Question

Helium is collected over water at \(25^{\circ} \mathrm{C}\) and 1.00 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

A total volume of 3.69 liters of gas must be collected to obtain 0.586 grams of helium at 25℃ and 1.00 atm total pressure.

Step-by-step solution

Find moles of helium gas

We know the mass of helium is 0.586 grams. To find the moles of helium gas, we'll use the molar mass of helium, which is approximately 4.00 g/mol. Moles of helium = (mass of helium) / (molar mass of helium) Moles of helium = \( \frac{0.586 \ \mathrm{g}}{4.00 \ \mathrm{g/mol}} \) Moles of helium = 0.1465 mol

Calculate the partial pressure of helium gas

Now we need to find the partial pressure of helium in the gas mixture. We subtract the vapor pressure of water (given as 23.8 torr) from the total pressure (given as 1.00 atm) to get the partial pressure of helium. First, we need to convert the total pressure from atm to torr: Total pressure in torr = 1.00 atm × (760 torr / 1 atm) = 760 torr Now we can find the partial pressure of helium gas: Partial pressure of helium = total pressure - vapor pressure of water Partial pressure of helium = 760 torr - 23.8 torr = 736.2 torr

Determine the volume of helium gas

We can now use the ideal gas law formula to determine the volume of helium gas collected: PV = nRT Where, P = pressure (in atm) V = volume (in L) n = moles of helium gas R = gas constant (0.0821 L atm/mol K) T = temperature (in Kelvin) First, we need to convert the temperature into Kelvin and the partial pressure of helium into atm: Temperature in Kelvin = 25℃ + 273.15 = 298.15 K Partial pressure of helium in atm = 736.2 torr × (1 atm / 760 torr) = 0.9684 atm Now we can determine the volume: V = \( \frac{nRT}{P} \) V = \( \frac{(0.1465 \ \mathrm{mol})(0.0821 \ \mathrm{L \ atm/mol \ K})(298.15 \ \mathrm{K})}{0.9684 \ \mathrm{atm}} \) V = 3.69 L So, a total volume of 3.69 liters of gas must be collected to obtain 0.586 grams of helium at 25℃ and 1.00 atm total pressure.

Key concepts

Molar Mass

The concept of molar mass is essential for converting between the mass of a substance and the amount in moles. Molar mass is the mass of one mole of a given substance or chemical element and is usually expressed in grams per mole (g/mol). For helium, which is a noble gas, its molar mass is approximately 4.00 g/mol. This means that one mole of helium atoms weighs 4.00 grams.

• To find the number of moles () of helium from a given mass, we use the relation: \( n = \frac{\text{mass}}{\text{molar mass}} \)
• In the exercise, with a mass of 0.586 grams of helium and a molar mass of 4.00 g/mol, the number of moles calculates to 0.1465 mol.

Understanding molar mass allows us to convert measurable quantities in chemistry and is vital for employing formulas like the ideal gas law.

Vapor Pressure

Vapor pressure refers to the pressure exerted by a vapor in equilibrium with its liquid or solid form at a given temperature. In our scenario, the vapor pressure of water at 25°C is given as 23.8 torr.

• This pressure represents the tendency of water molecules to escape into the gaseous phase. When collecting gases over water, one must consider this vapor pressure.
• To find the pressure exerted by the dry gas (helium), subtract the vapor pressure of water from the total pressure.

For example, the partial pressure of helium in the mixture was calculated by subtracting the vapor pressure of water from the total pressure, resulting in 736.2 torr. This step is crucial when calculating the behavior of gases collected over liquids.

Partial Pressure

Partial pressure is the pressure exerted by a single type of gas in a mixture of gases. Each gas in a mixture contributes to the total pressure. In this exercise, helium's partial pressure needed to be isolated from the mixture over water.

• The total pressure is the sum of the partial pressures of all the gases present. In the situation, the total pressure was given as 1.00 atm.
• To obtain the partial pressure of helium, the vapor pressure of water (23.8 torr) was subtracted from the total pressure, calculated as 760 torr in equivalent terms.
• This means that helium's partial pressure was 736.2 torr.

Partial pressure calculation is a necessary step in applying the ideal gas law to determine gas volumes correctly when mixtures are involved.

Temperature Conversion

Temperature conversion is a critical step when applying gas laws, as these equations require temperature values in Kelvin. The Kelvin scale is an absolute temperature scale used predominantly in scientific measurements.

• To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature.
• In this exercise, the temperature was given as 25℃, which converted to 298.15 K.

Conversions are necessary since the Kelvin scale is directly proportional to kinetic energy, which correlates with gas behavior changes. Always ensure to convert temperatures to Kelvin during gas law calculations to avoid errors.