Question

A 35.1 g sample of solid \(\mathrm{CO}_{2}(\) dry ice \()\) is added to a container at a temperature of 100 \(\mathrm{K}\) with a volume of 4.0 \(\mathrm{L} .\) If the container is evacuated (all of the gas removed), sealed and then allowed to warm to room temperature \((T=298 \mathrm{K})\) so that all of the solid \(\mathrm{CO}_{2}\) is converted to a gas, what is the pressure inside the container?

Short answer

= 0.797 mol \(CO_2\).

Step-by-step solution

Calculate the number of moles of CO₂

To determine the number of moles of CO₂, we will use the molar mass of CO₂. The molar mass of CO₂ = 12.01 + 2 × 16.00 = 44.01 g/mol. Using the mass of CO₂ and its molar mass, we can calculate the number of moles. Number of moles = mass / molar mass = (35.1 g) / (44.01 g/mol)

Key concepts

Molar Mass

Understanding molar mass is crucial when working with chemical compounds, as it plays a key role in determining the amount of substance present. Molar mass is defined as the mass of one mole of a substance and is expressed in grams per mole (g/mol).

In practical terms, molar mass serves as a conversion factor between the weight of a substance and the number of moles. The molar mass for any element can be found on the periodic table as it is the atomic weight of the element. For compounds like \( \mathrm{CO}_{2} \), you would add together the atomic weights of carbon (12.01 g/mol) and oxygen (16.00 g/mol, but since there are two oxygens in carbon dioxide, you must multiply this by two) to get the molar mass of the compound.

In our exercise about the gas pressure calculation, the molar mass of carbon dioxide is calculated by the equation:\[ \text{Molar Mass of } \mathrm{CO}_{2} = 12.01 + 2 \times 16.00 = 44.01 \text{ g/mol} \]This value is used to convert the mass of solid \( \mathrm{CO}_{2} \) to moles to further use in gas pressure calculation using the ideal gas law.

Moles Calculation

The moles calculation is a fundamental aspect of chemistry, as it allows scientists to quantify the amount of a substance. Moles are a measure of the number of particles, typically atoms or molecules, in a given mass of a substance.

To calculate the number of moles, divide the mass of the substance by its molar mass. For example, in the problem given, the moles of \( \mathrm{CO}_{2} \) are calculated using the equation:\[ \text{Number of moles} = \frac{\text{mass}}{\text{molar mass}}= \frac{35.1 \text{ g}}{44.01 \text{ g/mol}} \]This step is fundamental for understanding the quantity of \( \mathrm{CO}_{2} \) gas present, which in turn is necessary to calculate the pressure using the ideal gas law.

To enhance the understanding of this concept for students, we should emphasize the practical examples and encourage them to practice converting between mass and moles with various substances, reinforcing the idea that one mole of any substance contains Avogadro's number of particles, which is \(6.022 \times 10^{23}\) particles/mole.

Ideal Gas Law

The ideal gas law is a critical equation in chemistry and physics that relates the pressure, volume, temperature, and the amount of an ideal gas through the formula:\[ PV=nRT \]where:\[ \begin{align*} P & = \text{Pressure of the gas (in atmospheres)} \ V & = \text{Volume (in liters)} \ n & = \text{Number of moles of gas} \ R & = \text{Ideal gas constant (}\approx 0.0821 \text{ L·atm/(mol·K))} \ T & = \text{Temperature (in kelvins)} \end{align*} \]Understanding this law is key for predicting how a gas will behave under different conditions. For the exercise in question, the ideal gas law enables us to find the pressure inside the container after the solid \( \mathrm{CO}_{2} \) has been converted to gas and the container is warmed to room temperature.

To apply the ideal gas law, we must first ensure that all quantities are in the correct units, or convert them if necessary. For instance, the temperature must be in kelvins and volume in liters. It’s also useful for students to remember that the ideal gas constant \(R\) has different values depending on the units of pressure and volume used.

We should remind students that while the ideal gas law provides an excellent estimation, real gases may deviate from this behavior under certain conditions, particularly at high pressures or low temperatures where gas particles interact more.