Problem 21 - Chemistry Matter and Change

Question

How much carbon dioxide gas, in grams, is in a 1.0-L balloon at STP?

Step-by-Step Solution

Answer

There are approximately 1.96 grams of CO2 in a 1.0-L balloon at STP.

Step 1: Understanding Standard Temperature and Pressure

Standard temperature and pressure (STP) is defined as 0°C (273.15 K) and 1 atm pressure. Under these conditions, one mole of any ideal gas occupies 22.4 liters.

Step 2: Identify Molar Volume and Use It

At STP, 1 mole of a gas occupies 22.4 L. We will use this information to determine how many moles are in a 1.0 L gas sample. The relation is: \( \text{moles of gas} = \frac{\text{volume of gas}}{\text{molar volume}} = \frac{1.0 \, \text{L}}{22.4 \, \text{L/mol}} \).

Step 3: Calculate Moles of Gas

We calculate the moles of carbon dioxide: \( \text{moles of } CO_2 = \frac{1.0}{22.4} \approx 0.0446 \, \text{moles} \).

Step 4: Determine Molar Mass of Carbon Dioxide

The molar mass of\( CO_2 \) can be calculated as follows: \( C = 12.01 \, \text{g/mol} \) and \( O = 16.00 \, \text{g/mol} \times 2 \). This gives \( \text{molar mass of } CO_2 = 44.01 \, \text{g/mol} \).

Step 5: Calculate Mass of Carbon Dioxide

Multiply the number of moles by the molar mass to find the mass of the gas: \( \text{mass} = \text{moles} \times \text{molar mass} = 0.0446 \, \text{moles} \times 44.01 \, \text{g/mol} \approx 1.96 \, \text{grams} \).

Key Concepts

Standard Temperature and Pressure

When you hear about standard temperature and pressure, often abbreviated as STP, you're learning about a set of agreed conditions used for the comparison of gas volumes. At STP, the temperature is set at 0°C, or 273.15 Kelvin, and the pressure is set at 1 atmosphere (atm). These conditions provide a baseline that makes it easier to predict and calculate how gases behave. Under STP, one mole of any theoretical ideal gas occupies exactly 22.4 liters. This knowledge allows scientists and students to do calculations more easily, knowing that they're all working with the same set of standards.

Molar Volume

The concept of molar volume is based on how much space one mole of a gas occupies. At standard temperature and pressure (STP), this is known to be 22.4 liters. This value holds true for ideal gases, which are hypothetical gases that strictly follow the behaviors outlined by the gas laws. So, if you have a balloon with 1 liter of gas at STP, knowing the molar volume allows you to calculate how many moles of gas you have. Using the formula \( \text{moles of gas} = \frac{\text{volume of gas}}{\text{molar volume}} \), you can easily find the number of moles in less than the molar volume by simple division.

Molar Mass

Molar mass is a critical concept in chemistry because it gives the mass of one mole of a substance. For carbon dioxide \( CO_2 \), which consists of one carbon atom and two oxygen atoms, you can find its molar mass by adding up the average atomic masses. Carbon has a molar mass of 12.01 g/mol, and oxygen has a molar mass of 16.00 g/mol. Thus, the molar mass of \( CO_2 \) is \( 12.01 + (16.00 \times 2) = 44.01 \) g/mol. Once you know the amount in moles, converting to mass involves multiplying the number of moles by this molar mass value.

Ideal Gas Law

The ideal gas law is an important equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of an ideal gas. Expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is the volume of the gas
• \( n \) is the number of moles
• \( R \) is the ideal gas constant (~0.0821 L·atm/mol·K)
• \( T \) is the temperature in Kelvin

This equation is a cornerstone for problems involving gases, allowing you to solve for any one variable if the others are known. It's a very flexible tool used to understand and predict the behavior of gases under various conditions, embedding in its structure the assumptions of a perfect or "ideal" gas.