Question

A flask is first evacuated so that it contains no gas at all. Then, \(2.2 \mathrm{g}\) of \(\mathrm{CO}_{2}\) is introduced into the flask. On warming to \(22^{\circ} \mathrm{C}\), the gas exerts a pressure of \(318 \mathrm{mm}\) Hg. What is the volume of the flask?

Short answer

The volume of the flask is approximately 1.22 liters.

Step-by-step solution

Understand the Problem

We need to find the volume of a flask containing 2.2 g of \( \mathrm{CO}_2 \). The gas exerts a pressure of 318 mmHg at a temperature of 22°C.

Convert Units Appropriately

Convert the pressure from mmHg to atm for consistency in calculation. \[ 1 \text{ atm} = 760 \text{ mmHg} \] thus, \( 318 \text{ mmHg} = \frac{318}{760} \text{ atm} \).

Calculate Moles of CO2

Determine the moles of the \( \mathrm{CO}_2 \). The molar mass of \( \mathrm{CO}_2 \) is 44.01 g/mol. Use \( n = \frac{mass}{molar mass} \) to get \( n = \frac{2.2}{44.01} \text{ mol} \).

Convert Temperature to Kelvin

Convert temperature from Celsius to Kelvin using the formula: \( T(K) = 22 + 273.15 \). This gives \( T = 295.15 \text{ K} \).

Apply the Ideal Gas Law

Use the Ideal Gas Law formula \( PV = nRT \) where \( R = 0.0821 \text{ L atm/mol K} \). Substitute the known values: \( P = \frac{318}{760} \text{ atm} \), \( n = \frac{2.2}{44.01} \text{ mol} \), \( T = 295.15 \text{ K} \). Solve for \( V \).

Calculate Volume

Rearrange the Ideal Gas Law to find \( V \): \[ V = \frac{nRT}{P} = \frac{\left(\frac{2.2}{44.01}\right) \times 0.0821 \times 295.15}{\left(\frac{318}{760}\right)} \] and compute \( V \) to find the volume of the flask.

Key concepts

Mole Calculation

To calculate the moles of a gas, we use the relationship between mass, molar mass, and number of moles. In our exercise, we want to find the number of moles for carbon dioxide (CO\(_2\)) present, since knowing the amount of substance in moles is essential for using the Ideal Gas Law. Here are the simple steps:

• Identify the molar mass of CO\(_2\). Carbon dioxide, made up of one carbon and two oxygen atoms, has a molar mass of 44.01 g/mol.
• Use the formula to solve for moles: \( n = \frac{m}{M} \), where \( n \) is the number of moles, \( m \) is the mass of the gas in grams, and \( M \) is the molar mass.

In our example, the mass of the gas is 2.2 g, so plug in the values: \( n = \frac{2.2}{44.01} \). This calculation gives you the number of moles of CO\(_2\) available for further computations.

Pressure Conversion

In many chemistry problems, converting pressure into a common unit is crucial for applying the Ideal Gas Law. Pressure is often measured in different units like mmHg or atm. Here, we'll convert pressure from mmHg to atm, to ensure we use consistent units throughout.

The exercise gives us a pressure of 318 mmHg. Here’s how to proceed to convert it into atmospheres (atm):

• Remember the conversion factor: 1 atm = 760 mmHg.
• Calculate the pressure in atm by dividing: \( 318 \text{ mmHg} \times \frac{1 \text{ atm}}{760 \text{ mmHg}} \).

This calculation helps to express 318 mmHg as approximately 0.418 atm, a necessary conversion to use in the Ideal Gas Law formula.

Temperature Conversion

When dealing with gas laws, converting temperature from Celsius to Kelvin is a straightforward but vital step. The Kelvin scale is used in scientific calculations to avoid negative temperatures, as Kelvin starts from absolute zero.

To convert from Celsius to Kelvin, you simply add 273.15 to the Celsius temperature:

• Start with the Celsius temperature given in the problem, which is 22°C.
• Apply the conversion: \( T(\text{K}) = T(\degree \text{C}) + 273.15 \).

In this exercise, you will find \( T = 22 + 273.15 = 295.15 \; \text{K} \). Converting to Kelvin ensures that the temperature is always positive and compatible with the gas law formulas.

Volume Calculation

The Ideal Gas Law, expressed as \( PV = nRT \), is the key to calculating volume in this exercise. This formula relates the pressure \( P \), volume \( V \), and temperature \( T \) of a gas, along with the number of moles \( n \) and the ideal gas constant \( R \). Let's see how you can determine the volume:

• First, rearrange the Ideal Gas Law to solve for volume: \( V = \frac{nRT}{P} \).
• Plug in the values calculated earlier: \( n = \frac{2.2}{44.01} \) moles, \( T = 295.15 \; \text{K} \), and \( P = 0.418 \; \text{atm} \). Don't forget \( R = 0.0821 \; \text{L atm/mol K} \), the constant.

By substituting all known values into the equation, you calculate the volume \( V \) of the flask. This step not only yields the volume but also demonstrates how integrated each step is, from moles to pressure, temperature, and finally to volume!