Problem 57 - Physics

Question

A spherical balloon is made from a 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.

Step-by-Step Solution

Answer

The absolute pressure of the helium gas is approximately 62335 Pa.

Step 1 - Understand the Conditions for Floating

Since the balloon is floating, the buoyant force must equal the gravitational force on the balloon. According to Archimedes' Principle, \[F_b = \rho_{\text{air}} \cdot V_{\text{displaced}} \cdot g\]where \(F_b\) is the buoyant force, \(\rho_{\text{air}}\) is the density of air, \(V_{\text{displaced}}\) is the volume of the balloon (and helium), and \(g\) is the acceleration due to gravity.

Step 2 - Calculate the Buoyant Force

Next, calculate the volume of the balloon and thereby the volume of the displaced helium, using the formula for the volume of a sphere:\[ V = \frac{4}{3}\pi r^3 \]Substituting the radius \(r = 1.50\) m, we have:\[ V = \frac{4}{3}\pi (1.50)^3 \approx 14.137 \text{ m}^3 \]

Step 3 - Calculate the Weight of the Balloon

The weight \(W\) of the balloon (including helium gas and material) is given by:\[ W = m_{\text{material}} \cdot g + m_{\text{helium}} \cdot g \]Let's compute \(m_{\text{helium}}\):\[ m_{\text{helium}} = \rho_{\text{helium}} \cdot V \]where \(\rho_{\text{helium}}\) is the density of helium which is unknown but can be obtained once we solve for the pressure.

Step 4 - Equate Buoyant Force and Weight

Since the balloon just floats, we equate the buoyant force to the weight:\[ \rho_{\text{air}} \cdot V \cdot g = m_{\text{material}} \cdot g + \rho_{\text{helium}} \cdot V \cdot g \]After cancelling \(g\) and rearranging, we get:\[ \rho_{\text{helium}} = \rho_{\text{air}} - \frac{m_{\text{material}}}{V} \]Substitute \(\rho_{\text{air}} = 1.19 \text{ kg/m}^3\), \(m_{\text{material}} = 3\text{ kg}\), and \(V = 14.137 \text{ m}^3\), to find \( \rho_{\text{helium}} \).

Step 5 - Calculate the Density of Helium

Substitute the values into the equation:\[ \rho_{\text{helium}} = 1.19 - \frac{3}{14.137} \approx 0.978 \text{ kg/m}^3 \]

Step 6 - Use Ideal Gas Law to Find Pressure

Apply the Ideal Gas Law:\[ P = \frac{\rho_{\text{helium}} \cdot R \cdot T}{M_{\text{He}}} \]where \(P\) is the pressure we need to find, \(R = 8.314 \text{ J/mol K}\) is the universal gas constant, \(T = 305 \text{ K}\) is the helium temperature, and \(M_{\text{He}} = 4.002 \text{ g/mol}\) is the molar mass of helium. Convert molar mass into kg/mol: \(M_{\text{He}} = 0.004002 \text{ kg/mol}\).

Step 7 - Solve for Pressure

Substitute the known values into the Ideal Gas Law:\[ P = \frac{0.978 \cdot 8.314 \cdot 305}{0.004002} \approx 62335 \text{ Pa} \]

Step 8 - Conclusion

Hence, the absolute pressure of the helium gas in the spherical balloon is approximately \(62335 \text{ Pa}\).

Key Concepts

Archimedes' Principle

Archimedes' Principle is a fundamental concept in physics that explains why objects float. It states that any object submerged in a fluid (like water or air) experiences a buoyant force equal to the weight of the fluid displaced by the object. This principle is essential in understanding situations where objects either float, sink, or remain suspended at a particular level in the fluid.

• Buoyant force is directed upwards and is countered by the weight of the object.
• If the buoyant force equals the object's weight, the object will float, just like the balloon outlined in the original exercise.
• The buoyant force can be calculated using the formula: \[ F_b = \rho_{\text{air}} \times V_{\text{displaced}} \times g \]where \( \rho_{\text{air}} \) is the air's density, \( V_{\text{displaced}} \) is the volume displaced, and \( g \) is gravity.

Ideal Gas Law

The Ideal Gas Law is a vital equation in thermodynamics that describes the behavior of an ideal gas. It is often used to relate variables such as pressure, volume, and temperature to the amount of gas present. The law is expressed as:\[ PV = nRT \]where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume it occupies,
• \( n \) is the number of moles of gas,
• \( R \) is the ideal gas constant \(8.314 \text{ J/mol K}\),
• \( T \) is the temperature in Kelvin.

This law assumes that gases behave ideally, meaning that there are no intermolecular forces and that the volume occupied by gas molecules is negligible compared to the volume of the container. Although real gases do exhibit these forces and volumes, the Ideal Gas Law gives a close approximation for many cases, particularly at higher temperatures and lower pressures.

Density of Helium

Density is defined as mass per unit of volume. Helium is a light, inert gas often used in balloons due to its lower density compared to air, which provides buoyancy and lift. In this exercise, the density of helium was crucial to maintaining equilibrium.

• Its density can be calculated when both the mass and the volume are known.
• For gases, density can also be linked to their molar mass and the Ideal Gas Law.
• Density formula: \[ \rho = \frac{m}{V} \]where \( m \) is the total mass and \( V \) is the total volume.

The small density of helium means it can displace air in a balloon, creating a buoyant force sufficient to counteract its own weight plus the weight of the balloon material.

Buoyant Force

Buoyant force is an upward force exerted by a fluid that opposes the weight of an object immersed in the fluid. Critical to the concept of floating, the buoyant force explains why ships do not sink and submarines can be buoyant. It is governed by the principle explained by Archimedes.

• Calculated using the formula: \[ F_b = \rho_{\text{fluid}} \times V_{\text{submerged}} \times g \]
• The magnitude of the buoyant force is equivalent to the weight of the displaced fluid.
• If an object is completely submerged and is not rising or sinking, it means the buoyant force equals the gravitational pull (weight) of the object.

In real-life applications like hot air balloons and marine vessels, understanding buoyant force is critical to exert control over their movement in fluids.