Problem 83 - Chemistry An Atoms First Approach

Question

A piece of solid carbon dioxide, with a mass of \(7.8 \mathrm{g},\) is placed in a \(4.0\)-\(\mathrm{L}\) otherwise empty container at \(27^{\circ} \mathrm{C}\). What is the pressure in the container after all the carbon dioxide vaporizes? If \(7.8 \mathrm{g},\) solid carbon dioxide were placed in the same container but it already contained air at \(740\) torr, what would be the partial pressure of carbon dioxide and the total pressure in the container after the carbon dioxide vaporizes?

Step-by-Step Solution

Answer

The pressure in the container after all the carbon dioxide vaporizes is 3.44 atm (or 2614.4 torr), the partial pressure of carbon dioxide is 1874.4 torr, and the total pressure in the container when it already contained air at 740 torr is 2614.4 torr.

Step 1: 1. Convert temperature to Kelvin

To work with the Ideal Gas Law, we need to convert the given temperature from Celsius to Kelvin. We can do this using the formula: \(T(K)= T(°C) + 273.15\). \(T(K) = 27°C + 273.15 = 300.15 K\)

Step 2: 2. Convert mass of carbon dioxide to moles

We need to convert the mass of carbon dioxide (7.8 g) to moles using the molar mass of CO₂. The molar mass of CO₂ is approximately 44.01 g/mol. \(n = \frac{m}{M} = \frac{7.8 \, \text{g}}{44.01\, \text{g/mol}} = 0.177 \, \text{moles}\)

Step 3: 3. Calculate pressure before adding air

Now that we have all the necessary information, we can apply the Ideal Gas Law to find the pressure in the container after the carbon dioxide vaporizes before adding any air: \(P = \frac{nRT}{V} = \frac{0.177 \, \text{moles} \times 0.0821\, \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}} \times 300.15\, \text{K}}{4.0\, \text{L}} = 3.44 \, \mathrm{atm}\)

Step 4: 4. Convert pressure to torr

The asked pressure in the next step is given in torr, so we need to convert our pressure in atm to torr using the conversion factor: 1 atm = 760 torr. \(P_\text{before} = 3.44\, \text{atm} \times \frac{760 \, \text{torr}}{1 \, \text{atm}} = 2614.4 \, \text{torr}\)

Step 5: 5. Calculate partial pressure of carbon dioxide

The container already has a pressure due to air (740 torr). We can now find the partial pressure of carbon dioxide by subtracting this pressure from the total pressure calculated before. \(P_\text{CO₂} = P_\text{before} - P_\text{air} = 2614.4\, \text{torr} - 740\, \text{torr} = 1874.4\, \text{torr}\)

Step 6: 6. Calculate the total pressure

Finally, we will calculate the total pressure of the container after adding the initial air pressure (740 torr) to the partial pressure of carbon dioxide. \(P_\text{total} = P_\text{CO₂} + P_\text{air} = 1874.4\, \text{torr} + 740\, \text{torr} = 2614.4\, \text{torr}\) So, the pressure in the container after all the carbon dioxide vaporizes is 3.44 atm (or 2614.4 torr), the partial pressure of carbon dioxide is 1874.4 torr, and the total pressure in the container when it already contained air at 740 torr is 2614.4 torr.

Key Concepts

Carbon Dioxide

Carbon dioxide, often represented as CO₂, is a naturally occurring gas that is pivotal in various chemical and physical processes. It plays a significant role not only in nature but also in industrial applications and chemistry classrooms.
When dealing with CO₂ in the context of gas laws, it's important to remember its physical states. Solid carbon dioxide is commonly known as "dry ice," and sublimates directly from solid to gas, which makes it particularly useful in experiments that involve gas at room temperature settings.
In scenarios like this, where dry ice is placed in a sealed container, it will eventually vaporize and exert pressure within that container. This is described by the Ideal Gas Law:

• P is the pressure of the gas,
• V is the volume of the container,
• n is the number of moles of the gas,
• R is the Ideal Gas Constant (0.0821 L•atm/mol•K),
• T is the temperature in Kelvin.

Understanding how carbon dioxide transitions from solid to gas and calculates pressure involves a careful analysis of these variables and their interplay.

Pressure Calculation

Calculating gas pressure within a container is one of the crucial applications of the Ideal Gas Law, especially when discussing gases like carbon dioxide. To determine the pressure created by carbon dioxide after it vaporizes, we leverage the equation \(P = \frac{nRT}{V}\).
When carbon dioxide vaporizes, its gaseous form exerts force on the walls of the container. This force per unit area is what we term as 'pressure.' To arrive at the correct pressure figure, follow these steps:

• Calculate moles of CO₂. We've converted 7.8 grams of CO₂ to 0.177 moles using its molar mass (44.01 g/mol).
• Ensure temperature conversion to Kelvin for accuracy. Here, 27°C was converted to 300.15 K.
• Apply the Ideal Gas Law with the values. For a 4.0L container at 300.15K, the resulting pressure is 3.44 atm.

In practice, converting this pressure from atmospheres to other units, like torr, is often required (1 atm = 760 torr), ensuring comprehensibility across different scientific contexts.

Moles Conversion

To grasp chemistry concepts like gas laws, understanding how to convert mass into moles is essential. The conversion process is based on the fundamental relationship between mass and the number of particles, quantified as "moles."
For carbon dioxide, the conversion process begins by knowing the molar mass of CO₂, which is approximately 44.01 g/mol, meaning each mole of CO₂ has a mass of 44.01 grams.
Here's a step-by-step moles conversion process:

• Identify the mass of the substance in grams (7.8 g for CO₂ is given).
• Use the molar mass to convert grams to moles: \( n = \frac{7.8 \, \text{g}}{44.01 \, \text{g/mol}} = 0.177 \, \text{moles}\).

Understanding this conversion is foundational, as it informs the amount of substance available, which directly affects other calculations, like pressure using the Ideal Gas Law. This systematic conversion is repeated anytime mass needs to be evaluated in chemical equations or reactors.