Problem 54 - Chemistry and Chemical Reactivity

Question

You want to fill a cylindrical tank with \(\mathrm{CO}_{2}\) gas at \(865 \mathrm{mm}\) Hg and \(25^{\circ} \mathrm{C}\). The tank is \(20.0 \mathrm{m}\) long with a \(10.0-\mathrm{cm}\) radius. What mass of \(\mathrm{CO}_{2}\) (in grams) is required?

Step-by-Step Solution

Answer

Calculate volume, pressure, and temperature; apply ideal gas law to find moles, then convert to mass.

Step 1: Calculate the Volume of the Cylinder

The formula for the volume of a cylinder is given by \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length) of the cylinder.In our case, the radius \( r = 10.0 \) cm or \( 0.1 \) m (since we need to convert to meters for consistency with SI units), and the height \( h = 20.0 \) m.So, the volume of the cylinder is \[ V = \pi (0.1)^2 (20.0) = 0.02\pi \text{ m}^3. \]

Step 2: Convert Pressure to Pascals

The given pressure in the problem is in mm Hg, which we need to convert to Pascals (the SI unit for pressure). The conversion factor is \( 1 \) mm Hg = \( 133.322 \) Pa.Thus, the pressure \( P = 865 \times 133.322 = 115314.83 \) Pa.

Step 3: Convert Temperature to Kelvin

The temperature is given in degrees Celsius, so we convert it to Kelvin using the formula \( T(K) = T(^\circ C) + 273.15 \).Therefore, the temperature \( T = 25 + 273.15 = 298.15 \) K.

Step 4: Use Ideal Gas Law to Find Moles of CO2

The Ideal Gas Law is given by \( PV = nRT \), where \( n \) is the number of moles, \( R \) is the ideal gas constant \( 8.314 \) J/(mol·K), and we have already calculated \( P \), \( V \), and \( T \).Rearranging for \( n \), we have \( n = \frac{PV}{RT} \). Substituting values, we find \[ n = \frac{115314.83 \times 0.02\pi}{8.314 \times 298.15} .\]

Step 5: Calculate the Mass of CO2

Once we have calculated \( n \) in the previous step, we can find the mass by using the molar mass of \( CO_2 \), which is approximately \( 44.01 \) g/mol.The mass \( m \) is calculated by \( m = n \times 44.01 \). Use the value of \( n \) from Step 4 to find \( m \).

Key Concepts

Cylinder Volume Calculation

When dealing with gases in cylindrical containers, the first step is often calculating the volume of the cylinder. This can be done using the formula:

• \( V = \pi r^2 h \)

where:

• \( V \) is the volume of the cylinder,
• \( r \) is the radius, and
• \( h \) is the height or length of the cylinder.

In this exercise, the radius is given as 10 cm. However, as we perform scientific calculations, it's best to convert this into meters. Therefore, 10 cm becomes 0.1 meters. The height of the cylinder is 20 meters, as noted.
Using these values, the volume of the cylinder is calculated as:

• \[ V = \pi (0.1)^2 (20.0) = 0.02\pi \space \text{m}^3. \]

This volume will be used to determine how much space is available for the gas inside the cylinder.

Pressure Conversion

To work with the Ideal Gas Law, pressures usually need to be in Pascals, the SI unit for pressure. The problem gives the pressure in mm Hg (millimeters of mercury), a common unit in smaller-scale processes. To convert this:

• The conversion factor is: \( 1 \space \text{mm Hg} = 133.322 \space \text{Pa} \).

With a pressure of 865 mm Hg in this problem, you calculate:

• \[ P = 865 \times 133.322 = 115314.83 \space \text{Pa}. \]

Ensuring the pressure is in Pascals is crucial for accurate application of the Ideal Gas Law unless stated otherwise. This sets the stage for accurately finding other properties of the gas.

Temperature Conversion

Most scientific and engineering calculations use temperature in Kelvin, as it is the absolute temperature scale. If a temperature is given in degrees Celsius, it can be easily converted to Kelvin using:

• \( T(K) = T(^\circ C) + 273.15 \)

Here, the given temperature is 25°C, which converts to Kelvin as:

• \[ T = 25 + 273.15 = 298.15 \space \text{K}. \]

This conversion is important because the Ideal Gas Law requires temperatures in absolute terms. It helps provide consistent and accurate calculations of the gas properties under different conditions.

Molar Mass of CO2

Understanding the molar mass of CO2 is crucial for converting from moles to grams. The molar mass is the mass of one mole of a substance. For carbon dioxide (CO2), it's the sum of the atomic masses of its constituent elements:

• Carbon has an atomic mass of approximately 12.01 g/mol.
• Oxygen has an atomic mass of approximately 16.00 g/mol.

Since CO2 consists of one carbon atom and two oxygen atoms:

• \[ \text{Molar mass of } \mathrm{CO}_2 = 12.01 + 2 \times 16.00 = 44.01 \text{ g/mol}. \]

This value is key when you know the number of moles of CO2 and need to find the total mass. Once you calculate the moles using the Ideal Gas Law, the mass can be determined by multiplying the number of moles by the molar mass of CO2.