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Chapter 14: Problem 30

A spherical balloon is made from an amount of material whose mass is \(3.00 \mathrm{kg} .\) The thickness of the material is negligible compared to the 1.50-m radius of the balloon. The balloon is filled with helium (He) at a temperature of \(305 \mathrm{K}\) and just floats in air, neither rising nor falling. The density of the surrounding air is \(1.19 \mathrm{kg} / \mathrm{m}^{3} .\) Find the absolute pressure of the helium gas.

Short Answer

Expert verified
The absolute pressure of the helium gas is approximately 6.03 atm.

Step by step solution

01

Calculate the Volume of the Balloon

The balloon is spherical, so the volume \( V \) of the balloon can be calculated using the formula for the volume of a sphere: \[ V = \frac{4}{3} \pi r^3 \]Given that the radius \( r \) is \( 1.50 \, \text{m} \), the volume becomes: \[ V = \frac{4}{3} \pi (1.50)^3 \approx 14.137 \, \text{m}^3 \]
02

Apply Archimedes鈥� Principle

According to Archimedes鈥� Principle, the buoyant force is equal to the weight of the air displaced. The balloon floats, so the weight of the helium plus the weight of the balloon material must equal the buoyant force. The buoyant force \( F_b \) can be calculated as: \[ F_b = \rho_{\text{air}} \cdot V \cdot g \]where \( \rho_{\text{air}} = 1.19 \, \text{kg/m}^3 \) and \( g = 9.81 \, \text{m/s}^2 \).So: \[ F_b = 1.19 \times 14.137 \times 9.81 \approx 165.12 \, \text{N} \]
03

Calculate the Weight of the Balloon Material

The weight of the balloon material \( W_m \) is given by the mass of the material times gravitational acceleration:\[ W_m = m \times g \]Given \( m = 3.00 \, \text{kg} \), we calculate:\[ W_m = 3.00 \times 9.81 = 29.43 \, \text{N} \]
04

Calculate the Weight of the Helium

The total weight when the balloon floats is equal to the buoyant force, thus:\[ W_{\text{total}} = F_b = W_m + W_{\text{He}} \]where \( W_{\text{He}} = m_{\text{He}} \times g \).We rearrange for helium mass:\[ m_{\text{He}} \times g = F_b - W_m \]\[ m_{\text{He}} = \frac{F_b - W_m}{g} \]Calculating:\[ m_{\text{He}} = \frac{165.12 - 29.43}{9.81} \approx 13.82 \, \text{kg} \]
05

Use Ideal Gas Law to Find Pressure

The ideal gas law \( PV = nRT \) relates pressure \( P \), volume \( V \), and the number of moles \( n \) of the gas. To find the number of moles \( n \), \[ n = \frac{m_{\text{He}}}{M_{\text{He}}} \]where \( M_{\text{He}} = 4.00 \, \text{g/mol} = 0.004 \, \text{kg/mol} \).Now calculate \( n \):\[ n = \frac{13.82}{0.004} = 3455 \, \text{mol} \]Substitute into the ideal gas law:\[ P = \frac{nRT}{V} \]where \( R = 8.314 \, \text{J/(mol \cdot K)} \) and \( T = 305 \, \text{K} \).Calculating:\[ P = \frac{3455 \times 8.314 \times 305}{14.137} \approx 611716 \, \text{Pa} \]
06

Convert Pressure to ATM

Convert the pressure from pascals (Pa) to atmosphere (atm):\[ 1 \, \text{atm} = 101325 \, \text{Pa} \]So the pressure in atm is:\[ P = \frac{611716}{101325} \approx 6.03 \, \text{atm} \]

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Key Concepts

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

Archimedes鈥� Principle
This principle helps us understand why objects float or sink when placed in a fluid. Archimedes鈥� Principle states that the buoyant force acting on an object immersed in a fluid equals the weight of the fluid displaced by the object. In simpler words, any object submerged in a fluid experiences an upward force from the fluid, which is equal to the weight of the fluid that the object displaces.

