Question

What will be the pressure of the gas mixture of \(3.2 \mathrm{~g}\) methane and \(4.4 \mathrm{~g}\) carbon dioxide contained in a \(9 \mathrm{dm}^{3}\) flask at \(27^{\circ} \mathrm{C}\) ? (a) \(0.82\) atm (b) \(8.314 \times 10^{4} \mathrm{~atm}\) (c) \(1 \mathrm{~atm}\) (d) \(1.8 \mathrm{~atm}\)

Short answer

0.82 atm

Step-by-step solution

- Calculate the number of moles of methane

Use the molar mass of methane (CH4), which is approximately 16 g/mol, to calculate the number of moles (n) of CH4. The formula to find the number of moles is given by: n = mass (m) / Molar mass (M). In this case, n for CH4 = (mass of CH4) / (Molar mass of CH4) = 3.2 g / 16 g/mol.

- Calculate the number of moles of carbon dioxide

Repeat the process used in Step 1 for carbon dioxide (CO2), using its molar mass, which is approximately 44 g/mol. n for CO2 = (mass of CO2) / (Molar mass of CO2) = 4.4 g / 44 g/mol.

- Calculate the total number of moles in the gas mixture

Add the moles of CH4 obtained from Step 1 to the moles of CO2 obtained from Step 2 to get the total moles of gases in the mixture. Total moles = moles of CH4 + moles of CO2.

- Convert Celsius to Kelvin

Temperature must be in Kelvin for gas law calculations. Convert the temperature from Celsius to Kelvin by adding 273.15. T(K) = 27°C + 273.15.

- Use Ideal Gas Law to find the pressure

Apply the ideal gas law, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (0.0821 atm L/mol K), and T is the temperature in Kelvin. Solve the equation for P: P = (nRT)/V.

- Calculate the pressure of the gas mixture

Plug in the values for total moles (from Step 3), R (0.0821 atm L/mol K), T (from Step 4, in Kelvin), and V (9 dm³ which is the same as 9 L) into the equation obtained in Step 5 to calculate the pressure of the gas mixture.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a foundational principle in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as the formula: \( PV = nRT \), where

• \( P \) stands for pressure of the gas,
• \( V \) is the volume occupied by the gas,
• \( n \) is the number of moles,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature in Kelvin.

The law assumes that the gas being measured behaves ideally, meaning the gas particles are in random motion and do not interact with each other.

To apply this relationship to real-world problems, like calculating the pressure of a gas mixture in a flask, it's important to ensure that all the values are in the correct units. The ideal gas constant \( R \) typically has a value of \( 0.0821 \) atm L/mol K when working with the pressure in atmospheres, volume in liters, and temperature in Kelvin. If the temperature is given in degrees Celsius, conversion to Kelvin is mandatory before applying the Ideal Gas Law, which will be discussed in another section.

Molar mass

Molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It is a physical property that is useful for converting between the mass of a substance and the amount in moles, which is essential in chemical calculations and reactions.

For the substances in the given problem, methane (CH4) has a molar mass of approximately 16 g/mol, while carbon dioxide (CO2) has a molar mass of 44 g/mol. These values are determined by the sum of the atomic masses of the elements that make up each molecule. Knowing the molar mass allows us to compute how many moles are in a given mass of a substance using the formula: \( n = \frac{mass}{Molar~mass} \).
In this case, if you have 3.2 g of methane, you would divide this by the molar mass of methane to find the number of moles. This concept is vital for the problem at hand, as the next step involves calculating the number of moles of each gas before using the Ideal Gas Law.

Moles calculation

Calculating moles is a core skill in chemistry, used to quantify the amount of a substance. Moles link the microscopic scale of atoms and molecules to the macroscopic scale that we can measure. One mole of any substance contains Avogadro's number of particles, which is \( 6.022 \times 10^{23} \) particles/mol.

For our gas mixture problem, we begin by finding the number of moles of methane and carbon dioxide separately using their respective masses and molar masses. For methane:\[ n_{CH4} = \frac{mass~of~CH4}{Molar~mass~of~CH4} = \frac{3.2~g}{16~g/mol} \]
Likewise for carbon dioxide, we apply the same calculation with its molar mass:\[ n_{CO2} = \frac{mass~of~CO2}{Molar~mass~of~CO2} = \frac{4.4~g}{44~g/mol} \]
Once the number of moles for both gases is determined, they are simply added together to obtain the total number of moles in the gas mixture. This sum is crucial for the next steps, where we will apply the Ideal Gas Law to find the pressure of the mixed gases in the flask.

Temperature conversion

Temperature conversion is an essential process in gas law problems since the Ideal Gas Law requires temperature to be in Kelvin. To convert from Celsius to Kelvin, we add 273.15 to the Celsius temperature. The conversion formula is:\[ T(K) = T(°C) + 273.15 \]

For example, if the problem states that the gas mixture is at \(27°C\), we convert this to Kelvin as follows:\[ T(K) = 27°C + 273.15 = 300.15 K \]
Using Kelvin is important because it provides an absolute temperature scale where \(0 K\) represents absolute zero, the point at which particles have minimal thermal motion. This absolute scale ensures that all temperature-related calculations reflect the true kinetic energy of the particles in a substance.