Question

A 6.00 -ft cylinder has a radius of 26 in. It contains 189 lb of helium at \(25^{\circ} \mathrm{C}\). Express the volume of the cylinder \(\left(V=\pi r^{2} h\right)\) in liters, the amount of helium in moles, and the temperature in Kelvin.

Short answer

Answer: The volume of the cylinder is approximately 2531.84 L, the amount of helium is approximately 21433.06 moles, and the temperature is approximately 298.15 K.

Step-by-step solution

Calculate the volume of the cylinder

To find the volume of the cylinder (V), we can use the given formula: V = πr^2h, where r is the radius and h is the height of the cylinder. Given: Radius (r) = 26 inches Height (h) = 6.00 ft First, let's convert the radius from inches to feet by dividing it by 12 (1 ft = 12 inches): r = 26 in / 12 in/ft = 2.1667 ft. Now, let's plug in the values into the formula: V = π * (2.1667 ft)^2 * 6.00 ft Calculating the volume, we get: V ≈ 89.434 ft³

Convert the volume to liters

To convert the volume to liters, we can use the conversion factor 1 ft³ = 28.3168 L: V = 89.434 ft³ * 28.3168 L/ft³ ≈ 2531.84 L So the volume of the cylinder in liters is approximately 2531.84 L.

Calculate the amount of helium in moles

To find the amount of helium in moles, we need to use its mass and molar mass. Given the mass of helium is 189 lb, we first need to convert it to grams. The conversion factor is 1 lb = 453.592g: Mass of helium = 189 lb * 453.592g/lb ≈ 85780 g Now, let's calculate moles using the molar mass of helium (4.0026 g/mol): Moles of helium = 85780 g / 4.0026 g/mol ≈ 21433.06 mol So the amount of helium in the cylinder is approximately 21433.06 moles.

Convert the temperature to Kelvin

To convert the temperature from Celsius to Kelvin, we can use the conversion formula: Temperature (K) = Temperature (°C) + 273.15 Given that the temperature is 25°C: Temperature (K) = 25°C + 273.15 ≈ 298.15 K So the temperature in Kelvin is approximately 298.15 K. In summary, the volume of the cylinder is approximately 2531.84 L, the amount of helium is approximately 21433.06 moles, and the temperature is approximately 298.15 K.

Key concepts

Cylinder Volume Conversion

When working with the volume of a cylinder, it is essential to understand how to convert different units of measurement. The formula for finding the volume of a cylinder is (V = \( \pi r^2 h \)), where V is the volume, r is the radius, and h is the height. However, measurements are often not in the units convenient for the specific application. For example, in the explained exercise, the cylinder's dimensions were given in feet and inches, which were then converted to liters for the final volume measurement. This is a common practice in chemistry and other sciences, as liters are more regularly used when dealing with gases and liquids.
The conversion factor from cubic feet to liters is critical here: 1 ft³ = 28.3168 L. By using this conversion factor, students can easily switch from one volume unit to another and ensure that measurements are consistent with the international system of units, which is crucial for accurate scientific calculations.

Moles Calculation

The mole is a fundamental concept in chemistry that represents a specific number of particles, typically atoms or molecules. To calculate moles from a known mass of a substance, we use the formula moles = mass / molar mass. The molar mass is the mass of one mole of a substance and is measured in grams per mole (g/mol).
In the given exercise, the mass of the helium in pounds needed to be first converted to grams before we could apply this formula. This step is crucial because the molar mass is always referenced in grams. Once we have the mass in grams, dividing by the molar mass of helium (4.0026 g/mol) gives us the number of moles. Understanding the mole concept is imperative for stoichiometry and is essential for solving many types of chemistry problems, including reactions and gas law calculations.

Temperature Conversion

Temperature measurement is critical in chemistry as reaction rates, and various physical properties are affected by temperature. When dealing with temperature, it is common to convert values from Celsius to Kelvin for scientific calculations, as Kelvin is the SI unit for temperature. The conversion is straightforward: Temperature (K) = Temperature (°C) + 273.15.
In this exercise, beginning with a Celsius measurement, we added 273.15 to convert to Kelvin, which is necessary for gas law calculations, as these laws require an absolute temperature scale. Kelvin provides this scale because it starts at absolute zero, unlike Celsius or Fahrenheit, making it an integral part of chemistry calculations.

Stoichiometry

Stoichiometry is the section of chemistry that deals with the relative quantities of reactants and products in chemical reactions. It is based on the conservation of mass where the mass of the reactants equals the mass of the products. Key to stoichiometry is the concept of the mole, as it allows chemists to