Problem 41 - Chemistry The Central Science In Si Units

Question

A 50.0 g sample of solid \(\mathrm{CO}_{2}\) (dry ice) is added at \(-100^{\circ} \mathrm{C}\) to an evacuated (all of the gas removed) container with a volume of \(5.0 \mathrm{~L}\). If the container is sealed and then allowed to warm to room temperature \(\left(25^{\circ} \mathrm{C}\right)\) so that the entire solid \(\mathrm{CO}_{2}\) is converted to a gas, what is the pressure inside the container?

Step-by-Step Solution

Answer

The pressure inside the container is approximately 4.54 atm.

Step 1: Convert Mass to Moles

First, calculate the number of moles of carbon dioxide. The molar mass of CO₂ is approximately 44.01 g/mol. Use the formula: \[ n = \frac{\text{mass}}{\text{molar mass}} = \frac{50.0 \, \text{g}}{44.01 \, \text{g/mol}}. \] Calculate the moles of CO₂.

Step 2: Rearrange Ideal Gas Law for Pressure

The Ideal Gas Law equation is \( PV = nRT \). We want to solve for pressure \( P \), so rearrange the equation: \[ P = \frac{nRT}{V}. \]

Step 3: Calculate Temperature in Kelvin

Convert the temperature from Celsius to Kelvin using the formula: \[ T(K) = T(^\circ C) + 273.15. \] Given that the temperature is 25°C, substitute to find the temperature in Kelvin.

Step 4: Insert Values into Ideal Gas Law

Use the previously calculated moles and temperature. The volume \( V \) is given as 5.0 L, and the gas constant \( R \) is typically 0.0821 L·atm/mol·K. Add the values to the equation: \[ P = \frac{(n)(0.0821 \, \text{L·atm/mol·K})(298.15 \, \text{K})}{5.0 \, \text{L}}. \] Calculate \( P \) to find the pressure in atm.

Key Concepts

Moles Calculation

Understanding how to calculate moles is crucial in chemistry. Moles are a way to express the number of particles, such as atoms or molecules. In this problem, we're dealing with dry ice, which is the solid form of carbon dioxide \(\text{CO}_2\). To find out how many moles of \(\text{CO}_2\) we have, we need to know its mass and molar mass.
First, let's consider the mass given: 50.0 grams of \(\text{CO}_2\). The molar mass of \(\text{CO}_2\) is approximately 44.01 grams per mole. By dividing the mass by the molar mass, we determine the number of moles using the formula:

• \( n = \frac{\text{mass}}{\text{molar mass}} = \frac{50.0 \, \text{g}}{44.01 \, \text{g/mol}} \).

Thus, this calculation tells us how many moles of \(\text{CO}_2\) we have in the container. Calculating moles is an essential step when applying the Ideal Gas Law.

Temperature Conversion

For many calculations in physics and chemistry, especially those involving gases, it's important to convert temperatures from Celsius to Kelvin. Kelvin is the SI unit of temperature and helps in maintaining consistency across equations.
To convert Celsius to Kelvin, use the formula:

• \( T(\text{K}) = T(^{\circ}\text{C}) + 273.15 \).

In our exercise, the initial temperature of the dry ice is \-100^{\circ}\text{C}\, and it warms to room temperature, which is \25^{\circ}\text{C}\. By applying the formula, we obtain \[ T(\text{K}) = 25 + 273.15 = 298.15 \text{K} \].
This conversion is necessary to ensure all parameters match the units used in the Ideal Gas Law equation.

Pressure Calculation

Pressure is a vital parameter in the Ideal Gas Law. Pressure represents how much force the gas exerts on the walls of its container. In this exercise, we need to find the pressure inside a 5.0 L container after the dry ice sublimates into a gas.
Using the Ideal Gas Law \( PV = nRT \), we can rearrange it to solve for pressure \( P \):

• \( P = \frac{nRT}{V} \).

Where:

• \( n \) is the number of moles we calculated earlier,
• \( R \) is the universal gas constant \( 0.0821 \, \text{L}\cdot\text{atm/mol}\cdot\text{K} \),
• \( T \) is the temperature in Kelvin,
• \( V \) is the volume of the container.

Substitute the values into the equation to calculate \( P \,\) the pressure exerted by the \(\text{CO}_2\) gas. Accurate pressure calculation is essential in chemical reactions and engineering applications.

Dry Ice

Dry ice is the solid form of carbon dioxide \(\text{CO}_2\). It's called "dry" because it sublimates, which means it transitions directly from a solid to a gas without becoming a liquid. This unique property makes dry ice incredibly interesting and useful in various applications.
When dry ice sublimates at temperatures above \-78.5^{\circ}\text{C}\, it turns into \(\text{CO}_2\) gas. This process is what happens in our container as the temperature rises from \-100^{\circ}\text{C}\ to room temperature. The gas then exerts pressure within the closed container. Scientists and industries use dry ice for cooling, freezing, and creating special effects due to its properties.
Understanding the behavior of dry ice helps us appreciate its practical applications in real-world situations. Moreover, it provides valuable insights into phase transitions and gas behavior under different conditions.