Question

Calculate the total pressure (in atm) of a mixture of \(0.0200 \mathrm{~mol}\) of helium, He, and \(0.0100 \mathrm{~mol}\) of hydrogen, \(\mathrm{H}_{2}\), in a 2.50-L flask at \(10^{\circ} \mathrm{C}\). Assume ideal gas behavior. 5.82 Calculate the total pressure (in atm) of a mixture of \(0.0300 \mathrm{~mol}\) of helium, He, and \(0.0400 \mathrm{~mol}\) of oxygen, \(\mathrm{O}_{2},\) in a 4.00-L flask at \(20^{\circ} \mathrm{C}\). Assume ideal gas behavior.

Short answer

The total pressure of the gas mixture is 0.421 atm.

Step-by-step solution

Understanding the Ideal Gas Law

The Ideal Gas Law is given by \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant \( 0.0821 \frac{L \, atm}{mol \, K} \), and \( T \) is the temperature in Kelvin.

Convert Temperature to Kelvin

The temperature in Celsius must be converted to Kelvin for the ideal gas law. The conversion formula is \( T(K) = T(°C) + 273.15 \).\For the given problem, \( 20°C + 273.15 = 293.15 \, K \).

Calculate Pressure of Helium

Using the ideal gas law for helium: \[ P_{He} = \frac{n_{He}RT}{V} \] where \( n_{He} = 0.0300 \, mol \), \( V = 4.00 \, L \), and \( R \) and \( T \) are known. \[ P_{He} = \frac{0.0300 \times 0.0821 \times 293.15}{4.00} \approx 0.180 \, atm \].

Calculate Pressure of Oxygen

Using the ideal gas law for oxygen:\[ P_{O_2} = \frac{n_{O_2}RT}{V} \] where \( n_{O_2} = 0.0400 \, mol \). \[ P_{O_2} = \frac{0.0400 \times 0.0821 \times 293.15}{4.00} \approx 0.241 \, atm \].

Calculate Total Pressure

According to Dalton's Law of Partial Pressures, the total pressure is the sum of the partial pressures of the gases:\[ P_{total} = P_{He} + P_{O_2} \] \[ P_{total} = 0.180 \, atm + 0.241 \, atm = 0.421 \, atm \].

Key concepts

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a fundamental principle in chemistry that applies to gas mixtures. It states that in a mixture of non-reacting gases, the total pressure exerted by the mixture is the sum of the pressures that each gas would exert if it occupied the entire volume alone. This means you can calculate the total pressure by adding up the partial pressures of each gas in the mixture. For instance, when considering a flask containing helium and oxygen, the total pressure is the sum of the pressure exerted by the helium and the pressure exerted by the oxygen. Using this principle simplifies the calculation of the total pressure in gas mixtures and is especially useful when dealing with ideal gases, which conform perfectly to the Ideal Gas Law. Understanding Dalton's Law helps in predicting how gases will behave when mixed together in a confined space, an essential knowledge in fields like chemistry and meteorology.

Temperature Conversion

Temperature conversion is a basic yet crucial step when using the Ideal Gas Law. This is because the Ideal Gas Law requires temperature to be in Kelvin, the absolute temperature scale commonly used in scientific equations. The formula to convert Celsius to Kelvin is straightforward: add 273.15 to the Celsius temperature. For example, if you have a temperature of 20°C, you would convert it to 293.15 K using the formula. This conversion is important because Kelvin is a scale that starts at absolute zero, the theoretical point where all molecular motion stops. Using Celsius without conversion would give incorrect results, as the Ideal Gas Law's mathematical relationship relies on absolute temperature differences. Thus, always remember to convert temperature to Kelvin before substituting values into the Ideal Gas Equation.

Partial Pressure Calculation

Partial pressure calculation is an essential step when working with gas mixtures. According to the Ideal Gas Law, the partial pressure of a gas is directly proportional to the number of moles of the gas in the mixture. To calculate the partial pressure of a gas (e.g., helium or oxygen) in a container, you need to know:

• The number of moles of the gas (\( n \)).
• The volume of the container (\( V \)).
• The temperature in Kelvin (\( T \)).
• The ideal gas constant (\( R = 0.0821 \frac{L \, atm}{mol \, K} \)).

By substituting these values into the formula \( P = \frac{nRT}{V} \), you can find the partial pressure of each gas. For example, if you calculate helium's partial pressure using its specific moles, volume, and temperature, and repeat the same for oxygen, you'll find their respective pressures. Understanding how to calculate partial pressures enables you to find the total pressure in a gas mixture using Dalton’s Law, applying the computed partial pressures to derive the total.