Question

Helium is collected over water at \(25^{\circ} \mathrm{C}\) and \(1.00 \mathrm{~atm}\) total pressure. What total volume of gas must be collected to obtain \(0.586 \mathrm{~g}\) helium? (At \(25^{\circ} \mathrm{C}\) the vapor pressure of water is \(23.8\) torr.)

Short answer

The total volume of gas that needs to be collected to obtain 0.586 g of helium at \( 25^{\circ} \mathrm{C} \) and \( 1.00 \mathrm{~atm} \) total pressure is approximately \( 3.74 \mathrm{~L} \).

Step-by-step solution

Convert temperature to Kelvin

We need to convert the given temperature from Celsius to Kelvin, using the following formula: \( T(K) = T(°C) + 273.15 \) \( T(K) = 25°C + 273.15 = 298.15 K \)

Convert pressure units

We need to convert the total pressure and vapor pressure to the same units. The total pressure is given in atm, while the vapor pressure of water is given in torr. We'll convert both pressures to atm using the conversion factor 1 atm = 760 torr. Total pressure in atm: \( 1.00~atm \) Vapor pressure of water in atm: \( \frac{23.8~torr}{760~torr/atm} = 0.0313~atm \)

Calculate the partial pressure of helium

Using Dalton's law of partial pressures, we can calculate the partial pressure of helium in the mixture. Partial pressure of helium: \( P_{He} = P_{total} - P_{water} \) \( P_{He} = 1.00~atm - 0.0313~atm = 0.9687~atm \)

Find the number of moles of helium

Using the molar mass of helium (He = 4.00 g/mol), we can find the number of moles present in 0.586 g of helium: \( \text{moles of He} = \frac{\text{mass of He}}{\text{molar mass of He}} \) \( n_{He} = \frac{0.586~g}{4.00~g/mol} = 0.1465~mol \)

Use the Ideal Gas Law to solve for volume

The Ideal Gas Law is given by: \( PV = nRT \) We need to solve for volume, so the formula becomes: \( V = \frac{nRT}{P} \) Using the values calculated for the moles of helium (0.1465 mol), temperature (298.15 K), partial pressure of helium (0.9687 atm), and the universal gas constant (R = 0.0821 L atm/mol K), we can find the volume. \( V = \frac{(0.1465~mol)(0.0821~L~atm/mol~K)(298.15~K)}{0.9687~atm} \) \( V = 3.7358~L \) So, the total volume of gas that needs to be collected to obtain 0.586 g of helium is approximately 3.74 L.

Key concepts

Dalton's Law of Partial Pressures

When dealing with a mixture of gases, such as helium collected over water, Dalton's Law of Partial Pressures becomes highly relevant. This law states that the total pressure exerted by a mixture of non-reactive gases is equal to the sum of the partial pressures of individual gases.

For instance, in the presence of water vapor, the total pressure in a container is the sum of the pressure of helium and the pressure exerted by water vapor. To find the pressure of just the helium gas, you would subtract the vapor pressure of water from the total pressure of the gas mixture.

This step is crucial in solving for other gas properties, as demonstrated in the exercise where the partial pressure of helium was needed to calculate the volume of gas using the Ideal Gas Law. Accurate determination of partial pressures ensures that the calculations for the volume of helium are correct.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that provides a relationship between the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas. Expressed as PV = nRT, where R is the universal gas constant, this law allows us to calculate any one of these variables if the other three are known.

In our exercise, the Ideal Gas Law enables us to determine the volume of helium based on its pressure, the number of moles, and the temperature. By manipulating the equation, V = nRT/P, and inserting the known values, we can solve for V, the volume, which is crucial for practical lab applications such as collecting a specific amount of gas.

Molar Mass of Helium

Molar mass is the mass of one mole of a substance and is expressed in grams per mole (g/mol). For helium, a noble gas with the atomic number 2, the molar mass is approximately 4.00 g/mol. This value is essential when you are working with the physical amount of gas because it allows you to convert between grams and moles—a step that is necessary in the use of the Ideal Gas Law.

Knowing that the molar mass of helium is 4.00 g/mol, as used in our exercise, provides an understanding of how much actual helium (in grams) would correspond to a theoretical number of particles (moles). It also underscores the importance of standardizing units when dealing with chemical calculations, as discrepancies in units can lead to incorrect results.

Converting Temperature to Kelvin

Temperature is a critical variable in gas laws, and it must be measured in Kelvin for these equations to work correctly. Kelvin is the absolute temperature scale, which means it starts at absolute zero, the theoretical point where all molecular motion stops.

To convert Celsius to Kelvin, a simple addition of 273.15 to the Celsius temperature is all that's needed. This exercise included the conversion from 25°C to Kelvin, resulting in a temperature of 298.15 K. The use of Kelvin addresses the proportionality between volume and temperature of a gas, which is vital to accurate gas law calculations as seen in the Ideal Gas Law.