{-3}}{8.314 \times 295.15} \]Calculating the above expression:\[ n \approx 0.0371\, \text{mol} \]

Step 4: Mass of Air

We can find the mass using the molar mass of air, which is approximately \( 29.0\, \text{g/mol} \).\[ \text{Mass} = n \times \text{Molar Mass} = 0.0371 \times 29.0 = 1.076\, \text{g} \]Thus, the mass of air is approximately 1.076 gram.

Step 5: Mass of Helium

Repeat the calculation for helium. The molar mass of helium is \( 4.0\, \text{g/mol} \):\[ \text{Mass of helium} = n \times \text{Molar Mass} = 0.0371 \times 4.0 = 0.1484\, \text{g} \]Thus, the mass of helium is approximately 0.1484 gram.

Key Concepts

Balloon Burst Volume

When working with balloons and gases, knowing when a balloon will burst is important. A balloon bursts when its volume exceeds a certain limit. In this exercise, the critical volume is 0.900 liters. This means if you fill a balloon with any gas, be it air or helium, it will pop if the volume inside reaches or exceeds 0.900 liters. Understanding how much gas or air can cause a balloon to burst allows us to calculate the quantity of gas molecules needed. The gas keeps the balloon inflated, and if it expands too much from adding more gas at constant pressure, the balloon can't handle the strain and bursts. The burst volume is tied to the balloon's material and construction, but for this exercise, we only need that it can't hold more than 0.900 liters.

Temperature Conversion

When dealing with gases, especially using the Ideal Gas Law, it becomes necessary to work in Kelvin rather than Celsius. This is because Kelvin is an absolute temperature scale, which directly relates to the energy of particles.To convert temperature from Celsius to Kelvin, use the formula:

• Convert:
  
  
  
  

Applying this to our problem, with air at 22°C, we convert it to Kelvin:\[T = 22.0 + 273.15 = 295.15 \, \text{K}\] This conversion is crucial because it allows us to accurately apply the Ideal Gas Law in our calculations.

Molar Mass Calculation

To find out how much mass of a gas can be added to a balloon before it bursts, we need to calculate the moles of the gas and use the molar mass. The Ideal Gas Law, \(PV = nRT\), helps us determine the moles \(n\).Here's what each variable represents:

• \(P\): Pressure of the gas (converted to pascals if originally in atm)
• \(V\): Volume of the gas inside the balloon (preferably in cubic meters)
• \(R\): Ideal gas constant \(8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}\)
• \(T\): Temperature of the gas in Kelvin

Rearrange the law to solve for moles:\[n = \frac{PV}{RT}\]After finding \(n\), the next step involves using the molar mass to find the mass of the gas:

• For air, the molar mass is approximately \(29.0 \, \text{g/mol}\)
• For helium, it's \(4.0 \, \text{g/mol}\)

Multiply the moles by the molar mass to get the mass:\[\text{Mass} = n \times \text{Molar Mass}\]So, by understanding the molar mass calculation, you can see exactly how much gas can be safely added to the balloon without exceeding the burst volume. This method helps predict and measure how different gases behave when placed under the same conditions.