For instance, if you push a solid steel ball into the water, it feels lighter. This is because the water pushes up on it with a force equal to the weight of the water displaced. In the case of the balloon filled with helium, it floats because the upward buoyant force equals the combined weight of the balloon and helium. Just imagine the air as a sort of fluid in which the balloon is floating.
Buoyant Force
The buoyant force is that upward force exerted on objects when submerged in a fluid, such as liquid or gas. It plays a crucial role in determining whether an object will sink or float.
  • If buoyant force > weight of the object, the object floats.
  • If buoyant force < weight of the object, it sinks.
Calculating the buoyant force involves understanding the object's volume and the fluid's density. In the balloon's case, the buoyant force equals the volume of air displaced multiplied by the air's density and gravity (9.81 m/s虏).

This phenomenon is quite similar to how boats float on water. The boat displaces an amount of water equal to its weight, making it float precisely like our balloon in the air.
Volume of Sphere
A sphere is a three-dimensional shape, like a ball, and understanding its volume is essential for calculations involving anything round, such as the balloon in our exercise.

The formula for finding the volume of a sphere is \[V = \frac{4}{3} \pi r^3\]
where \( r \) is the radius of the sphere. It's beautiful how the three-dimensional round shape fills space. In our balloon problem, we used this formula to calculate the balloon鈥檚 volume, which was 14.137 cubic meters using a radius of 1.50 meters.

Grasping this concept also helps when dealing with planets, bubbles, or even fruit like oranges. They're all spheres, and their volume determines how much space, or air in the balloon's case, they can contain.
Pressure Conversion
Pressure is often measured in different units, and the ability to convert between units is necessary in science. Standard units like pascals (Pa) or atmospheres (atm) are commonly used.
  • 1 atm = 101325 Pa
In our exercise, after calculating the pressure using the ideal gas law in pascals, we converted it into atmospheres to provide a more understandable comparison. The conversion of 611716 Pa to atm resulted in approximately 6.03 atm.

This understanding is quite critical in practical applications, such as meteorology or any time we talk about atmospheric pressure or scuba diving depths. Converting properly ensures that data remains consistent and understandable across different contexts.

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Most popular questions from this chapter

A clown at a birthday party has brought along a helium cylinder, with which he intends to fill balloons. When full, each balloon contains \(0.034 \mathrm{m}^{3}\) of helium at an absolute pressure of \(1.2 \times 10^{5} \mathrm{Pa} .\) The cylinder contains helium at an absolute pressure of \(1.6 \times 10^{7} \mathrm{Pa}\) and has a volume of \(0.0031 \mathrm{m}^{3} .\) The temperature of the helium in the tank and in the balloons is the same and remains constant. What is the maximum number of balloons that can be filled?

The diffusion constant for the amino acid glycine in water has a value of \(1.06 \times 10^{-9} \mathrm{m}^{2} / \mathrm{s} .\) In a \(2.0-\mathrm{cm}\) -long tube with a cross-sectional area of \(1.5 \times 10^{-4} \mathrm{m}^{2},\) the mass rate of diffusion is \(m / t=4.2 \times 10^{-14} \mathrm{kg} / \mathrm{s}\) because the glycine concentration is maintained at a value of \(8.3 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{3}\) at one end of the tube and at a lower value at the other end. What is the lower concentration?

One assumption of the ideal gas law is that the atoms or molecules themselves occupy a negligible volume. Verify that this assumption is reasonable by considering gaseous xenon (Xe). Xenon has an atomic radius of \(2.0 \times 10^{-10} \mathrm{m} .\) For STP conditions, calculate the percentage of the total volume occupied by the atoms.

Two moles of an ideal gas are placed in a container whose volume is \(8.5 \times 10^{-3} \mathrm{m}^{3} .\) The absolute pressure of the gas is \(4.5 \times 10^{5} \mathrm{Pa} .\) What is the average translational kinetic energy of a molecule of the gas?

The average value of the squared speed \(\overline{v^{2}}\) does not equal the square of the average speed \((\bar{v})^{2} .\) To verify this fact, consider three particles with the following speeds: \(v_{1}=3.0 \mathrm{m} / \mathrm{s}, v_{2}=7.0 \mathrm{m} / \mathrm{s},\) and \(\quad v_{3}=9.0 \mathrm{m} / \mathrm{s} . \quad\) Calculate (a) \(\quad \overline{v^{2}}=\frac{1}{3}\left(v_{1}^{2}+v_{2}^{2}+v_{3}^{2}\right) \quad\) and (b) \((\bar{v})^{2}=\left[\frac{1}{3}\left(v_{1}+v_{2}+v_{3}\right)\right]^{2}\)
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