Question

A \(2.50-\mathrm{L}\) flask contains 0.60 \(\mathrm{g} \mathrm{O}_{2}\) at a temperature of \(22^{\circ} \mathrm{C}\) . What is the pressure (in atm) inside the flask?

Short answer

The pressure inside the flask is approximately \(0.184\,\text{atm}\).

Step-by-step solution

1. Convert the temperature from Celsius to Kelvin

To convert the temperature from Celsius to Kelvin: \(T_K = T_C + 273.15\) In this case, the transformation will be: \(22^{\circ}\mathrm{C} + 273.15\) K. So, \(T_K = 295.15\,\text{K}\).

2. Convert the given mass of gas to moles

To find the number of moles, we'll use the molecular weight of oxygen, O2 (32 g/mol): \(n = \frac{\text{mass}}{\text{molecular weight}}\) \(n = \frac{0.60\,\text{g}}{32\,\frac{\text{g}}{\text{mol}}}\) \(n \approx 0.01875\,\text{mol}\)

3. Calculate the pressure using the Ideal Gas Law

Now that we have the number of moles of gas (\(n \approx 0.01875\,\text{mol}\)), the temperature in Kelvin (\(T_K = 295.15\,\text{K}\)), and the flask volume (\(V = 2.50\,\text{L}\)), we can calculate the pressure using the Ideal Gas Law formula: \(P=\frac{nRT}{V}\) \(P=\frac{(0.01875\,\text{mol})(0.0821\,\frac{\text{L}\cdot\text{atm}}{\text{K}\cdot\text{mol}})(295.15\,\text{K})}{2.50\,\text{L}}\) \(P \approx 0.184\,\text{atm}\) So, the pressure inside the flask is approximately \(0.184\,\text{atm}\).

Key concepts

Pressure Calculation

Calculating the pressure of a gas requires understanding the Ideal Gas Law. This law is expressed with the formula \( P = \frac{nRT}{V}\), where \( P \) represents pressure, \( n \) stands for the number of moles, \( R \) is the gas constant (0.0821 \(\frac{L\cdot atm}{K\cdot mol}\)), \( T \) is temperature in Kelvin, and \( V \) is the volume in liters.

• Plug into the formula all known quantities.
• Ensure that temperature is in Kelvin and volume is in liters for consistency with \( R \).
• Calculate \( P \) by substituting the values for \( n \), \( R \), \( T \), and \( V \).

In our exercise, these values are: \( n \approx 0.01875\,\text{mol} \), \( R \approx 0.0821\,\frac{L\cdot atm}{K\cdot mol} \), \( T \approx 295.15\,\text{K} \), and \( V = 2.50\,\text{L} \). Calculating for \( P \) gives approximately \( 0.184\,\text{atm} \). As we can see, following these steps leads us directly to the solution.

Molar Mass

Understanding molar mass is crucial for converting a substance's mass into moles. Molar mass is the mass of one mole of a substance and is usually expressed in grams per mole (\(g/mol\)). For molecular oxygen (\( O_2 \)), the molar mass is \( 32\, g/mol \).

• To find moles, use the formula: \( n = \frac{\text{mass}}{\text{molar mass}} \).
• In this exercise, the mass of \( O_2 \) is \( 0.60\, g \).
• Dividing the mass by \( 32\, g/mol \) gives us \( \approx 0.01875\, mol \).

Accurate mole calculation is essential, as it is directly used in the Ideal Gas Law to find pressure. Recognizing the role of molar mass helps bridge the gap between mass and the Ideal Gas components.

Temperature Conversion

Converting temperature from Celsius to Kelvin is a common need in chemistry, especially when using the Ideal Gas Law. Since the law requires Kelvin, we follow a simple formula: \( T_K = T_C + 273.15 \).

• Kelvin is an absolute temperature scale based on absolute zero.
• Celsius is more commonly used in everyday contexts but needs conversion for scientific formulas.

In the given exercise, the temperature of \(22^{\circ} \text{C} \) is converted to Kelvin by simply adding \( 273.15 \), resulting in \( 295.15\,\text{K} \). Accurately converting temperatures ensures correct calculations in pressure estimation using the Ideal Gas Law.

Oxygen Gas

Oxygen gas (\( O_2 \)) is a diatomic molecule carrying significant importance in chemistry. Its properties, such as molar mass and reactivity, often make it a focus of study in gas calculations like those in the Ideal Gas Law.

• As a diatomic molecule, each consists of two oxygen atoms.
• Molar mass of \( O_2 \) is \( 32\, g/mol \), arising from two atoms of oxygen, each approximately \( 16\, g/mol \).

In this particular exercise, understanding \( O_2 \) helps us not only calculate the moles from grams but also appreciate its behavior in different conditions, affecting pressure and volume.

Gas Volume

Gas volume relates directly to how much space gas occupies, here measured in liters. When calculating aspects like pressure using the Ideal Gas Law, knowing the gas volume is pivotal.

• Gas volume largely depends on container size and is usually constant for the exercise.
• Ideal Gas Law assumes volume does not change with the gas inside.

In our context, the volume is \(2.50\,\text{L}\). When we plug this into the Ideal Gas Law, it helps us derive the pressure value per the model the law provides. Without an accurate measure of volume, the pressure read would undoubtedly be inaccurate.