Problem 101 - Chemistry Matter and Change

Question

Analyze A solid brick of dry ice \(\left(CO_{2}\right)\) weighs 0.75 kg. Once the brick has fully sublimated into \(\left(CO_{2}\right)\) gas, what would its volume be at STP?

Step-by-Step Solution

Answer

The volume of carbon dioxide gas produced when a 0.75 kg brick of dry ice fully sublimates is approximately 383.6 liters at STP.

Step 1: Convert mass of dry ice to moles

Given mass of dry ice is 0.75 kg. To convert it into moles, we need to use the molar mass of CO2. The molar mass of CO2 is 12.01 g/mol (for carbon) + 2 * 16.00 g/mol (for oxygen) = 44.01 g/mol. First, convert the mass of the dry ice brick from kg to g: 0.75 kg * 1000 g/kg = 750 g Now, we'll find the number of moles by dividing the mass of dry ice by the molar mass of CO2: \(n_{CO_{2}} = \frac{mass_{CO_{2}}}{molar\_mass_{CO_{2}}}\) \(n_{CO_{2}} = \frac{750 \ g}{44.01 \ g/mol}\) \(n_{CO_{2}} ≈ 17.04 \ mol\)

Step 2: Use the ideal gas law to find the volume

Now that we have the number of moles, we can use the ideal gas law equation and the given STP conditions to find the volume of CO2 gas: \(PV = nRT\) At STP, the pressure P is 1 atm, and the temperature T is 273.15 K. The gas constant R is 0.0821 L atm / (mol K). Now, we can plug the values and solve for the volume V: \(V = \frac{nRT}{P}\) \(V = \frac{(17.04 \ mol)(0.0821 \ L \cdot atm / (mol \cdot K))(273.15 \ K)}{1 \ atm}\) \(V ≈ 383.6 \ L\)

Step 3: State the final answer

At STP, the volume of carbon dioxide gas produced when a 0.75 kg brick of dry ice fully sublimates is approximately 383.6 liters.

Key Concepts

Molar Mass

Molar mass is a fundamental concept in chemistry, used to convert between the mass of a substance and its amount in moles. It is the mass of one mole of a given substance and is usually expressed in grams per mole \(g/mol\).

For carbon dioxide \(CO_2\), the molar mass is calculated by adding the atomic masses of one carbon atom and two oxygen atoms:- Carbon (C): 12.01 g/mol- Oxygen (O): 16.00 g/mol

Hence, the molar mass of \(CO_2\) is \(12.01 + 2 \times 16.00 = 44.01 \ g/mol\).

This value allows us to determine the number of moles in a sample by dividing the sample's mass by its molar mass, as shown in the given exercise with the dry ice conversion to moles.

STP Conditions

Standard Temperature and Pressure (STP) is a set of conditions used as a reference for gas calculations. At STP:

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

These conditions help simplify calculations by providing a common standard. Using STP, one mole of any ideal gas occupies 22.4 liters of volume.

In the exercise, these conditions are crucial because they allow us to use the ideal gas law directly to find the volume of carbon dioxide gas that would result from the sublimation of the dry ice brick.

Sublimation

Sublimation is a phase transition in which a substance goes directly from a solid to a gas state, bypassing the liquid phase.

Dry ice, the solid form of carbon dioxide, sublimates at temperatures above \(-78.5°C\) or \(-109.3°F\).
This property makes dry ice useful for various applications like refrigeration, as it transitions cleanly into a gas without leaving liquid residues.

In the exercise, sublimation describes how the solid dry ice turns directly into carbon dioxide gas, which is then analyzed for its volume.

Carbon Dioxide

Carbon dioxide \(CO_2\) is a colorless, odorless gas that is naturally present in the Earth's atmosphere. It consists of one carbon atom covalently double bonded to two oxygen atoms.

Despite being a minor component of the atmosphere, carbon dioxide plays a vital role in various processes:

• It is involved in the respiratory processes of plants and animals.
• It contributes to the greenhouse effect, which impacts Earth’s climate.
• It is used in industry for refrigeration and carbonation of beverages.

In the context of the exercise, understanding \(CO_2\) as a gas resulting from the sublimation of dry ice is essential for calculating its volume under different conditions like STP.