Problem 63 - University Physics with Modern Physics

Question

A balloon whose volume is 750 \(\mathrm{m}^{3}\) is to be filled with hydrogen at atmospheric pressure \(\left(1.01 \times 10^{5} \mathrm{Pa}\right) .\) (a) If the hydrogen is stored in cylinders with volumes of 1.90 \(\mathrm{m}^{3}\) at a gauge pressure of \(1.20 \times 10^{6} \mathrm{Pa},\) how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if the gas in the balloon and the surrounding air are both at \(15.0^{\circ} \mathrm{C}\) ? The molar mass of hydrogen \(\left(\mathrm{H}_{2}\right)\) is 2.02 \(\mathrm{g} / \mathrm{mol}\) . The density of air at \(15.0^{\circ} \mathrm{C}\) and atmospheric pressure is 1.23 \(\mathrm{kg} / \mathrm{m}^{3} .\) See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 \(\mathrm{g} / \mathrm{mol}\) ) instead of hydrogen, again at \(15.0^{\circ} \mathrm{C}\) ?

Step-by-Step Solution

Answer

(a) 32 cylinders. (b) ~8373 N. (c) ~7854 N.

Step 1: Calculate Total Moles of Hydrogen for the Balloon

Using the Ideal Gas Law, calculate the total moles of hydrogen required to fill the balloon. The volume is 750 m³ at atmospheric pressure (\(1.01 \times 10^5 \text{ Pa}\)).The Ideal Gas Law is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, and \( n \) is moles. Given that \( R = 8.314 \text{ J/mol K} \) and \( T = 288 \text{ K} \) (\( 15^{\circ} \text{C} \)), we have:\[n = \frac{PV}{RT} = \frac{(1.01 \times 10^5 \text{ Pa})(750 \text{ m}^3)}{(8.314)(288)} \approx 31,608 \text{ mol}\]

Step 2: Calculate Moles of Hydrogen Per Cylinder

Determine the moles of hydrogen in each cylinder using the Ideal Gas Law, given the cylinder's volume is 1.90 m³ and the gauge pressure is \(1.20 \times 10^6 \text{ Pa}\). The total pressure is \(1.21 \times 10^6 \text{ Pa}\) (including atmospheric pressure).\[n = \frac{(1.21 \times 10^6 \text{ Pa})(1.90 \text{ m}^3)}{(8.314)(288)} \approx 1008 \text{ mol}\]

Step 3: Calculate Number of Cylinders Needed

Find the number of cylinders required by dividing the total moles of hydrogen needed (from Step 1) by the moles per cylinder (from Step 2).\[\text{Number of cylinders} = \frac{31608}{1008} \approx 31.36\]Since you can't have a fraction of a cylinder, round up to 32 cylinders.

Step 4: Determine Lifting Capacity of Hydrogen Balloon

Calculate the buoyant force using the density of air (1.23 kg/m³) minus the density of hydrogen and multiply by the volume of the balloon.The density of hydrogen \((\text{H}_2)\) is \(\frac{2.02 \text{ g/mol}}{22.4 \text{ L/mol}} \approx 0.08988 \text{ kg/m}^3\), corrected for temperature gives \(\frac{2.02 \text{ g/mol}}{24.789 \text{ L/mol}} \approx 0.0838 \text{ kg/m}^3\) (accounting for L to m³).The buoyant force is:\[F_b = V(\rho_{\text{air}} - \rho_{\text{H}_2})g\]\[F_b = 750\text{ m}^3(1.23 \text{ kg/m}^3 - 0.0838 \text{ kg/m}^3)(9.81 \text{ m/s}^2) \approx 8463 \text{ N}\]The weight supported, minus the weight of hydrogen, is approximately 8373 N (8463 N - the weight of hydrogen in the balloon).

Step 5: Determine Lifting Capacity of Helium Balloon

Now calculate the lifting capacity if the balloon is filled with helium. The molar mass of helium is 4.00 g/mol, giving a density of approximately 0.166 kg/m³.\[F_b = 750\text{ m}^3(1.23 \text{ kg/m}^3 - 0.166 \text{ kg/m}^3)(9.81 \text{ m/s}^2) \approx 7854 \text{ N}\]Hence, the supported weight, minus the weight of helium, is slightly less than hydrogen, considering helium is denser.

Key Concepts

Buoyant Force

When an object is submerged in a fluid, it gets pushed upward by a force called the buoyant force. This force is equivalent to the weight of the fluid displaced by the object. For balloons filled with lighter gases such as hydrogen or helium, this concept is important as it determines how much additional weight the balloon can carry.
For example, when calculating the buoyant force on a hydrogen-filled balloon, we subtract the density of hydrogen from the density of air. This gives us the effective lift per unit volume of the balloon. The formula for buoyant force, denoted as \( F_b \), is:

• \( F_b = V(\rho_{\text{air}} - \rho_{\text{gas}})g \)

where \( V \) is the volume of the balloon, \( \rho_{\text{air}} \) is the density of the air, \( \rho_{\text{gas}} \) is the density of the gas inside the balloon, and \( g \) is the acceleration due to gravity. This principle helps us understand the lifting capacity of balloons filled with different gases under the same conditions.

Moles of Gas

The concept of moles is fundamental in understanding how gases behave under different conditions. A mole represents a specific number of particles or molecules of a substance, approximately \( 6.022 \times 10^{23} \). To determine the number of moles in a given volume of gas using the Ideal Gas Law, you can use the equation:
\[ n = \frac{PV}{RT} \]
This equation relates pressure (\( P \)), volume (\( V \)), and temperature (\( T \)) of the gas, with \( R \) being the Ideal Gas Constant.
Let's break it down:

• When you know the pressure, volume, and temperature, you can calculate the total moles of gas present.
• An increase in pressure increases the number of moles for the same volume if temperature stays constant.

Understanding moles helps in comparing the quantity of gas stored under different pressures and volumes, such as the hydrogen in cylinders for inflating a balloon.

Density of Gases

Density is a measure of mass per unit volume and in gases, it can vary significantly with changes in pressure and temperature. The density \( \rho \) of a gas can be calculated using:

• \( \rho = \frac{m}{V} \)

Here, \( m \) is the mass of the gas and \( V \) is its volume. For gases, density can also be calculated using molar mass and the volume occupied by one mole (at standard temperature and pressure).
For instance, hydrogen has a very low density, making it ideal for filling balloons as it provides high buoyant force compared to heavier gases.

• Helium, though denser than hydrogen, is often used due to safety concerns related to hydrogen’s flammability.

To calculate the density of hydrogen, we use its molar mass and adjustments for actual temperature and pressure conditions to convert to kg/m³.

Cylinder Calculations

In this exercise, cylinders with hydrogen gas are used to fill a large balloon. Calculating how many cylinders are required involves a few steps. Using the Ideal Gas Law, we determine the total moles of gas needed to fill the balloon and how many moles are present in each cylinder.

• A cylinder's contents are often under high pressure, which greatly increases the number of moles of gas it can hold compared to the same volume at atmospheric pressure.
• For each cylinder, apply the Ideal Gas Law to find the moles of gas contained at the given pressure, then use this information to determine how many cylinders are needed by dividing the total moles required by the moles per cylinder.

This practical application of gas laws illustrates how pressurized storage allows for transportation of large amounts of gas in convenient form factors.