Question

A 2.10 - \(\mathrm{L}\) vessel contains \(4.65 \mathrm{~g}\) of a gas at \(1.00 \mathrm{~atm}\) and \(27.0^{\circ} \mathrm{C}\). (a) Calculate the density of the gas in \(\mathrm{g} / \mathrm{L}\). (b) What is the molar mass of the gas?

Short answer

(a) Density = 2.214 g/L, (b) Molar mass = 54.71 g/mol

Step-by-step solution

Convert Temperature to Kelvin

To calculate the density and molar mass, we first need the temperature in Kelvin. Convert the given temperature \(27.0^{\circ}C\) to Kelvin using the formula: \(T(K) = T(^{\circ}C) + 273.15\). Thus, \(T = 27.0 + 273.15 = 300.15 \, K\).

Calculate Density

Density \(\rho\) is calculated as mass \(m\) divided by volume \(V\). Here, mass \(m = 4.65\, \mathrm{g}\) and volume \(V = 2.10\, \mathrm{L}\). Therefore, \[\rho = \frac{4.65\, \mathrm{g}}{2.10\, \mathrm{L}} = 2.214\, \mathrm{g/L}.\]

Use Ideal Gas Law to Find Moles

We will use the ideal gas law \(PV = nRT\) to find the number of moles \(n\). Rearrange for \(n\): \[n = \frac{PV}{RT}.\] Here, \(P = 1.00\, \mathrm{atm}\), \(V = 2.10\, \mathrm{L}\), \(R = 0.0821\, \mathrm{L\cdot atm\cdot mol^{-1}\cdot K^{-1}}\), and \(T = 300.15\, \mathrm{K}\). Substitute these into the equation: \[n = \frac{1.00 \times 2.10}{0.0821 \times 300.15} = 0.0850\, \mathrm{mol}.\]

Calculate Molar Mass of the Gas

Molar mass \(M\) is calculated by dividing the mass of the substance by the amount in moles. Thus, \[M = \frac{4.65 \, \mathrm{g}}{0.0850 \, \mathrm{mol}} = 54.71\, \mathrm{g/mol}.\]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation used to relate the properties of gases. It is written as \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This law is exceptionally useful for performing calculations when the state of a gas changes or when you need to find an unknown property.
To use the ideal gas law effectively, you must ensure all units are compatible with \( R \), which is usually \( 0.0821 \, L \cdot atm \cdot mol^{-1} \cdot K^{-1} \). For instance, if your pressure is in atmospheres, volume in liters, and temperature in Kelvin, then \( R \) remains consistent.
The equation helps us to calculate the number of moles of a gas present in a container if we know the other parameters, reinforcing the interconnected nature of gas properties.

Molar Mass Calculation

The molar mass of a substance is a measure of the mass of one mole of that substance and is usually expressed in grams per mole (\( g/mol \)). To calculate the molar mass of a gas using experimental data, you can divide the mass of the gas by the number of moles.
In an exercise like the one provided, you have already determined the number of moles from the Ideal Gas Law. The molar mass \( M \) is given by the formula: \[ M = \frac{m}{n} \] where \( m \) is the mass of the gas and \( n \) is the number of moles.
For example, if you have 4.65 grams of gas and have calculated 0.0850 moles, the molar mass will be 54.71 \( g/mol \). Remember, knowing the molar mass can help identify the gas or verify its identity.

Temperature Conversion

Temperature plays a crucial role in gas calculations, and it is essential to use the correct temperature scale. In the context of gas laws, temperature must always be in Kelvin. The Kelvin scale begins at absolute zero, making it an absolute scale which is necessary for the correct interpretation of gas behaviors.
To convert a temperature from degrees Celsius (\( ^\circ C \)) to Kelvin (\( K \)), you use the formula: \[ T(K) = T(^\circ C) + 273.15 \] This conversion ensures that all the calculations involving temperature, such as those in the Ideal Gas Law, are accurate and meaningful.
For instance, converting 27.0\(^\circ C\) to Kelvin, you add 273.15, resulting in 300.15\( K \). Keeping these details in mind prevents errors that could arise from temperature miscalculations.

Gas Density Calculation

Gas density is an important property that gives us information about how much mass is contained in a given volume. The density of a gas \( \rho \) can be calculated using the formula: \[ \rho = \frac{m}{V} \] where \( m \) is the mass and \( V \) is the volume. This relationship shows how the mass of a gas is spread over its container’s volume, which helps in understanding and comparing different gases.
For example, if you have a gas mass of 4.65 grams contained in a 2.10-liter tank, the density would be 2.214 \( g/L \).
Understanding gas density lets us predict how a gas might behave under different conditions, as denser gases like those containing heavier molecules will have different applications compared to lighter gases, which might be used, for instance, in inflating balloons or in lighter-than-air applications.