Problem 72 - College Physics

Question

An ideal gas sample occupies a container of volume 0.75 Lat STP. Find (a) the number of moles and (b) the number of moleculesin in the sample. (c) If the gas is carbon monoxide (CO), what is the sample's mass?

Step-by-Step Solution

Answer

0.0335 mol; \(2.02 \times 10^{22}\) molecules; 0.938 g.

Step 1: Understand STP Conditions

Standard Temperature and Pressure (STP) refers to a temperature of 273.15 K (0°C) and a pressure of 1 atm. Under these conditions, 1 mole of an ideal gas occupies 22.4 L.

Step 2: Calculate the Number of Moles

Use the molar volume under STP conditions. The formula is \( n = \frac{V}{V_m} \), where \( V \) is the volume of the gas and \( V_m = 22.4 \text{ L/mol} \) is the molar volume. Thus, \( n = \frac{0.75 \text{ L}}{22.4 \text{ L/mol}} \approx 0.0335 \text{ mol} \).

Step 3: Calculate the Number of Molecules

Use Avogadro's number, which is \( 6.022 \times 10^{23} \text{ molecules/mol} \). Multiply it by the number of moles: \( N = 0.0335 \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 2.02 \times 10^{22} \text{ molecules} \).

Step 4: Determine the Mass of CO

Find the molar mass of carbon monoxide (CO): Carbon has an atomic mass of 12.01 g/mol, and oxygen has an atomic mass of 16.00 g/mol. Thus, \( M = 12.01 + 16.00 = 28.01 \text{ g/mol} \). Multiply the molar mass by the number of moles: \( m = 0.0335 \text{ mol} \times 28.01 \text{ g/mol} \approx 0.938 \text{ g} \).

Key Concepts

Moles Calculation

Calculating moles is an essential aspect of understanding chemical reactions and properties of gases. With the Ideal Gas Law, finding the number of moles is relatively straightforward, especially under standard conditions. To calculate moles when a gas is at Standard Temperature and Pressure (STP), use the formula:

• \( n = \frac{V}{V_m} \)

In this formula, \( n \) represents the number of moles, \( V \) is the volume of the gas, and \( V_m \) is the molar volume at STP, which is 22.4 L/mol. For example, if our gas occupies a volume of 0.75 L at STP, we can calculate the moles as follows:

• \( n = \frac{0.75 \text{ L}}{22.4 \text{ L/mol}} \approx 0.0335 \text{ mol} \)

This simple division gives us the number of moles in the sample, which is crucial for further calculations.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry. It represents the number of particles, usually atoms or molecules, in one mole of a substance. The value of Avogadro's Number is:

• \( 6.022 \times 10^{23} \text{ molecules/mol} \)

This value is essential when converting between moles and the actual number of molecules or atoms. For instance, in our example with 0.0335 moles of a gas, you can compute the number of molecules using this number by multiplying:

• \( N = 0.0335 \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol} \approx 2.02 \times 10^{22} \text{ molecules} \)

This conversion helps to bridge the gap between the macroscopic scale of moles and the microscopic scale of individual molecules.

Molar Mass Calculation

The molar mass of a substance is the mass of one mole of its particles. It's typically measured in grams per mole (g/mol) and is obtained by summing the atomic masses of all atoms in a molecule. For example, when calculating the molar mass of carbon monoxide (CO), you add:

• Carbon's atomic mass: 12.01 g/mol
• Oxygen's atomic mass: 16.00 g/mol
• \( ext{Molar Mass of CO} = 12.01 + 16.00 = 28.01 \text{ g/mol} \)

To find the mass of the sample, multiply the molar mass by the number of moles, \( n \):

• \( m = 0.0335 \text{ mol} \times 28.01 \text{ g/mol} \approx 0.938 \text{ g} \)

Understanding molar mass is crucial for relating the mass of a substance to the number of moles, which in turn is essential for chemical calculations.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) are reference conditions for measuring gases. They allow us to simplify calculations and make meaningful comparisons between different gas samples. At STP:

• The temperature is 273.15 K (0°C).
• The pressure is 1 atm.

Under these conditions, a neat property is that one mole of any ideal gas occupies 22.4 liters. This applies universally, allowing us to easily calculate moles or volumes of gases from one to another when they are at STP. This predefined state simplifies many gas calculations and is a common benchmark used in chemistry for analyzing gas behaviors.