Problem 109 - 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 acetylene, \(\mathrm{C}_{2} \mathrm{H}_{2}\) (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) C2H2 cylinder has greater pressure. (b) C2H2 cylinder contains more molecules.

Step 1: Determine Molar Mass of CO and C2H2

First, we must find the molar mass of each gas. The molar mass of CO is the sum of the atomic masses of carbon (C) and oxygen (O): - Carbon (C): 12.01 g/mol - Oxygen (O): 16.00 g/mol Thus, the molar mass of CO is 28.01 g/mol. For acetylene, C2H2, the molar mass is the sum of the atomic masses of two carbons and two hydrogens: - 2 x Carbon (C): 2 x 12.01 g/mol = 24.02 g/mol - 2 x Hydrogen (H): 2 x 1.01 g/mol = 2.02 g/mol Thus, the molar mass of C2H2 is 26.04 g/mol.

Step 2: Calculate Moles of Each Gas

Using the molar masses, find the number of moles in each cylinder. For CO (given mass = 1.0 kg = 1000 g):\[\text{Moles of CO} = \frac{1000 \text{ g}}{28.01 \text{ g/mol}} \approx 35.7 \text{ moles}\]For C2H2 (given mass = 1.0 kg = 1000 g):\[\text{Moles of C2H2} = \frac{1000 \text{ g}}{26.04 \text{ g/mol}} \approx 38.4 \text{ moles}\]

Step 3: Use Ideal Gas Law to Compare Pressures

According to the ideal gas law, \( PV = nRT \), where \(P\) is pressure, \(V\) is volume, \(n\) is moles, \(R\) is the gas constant, and \(T\) is temperature in Kelvin.Since both gases are at the same temperature and volume:\[ P \propto n\]Therefore, the pressure is directly proportional to the number of moles.Since C2H2 has more moles (38.4) than CO (35.7), the pressure in the C2H2 cylinder is greater.

Step 4: Compare Number of Molecules

The number of molecules can be compared using Avogadro's number, \(6.022 \times 10^{23}\) molecules per mole.The number of molecules in each gas is given by the formula:\[ \text{Number of molecules} = \text{moles} \times \text{Avogadro's number}\]For CO:\[ 35.7 \text{ moles} \times 6.022 \times 10^{23} \approx 2.15 \times 10^{25} \text{ molecules}\]For C2H2:\[ 38.4 \text{ moles} \times 6.022 \times 10^{23} \approx 2.31 \times 10^{25} \text{ molecules}\]Thus, the cylinder with C2H2 contains the greater number of molecules.

Key Concepts

Molar Mass Calculation

Molar mass is an essential concept in chemistry. It helps us understand the mass of one mole of a substance, and it serves as the bridge between mass in grams and the amount in moles.
To calculate molar mass, you simply add up the atomic masses of all the atoms in a molecule. Let's break it down using carbon monoxide (CO) and acetylene \((C_2H_2)\) as examples.
Carbon monoxide is made up of one carbon atom and one oxygen atom. The atomic mass of Carbon (C) is 12.01 g/mol, and Oxygen (O) is 16.00 g/mol. By adding these together, we get the molar mass of CO:

• CO molar mass: 28.01 g/mol

Similarly, for acetylene \((C_2H_2)\), we sum up the masses of its atoms. It has two carbon atoms and two hydrogen atoms.

• 2 x Carbon: 24.02 g/mol (because 2 times 12.01 g/mol)
• 2 x Hydrogen: 2.02 g/mol (because 2 times 1.01 g/mol)
• Total C\(_2\)H\(_2\) molar mass: 26.04 g/mol

Understanding molar mass allows you to calculate how much a certain amount of substance would weigh, which is crucial for stoichiometry and understanding gas behaviors like pressure, as seen here. The lower the molar mass for a fixed mass of gas, the more moles and consequently, molecules, it contains.

Avogadro's Number

Understanding Avogadro's number is key to grasping the molecular scale of gases. This number, \(6.022 \times 10^{23}\), represents the number of atoms, ions, or molecules in one mole of a substance. It's an incredibly large number due to the small size of molecules.
When comparing carbon monoxide and acetylene, Avogadro's number allows us to relate the moles of a gas to the actual count of molecules.

• For CO, we found 35.7 moles. Multiplying by Avogadro's number gives us about \(2.15 \times 10^{25}\) molecules.
• For C\(_2\)H\(_2\), there are 38.4 moles. This results in approximately \(2.31 \times 10^{25}\) molecules.

At equivalent mass, the acetylene cylinder has more moles because it has a lower molar mass. This translates to more molecules, as shown by the result when using Avogadro's number. Recognizing this connection helps you understand why, despite being the same mass, the number of molecules—and therefore the pressure—can differ based on molar mass.

Gas Pressure Comparison

Why does the pressure vary between different gases in identical containers? This boils down to the ideal gas law, formulated as \(PV = nRT\). Here, \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is temperature.
In situations where the volume \(V\) and temperature \(T\) are constant for both gases, pressure \(P\) is directly proportional to the number of moles \(n\). For our example:

• CO has about 35.7 moles.
• C\(_2\)H\(_2\) has about 38.4 moles.

Since acetylene \((C_2H_2)\) contains more moles than carbon monoxide \((CO)\) at the same temperature and volume, it has a higher pressure in its container. This relationship highlights how important the number of moles is in determining gas behavior under constant conditions.
Recognizing the influence of moles on pressure, it becomes clear why pressure differences arise between gases like CO and C\(_2\)H\(_2\) even when they occupy the same space and mass, further showing where the ideal gas law comes into play in practical scenarios.