Question

Suppose two \(200.0-\mathrm{L}\) tanks are to be filled separately with the gases helium and hydrogen. What mass of each gas is needed to produce a pressure of 2.70 atm in its respective tank at \(24^{\circ} \mathrm{C} ?\)

Short answer

To produce a pressure of 2.70 atm in their respective tanks at \(24^{\circ} \mathrm{C}\), approximately \(88.00 \: \text{g}\) of helium and \(44.44 \: \text{g}\) of hydrogen are needed.

Step-by-step solution

Convert the temperature from Celsius to Kelvin

Before we begin, we need to convert the temperature from Celsius to Kelvin since the Ideal Gas Law requires the temperature to be in Kelvin. The conversion formula is: Temperature (K) = Temperature (°C) + 273.15 For our problem, our temperature in Kelvin will be: Temperature (K) = 24°C + 273.15 = 297.15 K

Find the number of moles of each gas using the Ideal Gas Law

Using the Ideal Gas Law formula (PV = nRT), we can find the number of moles (n) of each gas by rearranging the formula: n = PV / RT For both gases, we are given: - Pressure (P) = 2.70 atm - Volume (V) = 200.0 L - Temperature (T) = 297.15 K - Ideal Gas Constant (R) = 0.0821 L atm/mol K Now, we can plug these values into the Ideal Gas Law formula: n(He) = n(H₂) = (2.70 atm) × (200.0 L) / (0.0821 L atm/mol K × 297.15 K) n(He) = n(H₂) ≈ 22.00 mol

Convert the moles of each gas to mass

The mass of a gas can be found by multiplying the number of moles of the gas by its molar mass. The molar mass of helium (He) is 4.00 g/mol, while the molar mass of hydrogen (H₂) is 2.02 g/mol. Mass (He) = Moles (He) × Molar Mass (He) Mass (He) ≈ 22.00 mol × 4.00 g/mol = 88.00 g Mass (H₂) = Moles (H₂) × Molar Mass (H₂) Mass (H₂) ≈ 22.00 mol × 2.02 g/mol ≈ 44.44 g Thus, the mass needed for each gas is approximately 88.00 g of helium and 44.44 g of hydrogen to produce a pressure of 2.70 atm in their respective tanks at 24°C.

Key concepts

Gas Laws

The gas laws are essential principles in chemistry that describe the behavior of gases. These laws explain how pressure (P), volume (V), and temperature (T) are interrelated in a gas sample. The Ideal Gas Law is a key equation: \[ PV = nRT \] where P is the pressure in atmospheres, V is the volume in liters, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. These laws assume that gases are made of tiny particles moving in straight lines with elastic collisions.This means they can change direction or speed if they hit something else, but don't lose energy in the process.This helps predict how gases will behave when conditions like volume or temperature change.Understanding these principles is crucial for disciplines like physics and engineering, as well as various real-life applications such as predicting weather patterns and designing respiratory equipment.

Temperature Conversion

Temperature conversion is a fundamental step when working with the Ideal Gas Law, as it requires temperatures to be in Kelvin.Celsius is commonly used, but to convert it to Kelvin, add 273.15 to the Celsius temperature: \[ \text{Temperature (K) = Temperature (°C) + 273.15} \]For example, the conversion of 24^{\circ} C = 24 + 273.15 = 297.15 K.Kelvin is the absolute temperature scale, starting from absolute zero, the point where atomic motion stops.It is essential to use Kelvin in calculations to ensure consistency with the Ideal Gas Law.By using Kelvin, we ensure that all temperature-related calculations are accurate and compatible within the physical laws governing gas behavior.This practice is important not only for academic exercises but also in scientific research where precision is critical.

Molar Mass Calculation

Calculating the molar mass of a gas is critical for converting moles to mass.Moles tell us how many molecules of gas there are, but practical applications require a mass measurement, usually in grams.You convert moles to mass using the molar mass (M), the weight of one mole of a substance.For helium (\text{He}), the molar mass is 4.00 g/mol.For hydrogen (\text{H}_2), it is 2.02 g/mol.The formula to convert moles to mass is: \[\text{Mass} = \text{Moles} \times \text{Molar Mass} \]By applying this for 22.00 moles of each gas:

• \text{Mass (He)} = 22.00 \text{ mol} \times 4.00 \text{ g/mol} = 88.00 \text{ g}
• \text{Mass (H}_2) = 22.00 \text{ mol} \times 2.02 \text{ g/mol} \approx 44.44 \text{ g}

This calculation is vital in chemical reactions and many industrial processes, where precise measurements of materials are required.

Pressure in Gases

Pressure in gases is a measure of how often and how forcefully gas molecules collide with the walls of their container. In the context of the Ideal Gas Law, pressure is measured in atmospheres (atm). For the given exercise, the pressure is specified as 2.70 atm. When you're working with gases, it's important to understand how changes in volume and temperature affect pressure. According to Boyle’s Law, pressure is inversely proportional to volume, meaning if one increases, the other decreases at constant temperature. Charles's Law states that pressure is directly proportional to temperature at constant volume. These relationships help predict changes in gas behavior under different conditions and are applied in fields such as respiratory therapy and meteorology. Understanding these concepts ensures that we can control and utilize gases effectively in both everyday applications and specialized scientific experiments.