Question

Suppose two separate 100.0 - \(\mathrm{L}\) tanks are to be filled, one with helium and one with hydrogen. What mass of each gas is needed to produce a pressure of 2.70 atm in its respective tank at 24 C ?

Short answer

To fill one tank with helium and the other with hydrogen and create a pressure of 2.70 atm at 24 C, the mass of helium required is approximately 0.44 g and the mass of hydrogen required is approximately 0.22 g.

Step-by-step solution

Recall the Ideal Gas Law Formula

The Ideal Gas Law formula is given as: \(PV = nRT\), where P is the pressure, V is the volume, n is the amount of gas in moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Convert given temperature to Kelvin

The temperature given is in Celsius. We need to convert it to Kelvin using the formula: \(T_K = T_C + 273.15\). Here, \(T_C = 24\) C, therefore \(T_K = 24 + 273.15 = 297.15\) K.

Rearrange the Ideal Gas Law formula for moles of gas

We need to find the mass of each gas, so first, rearrange the Ideal Gas Law formula for moles (n): \[n = \frac{PV}{RT}\]

Calculate the moles of helium and hydrogen required

Given the pressure \(P = 2.70\) atm, volume \(V = 100.0 \: L = 0.1\: m^3\), gas constant \(R = 0.0821 \: \frac{L \: atm}{K \: mol}\), and the temperature in Kelvin \(T_K = 297.15\: K\), we can plug these values into the rearranged formula for the moles of each gas: For helium and hydrogen: \[n = \frac{2.70 \times 0.1}{0.0821 \times 297.15}\] Calculate n for both gases: \[n \approx 0.110 \: mol\]

Calculate the mass of helium and hydrogen required

To calculate the mass of helium and hydrogen, we need to use the molar masses for helium and hydrogen. The molar mass of helium is approximately 4 g/mol and that for hydrogen is approximately 2 g/mol. For helium: \[mass_{He} = n_{He} \times molar \:mass_{He} \approx 0.110 \times 4 \approx 0.44 \: g\] For hydrogen: \[mass_{H_2} = n_{H_2} \times molar \:mass_{H_2} \approx 0.110 \times 2 \approx 0.22 \: g\]

Conclusion

To fill one tank with helium and the other with hydrogen and create a pressure of 2.70 atm at 24 C, the mass of helium required is approximately 0.44 g and the mass of hydrogen required is approximately 0.22 g.

Key concepts

Pressure Volume Temperature Relationship

The relationship between the pressure, volume, and temperature of a gas is elegantly summarized in the Ideal Gas Law. At the heart of this law is the understanding that gases have properties that are predictably interrelated. When any two of these properties are altered in a closed system, there is a corresponding change in the third.

Imagine gas particles moving inside a container. They zoom around, hitting the walls, and exerting pressure. If the temperature rises, their movement becomes more frenetic, leading to more frequent and forceful wall collisions, thus increasing the pressure if the container's volume remains unchanged. Alternatively, expanding the container's volume gives the particles more space to move about, decreasing their impact on the walls, thereby reducing the pressure if the temperature remains constant.

Applying the Ideal Gas Law

The formula connects these variables in a simple expression: \(PV = nRT\). Here, \(P\) represents pressure, \(V\) is the volume, and \(T\) denotes temperature measured in Kelvin. They are all state functions, meaning their values depend only on the current state of the gas, not on the process used to get there. This law is based on the ideal gas assumption, which works well under typical conditions but deviates at high pressures or low temperatures, where real gas behavior begins to show nuances not accounted for in the idealized version.

Gas Moles Calculation

Determining the number of moles in a gas sample is essential for chemical studies and calculations. The quantity \(n\), representing moles, can be directly calculated using the rearranged Ideal Gas Law: \[n = \frac{PV}{RT}\]. In essence, this equation provides a clear method to find the amount of gas (in moles) when the pressure (\(P\)), volume (\(V\)), and temperature (\(T\)) of the system are known.

When we apply this to a fixed volume and temperature - as in our textbook example with the 100.0-L tanks at 24°C (or 297.15 K) - we can use the known constants (such as the ideal gas constant \(R\)) and measured variables (\(P\) and \(V\)) to find the number of moles without concern for the gas's identity. This step is pivotal because it translates physical parameters into a quantity that can be used in further calculations, such as finding the mass of the gas, and leads to an understanding of the amount of substance involved in reactions.

Molar Mass

Molar mass links the microscopic properties of molecules to the macroscopic measurements we can observe and calculate. It is defined as the mass of one mole (6.02 x 10^23 particles) of a substance and is expressed in grams per mole (g/mol).

For gases, this value is of paramount importance, as it allows us to transition from the abstract world of moles, often a non-intuitive quantity, to the tangible realm of mass - something that can be measured on a scale. The molar mass can vary widely from element to element; for instance, helium has a molar mass of approximately 4 g/mol while hydrogen's is about 2 g/mol.

Application in Calculations

To move from moles to the mass of a gas, the formula \(mass = molar \_ mass \times n\) is used, where \(n\) stands for the number of moles. This concept was crucial in solving our textbook problem, as it allowed us to calculate the exact mass of helium and hydrogen needed to achieve the specified pressure in the 100.0-L tanks at the given temperature.