Question

Suppose two 200.0 - \(L\) 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°C, approximately 87.954 g of helium and 43.977 g of hydrogen are needed.

Step-by-step solution

Convert temperature to Kelvin

Convert the given temperature from Celsius to Kelvin: T = 24°C + 273 = 297 K

Arrange the ideal gas law equation to solve for n

We know that PV = nRT. We want to find the value of n for each gas, so we can rearrange the equation as follows: n = PV / RT Now we will find the number of moles for each gas separately. For helium:

Find the number of moles (n) of helium

Using the rearranged ideal gas law equation, plug in the values for helium: n(He) = (2.70 atm) * (200.0 L) / (0.0821 L.atm / (K.mol) * 297 K) = 21.9885 moles For hydrogen:

Find the number of moles (n) of hydrogen

Using the same rearranged ideal gas law equation, plug in the values for hydrogen: n(H2) = (2.70 atm) * (200.0 L) / (0.0821 L.atm / (K.mol) * 297 K) = 21.9885 moles Now, we have the number of moles for both helium and hydrogen. Next, we will find the mass of each gas using their molar masses. Molar masses: - Helium (He) = 4 g/mol - Hydrogen (H2) = 2 g/mol (since it's diatomic)

Calculate the mass of helium

Using the number of moles and molar mass, find the mass of helium: Mass(He) = n(He) * Molar mass(He) = 21.9885 moles * 4 g/mol = 87.954 g

Calculate the mass of hydrogen

Using the number of moles and molar mass, find the mass of hydrogen: Mass(H2) = n(H2) * Molar mass(H2) = 21.9885 moles * 2 g/mol = 43.977 g To produce a pressure of 2.70 atm in their respective tanks at 24°C: - The mass of helium required is approximately 87.954 g. - The mass of hydrogen required is approximately 43.977 g.

Key concepts

Gas Pressure and Volume

Understanding the relationship between gas pressure and volume is essential for working with gases in a closed system. Let's explore this relationship within the context of the Ideal Gas Law, which provides a clear equation involving pressure (P), volume (V), moles (n), temperature (T), and the ideal gas constant (R).

The law is expressed as PV = nRT. Through this equation, we see that at a constant temperature, if the volume of the gas increases, the pressure decreases, and vice versa, which is known as Boyle's Law. When solving our example problem where we are given the pressure, volume, and temperature, and need to find the mass, we first needed to determine the number of moles of gas using the Ideal Gas Law. In our case, both gases were set to have the same pressure and volume in separate tanks, which led us to the same number of moles for each, demonstrating this aspect of the gas law in a practical sense.

Role of Pressure and Volume in Calculations

When using the ideal gas law, always remember you are dealing with a fixed amount of gas where other variables might change. For our calculations, pressure was given in atmospheres (atm) and volume was in liters (L), ensuring these units were consistent with the value of R (0.0821 L.atm / (K.mol)). After determining the moles, we could calculate the mass of the gases, which connected us to another core concept: moles and molar mass.

Moles and Molar Mass

In our exercise, moles and molar mass play a crucial role in transitioning from the conceptual gas parameters to tangible quantities. The number of moles (n) expresses the quantity of a substance, and molar mass (atomic or molecular weight) represents the mass of one mole of that substance.

To determine the mass of gas required for our tanks, once the moles were calculated using the ideal gas law, we multiplied the number of moles by the molar mass of each gas. The molar mass is unique to each substance; for helium, it's 4 g/mol, and for hydrogen, as a diatomic molecule (H2), it's 2 g/mol. These molar masses are constants and taken from the periodic table.

Finding Mass from Moles

In practice, the actual mass of gas you would need, in grams, is the product of the number of moles and the molar mass. As seen in the steps for the exercise, multiplying the number of moles of helium by the molar mass of helium gives the required mass of helium gas, and the same process is used for hydrogen. This step is an application of the concept of molar mass which bridges the gap between the microscopic scale of atoms and molecules and the macroscopic world we can measure.

Converting Temperature to Kelvin

One key aspect of working with the Ideal Gas Law is using the correct temperature scale. All gas law calculations require temperature to be in Kelvin rather than Celsius or Fahrenheit.

The conversion is straightforward: To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. This shift is because 0 Kelvin is absolute zero, the point at which no more thermal energy can be removed from a substance, and the Kelvin scale does not use degrees, merely 'Kelvin' (although temperatures on this scale are often informally referred to as 'degrees Kelvin').

Significance of Kelvin in Calculations

The importance of using Kelvin in gas law calculations lies in the nature of temperature as a measure of kinetic energy within a substance. The Kelvin scale provides a true zero point, allowing for proportional changes in gas properties to be represented more accurately. In our exercise, converting the given Celsius temperature (24°C) to Kelvin was the first critical step. Without it, the subsequent calculations using the Ideal Gas Law would not have provided the correct results because the values for the gas constant (R) rely on the use of the Kelvin scale for temperature.