A 20-cm-diameter cylinder that is 40 cm long contains 50 g of oxygen gas at 20°C. What is the number density of the oxygen?

검증된 단계별 안내

Step 1: Start by understanding the concept of number density. Number density is defined as the number of particles (in this case, molecules of oxygen) per unit volume. Mathematically, it is given by \( n = \frac{N}{V} \), where \( n \) is the number density, \( N \) is the total number of molecules, and \( V \) is the volume of the container.

Step 2: Calculate the volume of the cylinder. The formula for the volume of a cylinder is \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the height (or length) of the cylinder. Convert the diameter (20 cm) to radius (10 cm) and the height (40 cm) to meters before substituting into the formula.

Step 3: Use the molar mass of oxygen to determine the number of moles of oxygen gas. The molar mass of oxygen (O₂) is approximately 32 g/mol. The number of moles \( n_{\text{moles}} \) is given by \( n_{\text{moles}} = \frac{m}{M} \), where \( m \) is the mass of the gas (50 g) and \( M \) is the molar mass.

Step 4: Use Avogadro's number to find the total number of molecules \( N \). Avogadro's number is \( 6.022 \times 10^{23} \) molecules/mol. Multiply the number of moles \( n_{\text{moles}} \) by Avogadro's number to get \( N \).

Step 5: Finally, calculate the number density \( n \) by dividing the total number of molecules \( N \) by the volume \( V \) of the cylinder. Ensure that the units are consistent (e.g., volume in cubic meters) to obtain the correct number density in molecules per cubic meter.

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주요 개념

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Number Density

Number density is defined as the number of particles (atoms, molecules, etc.) per unit volume. It is typically expressed in units such as particles per cubic meter (m³). In this context, calculating the number density of oxygen gas involves determining how many molecules are present in the given volume of the cylinder.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is essential for understanding the behavior of gases under various conditions and can be used to find the number of moles of oxygen in the cylinder, which is necessary for calculating number density.

Volume of a Cylinder

The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height. In this problem, the diameter of the cylinder is given, allowing us to find the radius and subsequently the volume, which is crucial for determining the number density of the oxygen gas contained within.