Question

Calculate the ratio of the lifting powers of helium (He) gas and hydrogen (H \(_{2}\) ) gas under identical circumstances. Assume that the molar mass of air is \(29.5 \mathrm{~g} / \mathrm{mol}\).

Short answer

The ratio of the lifting powers of helium and hydrogen under identical conditions is approximately 0.926.

Step-by-step solution

Find the molar masses of He and H\(_{2}\)

To find the lifting power of each gas, first, we need the molar masses of helium (He) and hydrogen (H\(_{2}\)). Using the periodic table, we can find that the molar mass of He is 4 g/mol, while the molar mass of H\(_{2}\) is 2 g/mol (since there are two hydrogen atoms in a H\(_{2}\) molecule).

Calculate the lifting force of helium and hydrogen gas

The lifting force of a gas is determined by the difference in weight between the gas and the air it displaces. We can use the ideal gas law, \(PV = nRT\), to find the mass of the gas in a specific volume, where P, V, n, R, and T represent pressure, volume, moles, gas constant, and temperature, respectively. Assuming identical conditions for both gases, we can say that \(m_{He} = n_{He} \cdot M_{He}\) and \(m_{H_2} = n_{H_2} \cdot M_{H_2}\), where \(m_{He}\) and \(m_{H_2}\) are the masses of helium and hydrogen in the specific volume, respectively, and \(M_{He}\) and \(M_{H_2}\) are their molar masses. Using the ideal gas law and rearranging, we have \(n_{He}= \frac{PV}{RT}\) and \(n_{H_2} = \frac{PV}{RT}\). Since both gases are under identical conditions, we can find the ratio of moles of each gas without knowing the specific values of P, V, R, and T.

Determine the ratio of lifting forces

To determine the ratio of lifting forces, we need to find the difference in mass between each gas and the air it displaces and then find the ratio of these differences. We will use \(M_{air}\), which is given as \(29.5 \mathrm{~g} / \mathrm{mol}\). We have \(m_{air} = n_{air} \cdot M_{air}\), where \(n_{air}\) is the number of moles of air. Since identical volumes of He, H\(_{2}\), and air are being compared under the same conditions, we have \(n_{air} = n_{He} = n_{H_2}\). Now, we can find the differences in mass and the ratio of the lifting forces: \(\frac{L_{He}}{L_{H_2}} = \frac{m_{air} - m_{He}}{m_{air} - m_{H_2}}\) Substituting the expressions for \(m_{air}\), \(m_{He}\), and \(m_{H_2}\): \(\frac{L_{He}}{L_{H_2}} = \frac{n_{air}M_{air} - n_{He}M_{He}}{n_{air}M_{air} - n_{H_2}M_{H_2}}\) Since \(n_{air} = n_{He} = n_{H_2}\), we have: \(\frac{L_{He}}{L_{H_2}} = \frac{M_{air} - M_{He}}{M_{air} - M_{H_2}}\)

Calculate the ratio of lifting powers

Now, we can plug in the values of the molar masses and find the ratio of lifting powers of helium and hydrogen: \(\frac{L_{He}}{L_{H_2}} = \frac{29.5 - 4}{29.5 - 2}\) \(\frac{L_{He}}{L_{H_2}} = \frac{25.5}{27.5}\) \(\frac{L_{He}}{L_{H_2}} = 0.926\) The ratio of the lifting powers of helium and hydrogen under identical conditions is approximately 0.926.

Key concepts

Lifting Force

Lifting force is an important concept when dealing with gases like helium and hydrogen. It refers to the ability of a gas to "lift" or displace air, which in practical terms means how much weight a certain volume of gas can carry. The lifting force is based on Archimedes' principle. This principle tells us that the force exerted on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. When it comes to gases, the "fluid" is the air itself. For gases, the lifting force depends on the difference in density between the gas and the surrounding air:

• If a gas is less dense than air, it will exert a lifting force.
• The greater the density difference, the greater the lifting force.

Using the ideal gas law, we can predict the lifting power by calculating the differences in weight between a specific volume of the gas and the same volume of air under the same conditions. This means evaluating molar masses can give insights into how well a gas can lift under certain circumstances.

Molar Mass

Molar mass is the weight of one mole of a substance, given in grams per mole. It plays a crucial role in calculating the lifting ability of gases like helium and hydrogen. When comparing lifting forces of gases, the molar mass helps determine how much of a gas is present in a given volume, set under standard conditions. - To find the molar mass of a compound, you add up the molar masses of each individual element. - For instance, helium has a molar mass of 4 g/mol, because it is an element with one atom per molecule. - Hydrogen, as found in nature as the diatomic molecule H 2, has a molar mass of 2 g/mol (since each hydrogen atom has an approximate molar mass of 1 g/mol). The molar mass is essential because it reflects how many grams one mole of a gas weighs. The lighter the molar mass, the less dense the gas and the better it is at lifting compared to heavier gases under the same temperature and pressure.

Helium and Hydrogen

Helium and hydrogen are two of the lightest gases found on Earth. Because of their low density, they are often used in balloons and airships for lifting purposes. - Helium is an inert, non-flammable gas with a molar mass of 4 g/mol. It is often chosen for lifting due to its safety, stemming from its lack of reactivity. - On the other hand, hydrogen has a molar mass of 2 g/mol, making it even lighter than helium. This makes hydrogen a more efficient lifting gas. However, hydrogen is highly flammable, like the famous Hindenburg disaster illustrates, which means it poses a safety risk more considerable than helium. In floating applications, the choice between helium and hydrogen often hinges on the need to balance efficiency and safety. These gases' lightweight, in conjunction with the ideal gas law, allows us to relate their molar masses directly to their lifting capabilities, helping us calculate precisely how well they can perform in practical applications like inflating balloons or lifting other objects.