Question

A balloon filled with He gas and another balloon filled with \(\mathrm{H}_{2}\) gas have the same values for \(P\) and \(\bar{T}\). (a) The density of the He gas is greater than the density of the \(\mathrm{H}_{2}\) gas. How can you prove this using the ideal gas law? (b) How much more dense than the \(\mathrm{H}_{2}\) gas is the He gas?

Short answer

(a) Using the ideal gas law, we find the density formula: \(\rho = \frac{PM}{RT}\). The density of a gas is directly proportional to its molar mass (M). Since He has a greater molar mass (4 g/mol) than H₂ (2 g/mol), the density of He is greater than the density of H₂. (b) To compare the densities, we calculate the ratio \(\frac{\rho_{He}}{\rho_{H_2}} = \frac{M_{He}}{M_{H_2}}\). Plugging in the molar masses, we find \(\frac{\rho_{He}}{\rho_{H_2}} = 2\). So, the He gas is 2 times more dense than the H₂ gas under the same pressure and temperature conditions.

Step-by-step solution

(a) Proving the density of He is greater than the density of H₂ with the ideal gas law

First, let's write down the ideal gas law formula: \[PV = nRT\] where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature. To find the density (ρ), we need to relate the mass (m) to the volume (V). We can write the number of moles (n) as the mass (m) divided by the molar mass (M): \[n = \frac{m}{M}\] Now we can rewrite the ideal gas law formula: \[PV = \frac{m}{M}RT\] Now, solve for the density (ρ), which is mass (m) divided by volume (V): \[\rho = \frac{m}{V} = \frac{PM}{RT}\] From this equation, we can see that the density of a gas is directly proportional to its molar mass (M) under the same conditions of pressure and temperature. Since the molar mass of He is approximately 4 g/mol and the molar mass of H₂ is approximately 2 g/mol, it is evident that the density of He is greater than the density of H₂.

(b) Comparing the density of He and H₂

To find how much more dense the He gas is compared to the H₂ gas, we'll calculate the ratio of their densities. Using the density formula from part (a) and assuming P and T are constant for both gases: \[\frac{\rho_{He}}{\rho_{H_2}} = \frac{\frac{P \times M_{He}}{RT}}{\frac{P \times M_{H_2}}{RT}}\] The pressure, temperature, and universal gas constants cancel out in this equation: \[\frac{\rho_{He}}{\rho_{H_2}} = \frac{M_{He}}{M_{H_2}}\] Now we plug in the molar masses of He (4 g/mol) and H₂ (2 g/mol): \[\frac{\rho_{He}}{\rho_{H_2}} = \frac{4}{2}\] \[\frac{\rho_{He}}{\rho_{H_2}} = 2\] So, the helium gas is 2 times more dense than the hydrogen gas under the same pressure and temperature conditions.

Key concepts

Gas Density

Gas density refers to the mass of gas per unit of volume. It is an important property in the study of gases. Understanding gas density can help us determine how gases behave under different conditions. To find the density of a gas, we use the formula \( \rho = \frac{m}{V} \), where \( \rho \) is the density, \( m \) is the mass, and \( V \) is the volume.

Using the ideal gas law, \( PV = nRT \), we can relate density to other properties of a gas. By substituting \( n = \frac{m}{M} \) into the ideal gas law, we derive \( \rho = \frac{PM}{RT} \). This equation shows that gas density is directly proportional to its molar mass when pressure and temperature remain constant.

In the context of the example with helium (He) and hydrogen (H₂) gases, since helium has a larger molar mass than hydrogen, it results in a greater density.

Molar Mass

Molar mass is the mass of one mole of a given substance. It is usually expressed in grams per mole (g/mol). Molar mass is essential for determining several properties of gases, like density. Calculating molar mass helps us make meaningful comparisons between different gases.

To find the molar mass, you sum up the atomic masses of all atoms present in a molecule of the substance. For example, helium (He) has a molar mass of approximately 4 g/mol, while hydrogen (H₂) has a molar mass of about 2 g/mol.

The higher the molar mass of a gas, the greater the density, if pressure and temperature stay the same. As demonstrated in the solution with He and H₂, the molar mass ratio directly influences density ratio: \( \frac{\rho_{He}}{\rho_{H_2}} = \frac{M_{He}}{M_{H_2}} \). Hence, He is twice as dense as H₂ because its molar mass is twice as much.

Avogadro's Law

Avogadro's law states that equal volumes of all gases, at the same temperature and pressure, have the same number of molecules. This principle is fundamental in understanding and comparing the behaviors of different gases under identical conditions.

While Avogadro's law doesn't directly calculate density, it remains crucial in studying gas volume relations. By introducing Avogadro's law into density discussions, it emphasizes that the effects on density arise from variations in molar mass rather than variations in the number of molecules.

For example, helium and hydrogen having equal volumes might suggest they contain the same number of molecules, supporting the use of molar mass to determine differing densities. This principle helps simplify conditions when using the ideal gas law. It supports why, under the same conditions, gases with different molar masses exhibit varying densities.