Question

If the molar mass of monoatomic deuterium (d) is \(2.0141 \mathrm{~g}\) then what is the density of diatomic deuterium \(\left(\mathrm{D}_{2}\right.\) ) gas at \(25^{\circ} \mathrm{C}\) and \(1.00\) atmospheric pressure? (a) \(0.165 \mathrm{~g} / \mathrm{lit}\) (b) \(5.125 \mathrm{~g} / \mathrm{lit}\) (c) \(1.565 \mathrm{~g} / \mathrm{lit}\) (d) \(3.698 \mathrm{~g} / \mathrm{lit}\)

Short answer

The density of diatomic deuterium is approximately 0.165 g/L, which corresponds to option (a).

Step-by-step solution

Understand the Problem

We are asked to find the density of diatomic deuterium gas (\(\mathrm{D}_2\)) under given conditions, using the molar mass of monoatomic deuterium and standard atmospheric conditions.

Calculate the Molar Mass of Diatomic Deuterium

Since diatomic deuterium gas consists of two deuterium atoms, we need to double the molar mass of monoatomic deuterium. The molar mass of monoatomic deuterium is given as \(2.0141 ~\mathrm{g/mol}\). Thus, the molar mass of diatomic deuterium \(\mathrm{D}_2\) is:\[2 \times 2.0141 ~\mathrm{g/mol} = 4.0282 ~\mathrm{g/mol}\]

Apply the Ideal Gas Law

The formula for the density \(\rho\) of a gas using the ideal gas law is:\[\rho = \frac{PM}{RT}\]Where \(P\) is the pressure, \(M\) is the molar mass, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.

Convert Temperature to Kelvin

The temperature given is \(25^{\circ}\mathrm{C}\). To convert it to Kelvin, use the formula:\[T = 25 + 273.15 = 298.15 ~\mathrm{K}\]

Substitute Values and Solve

Use the values:- Pressure \(P = 1.00 ~\mathrm{atm}\)- Molar mass \(M = 4.0282 ~\mathrm{g/mol}\)- Temperature \(T = 298.15 ~\mathrm{K}\)- Universal gas constant \(R = 0.0821 ~\text{L atm K}^{-1}\text{mol}^{-1}\)Now, substitute these into the formula:\[\rho = \frac{(1.00 ~\text{atm})(4.0282 ~\text{g/mol})}{(0.0821 ~\text{L atm K}^{-1}\text{mol}^{-1})(298.15 ~\text{K})}\]Calculate \(\rho\) to find the density.

Calculate the Density

Calculate the expression:\[\rho = \frac{4.0282}{0.0821 \times 298.15} \approx 0.165 \text{ g/L}\]

Key concepts

Ideal Gas Law

The Ideal Gas Law is an essential formula in chemistry that helps us understand how gases behave under various conditions. This formula is written as \(PV = nRT\), where \(P\) stands for pressure, \(V\) for volume, \(n\) for the number of moles, \(R\) for the universal gas constant, and \(T\) for temperature in Kelvin. In the context of gas density calculations, this law allows us to find the density \(\rho\) of a gas by rearranging it to the form:

• \(\rho = \frac{PM}{RT}\)

By doing this, you can calculate the density of a gas using its molar mass \(M\), pressure \(P\), temperature \(T\), and the universal gas constant \(R\). Understanding the Ideal Gas Law is crucial as it links physical properties of gases, offering real-world applications ranging from predicting weather patterns to designing airbags.

Molar Mass

Molar Mass is the weight of one mole of a given substance, measured in grams per mole (g/mol). It acts as a conversion factor between the mass of a substance and the amount of substance in moles. In the context of gases, knowing the molar mass is vital since it is directly used in the Ideal Gas Law to find density.

• For monoatomic deuterium, the molar mass is given as \(2.0141 \,\mathrm{g/mol}\).
• Since diatomic deuterium \(\mathrm{D}_2\) consists of two atoms, we find its molar mass by doubling that of the monoatomic deuterium: \(2 \times 2.0141 \,\mathrm{g/mol} = 4.0282 \,\mathrm{g/mol}\).

Understanding molar mass helps you determine how much of a substance you have in a chemical reaction or process, and it is essential for calculating gas density using the Ideal Gas Law.

Temperature Conversion

Temperature Conversion is critical when dealing with the Ideal Gas Law, because the temperature variable \(T\) must be in Kelvin. To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature.

• Given: \(25^{\circ}\mathrm{C}\)
• Convert: \(T = 25 + 273.15 = 298.15 \,\mathrm{K}\)

Kelvin is the absolute temperature scale commonly used in scientific calculations since it starts at absolute zero, where theoretically, all kinetic energy ceases. Converting temperature correctly ensures the accuracy of the calculations, particularly when applying the Ideal Gas Law.

Universal Gas Constant

The Universal Gas Constant \(R\) is a crucial part of the Ideal Gas Law, serving as the link between the other variables. Its value depends on the units used in the equation. For calculations involving pressure in atmospheres and volume in liters, \(R\) is typically \(0.0821 \,\text{L atm K}^{-1}\text{mol}^{-1}\).

• The constant \(R\) relates energy to temperature and moles, making it fundamental for calculating gas properties.
• It ensures consistency among various gas law equations, allowing conversions between different units.

Understanding \(R\) is essential for utilizing the Ideal Gas Law appropriately, making calculations involving gases both reliable and standardized across various scientific fields.