Problem 99 - Chemistry and Chemical Reactivity

Question

You have two pressure-proof steel cylinders of equal volume, one containing \(1.0 \mathrm{kg}\) of \(\mathrm{CO}\) and the other containing \(1.0 \mathrm{kg}\) of acetylenc, \(\mathrm{C}_{2} \mathrm{H}_{2}\) (a) In which cylinder is the pressure greater at \(25^{\circ} \mathrm{C} ?\) (b) Which cylinder contains the greater number of molecules?

Step-by-Step Solution

Answer

(a) Acetylene cylinder has greater pressure. (b) Acetylene cylinder has more molecules.

Step 1: Calculate Molar Mass of CO and C2H2

First, we need to determine the molar mass for each gas. The molar mass of carbon monoxide (CO) is calculated by adding the atomic mass of carbon (12.01 g/mol) and oxygen (16.00 g/mol), which gives us: \[ \text{Molar Mass of CO} = 12.01 + 16.00 = 28.01 \ \text{g/mol} \]For acetylene (C2H2), the molar mass is given by: \[ \text{Molar Mass of C}_{2}\text{H}_{2} = 2(12.01) + 2(1.008) = 26.036 \ \text{g/mol} \]

Step 2: Determine the Number of Moles

Using the known weights of the gases (1.0 kg = 1000 g) and their respective molar masses calculated above, we calculate the number of moles of each gas:For CO:\[ \text{Number of moles of CO} = \frac{1000}{28.01} = 35.7 \ \text{moles} \]For C2H2:\[ \text{Number of moles of C}_{2}\text{H}_{2} = \frac{1000}{26.036} = 38.4 \ \text{moles} \]

Step 3: Compare the Pressures

Both gases are in equal volumes at the same temperature. According to the ideal gas law \(PV = nRT\), since the volumes (V) and the temperature (T) remain constant, and assuming R is a constant, the pressure (P) will vary with the number of moles (n). Hence, the cylinder with more moles will have greater pressure.Since acetylene has more moles (38.4) compared to CO (35.7), the pressure in the acetylene cylinder is greater.

Step 4: Count the Molecules

To find which cylinder contains the greater number of molecules, we need to use Avogadro's number (\(6.022 \times 10^{23}\) molecules/mol).Number of molecules in CO:\[ 35.7 \times 6.022 \times 10^{23} = 2.15 \times 10^{25} \ \text{molecules} \]Number of molecules in C2H2:\[ 38.4 \times 6.022 \times 10^{23} = 2.31 \times 10^{25} \ \text{molecules} \]Thus, the cylinder with acetylene has a greater number of molecules.

Key Concepts

Molar Mass

The molar mass of a substance is central to understanding chemical quantities. It is the mass of one mole of molecules or atoms. This is expressed in grams per mole (g/mol). The molar mass is obtained by summing the atomic mass units of all the atoms present in a molecular formula.

For example, to find the molar mass of carbon monoxide (CO), we add the atomic mass of carbon (12.01 g/mol) with oxygen (16.00 g/mol), giving us a molar mass of 28.01 g/mol for CO. Similarly, for acetylene (C₂H₂), it involves summing the masses of its atoms: two carbon atoms and two hydrogen atoms. Thus, its molar mass is 26.036 g/mol.

This molar mass is crucial when calculating the number of moles, which helps us translate mass into a count of entities like molecules, enabling stoichiometric calculations. Understanding molar mass is foundational in comparing the quantities of different substances in chemical reactions.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry, which helps us to relate the pressure, volume, temperature, and number of moles of a gas. It is expressed by the equation: \[ PV = nRT \]where:

• P is the pressure
• V is the volume
• n is the number of moles
• R is the universal gas constant
• T is the temperature in Kelvin

In this law, if any three of these variables are known, the fourth can be calculated.

For the gases in our cylinders, because they are at the same temperature and volume, the pressure becomes directly proportional to the number of moles. Hence, more moles mean higher pressure due to greater particle collisions within the same volume. This concept helps us determine why the acetylene cylinder has greater pressure than that containing carbon monoxide when both contain the same mass of gas.

Avogadro's Number

Avogadro's Number is a key concept for understanding the amount of substance in chemistry. It is the number of entities (usually atoms or molecules) in one mole of a substance, specifically: \[ 6.022 imes 10^{23} \] entities/mol.

This means one mole of any substance contains this vast number of particles, whether it's atoms in an element or molecules in a compound. This constant is essential for converting moles into an actual number of particles.

In our example problems, even though the mass of the gases is the same, converting to moles and then using Avogadro's Number allows us to determine which cylinder holds more molecules. By multiplying the number of moles found for each gas by Avogadro's Number, we see the acetylene cylinder holds more molecules than the carbon monoxide cylinder, because it has more moles. This is a wonderful illustration of Avogadro's principle that equal volumes of gases (at the same temperature and pressure) contain an equal number of molecules.