Question

A sample of unknown gas has a mass of \(1.95 \mathrm{~g}\) and occupies \(3.00 \mathrm{~L}\) at \(1.25 \mathrm{~atm}\) and \(20^{\circ} \mathrm{C} .\) What is the molar mass of the unknown gas?

Short answer

The molar mass of the unknown gas is approximately \(12.53 \mathrm{~g/mol}\).

Step-by-step solution

Understand the Given Information

We are given the mass of the gas as \(1.95 \mathrm{~g}\), the volume as \(3.00 \mathrm{~L}\), the pressure of the gas as \(1.25 \mathrm{~atm}\), and the temperature as \(20^{\circ} \mathrm{C}\). The unknown we need to find is the molar mass of the gas.

Convert Temperature to Kelvin

To use the ideal gas law, we need to convert the temperature from Celsius to Kelvin. The conversion formula is \(T(K) = T(°C) + 273.15\). Thus, at \(20^{\circ} \mathrm{C}\), \(T(K) = 20 + 273.15 = 293.15 \mathrm{~K}\).

Use the Ideal Gas Law

The ideal gas law is \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin. Rearranging for \(n\), we have \(n = \frac{PV}{RT}\).

Substitute Known Values into Ideal Gas Law

Substitute \(P = 1.25 \mathrm{~atm}\), \(V = 3.00 \mathrm{~L}\), \(R = 0.0821 \mathrm{~L \cdot atm \cdot K^{-1} \cdot mol^{-1}}\), and \(T = 293.15 \mathrm{~K}\) into the equation: \(n = \frac{1.25 \times 3.00}{0.0821 \times 293.15}\).

Calculate Number of Moles

Calculate \(n = \frac{3.75}{24.084315} \approx 0.1556 \mathrm{~moles}\).

Calculate Molar Mass

The molar mass is calculated using the formula: \(\text{Molar Mass} = \frac{\text{mass}}{\text{moles}}\). Substitute the mass \(1.95 \mathrm{~g}\) and the number of moles \(0.1556\) to find the molar mass: \(\frac{1.95}{0.1556} \approx 12.53 \mathrm{~g/mol}\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that provides a valuable connection between the physical properties of gases. Its formula is expressed as \(PV = nRT\), where:

• \(P\) represents the pressure of the gas
• \(V\) stands for the volume that the gas occupies
• \(n\) is the number of moles of the gas
• \(R\) is the ideal gas constant (typically \(0.0821 \text{ L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\))
• \(T\) is the temperature of the gas in Kelvin

The Ideal Gas Law allows us to determine one of these variables if the others are known. In scenarios involving molar mass calculations, this equation is invaluable, helping to connect the mass of a gas to its molecular behavior under varying conditions of temperature and pressure. By using the Ideal Gas Law in the process of calculating molar mass, we directly determine the number of moles from the given conditions, which can further be used to compute the molar mass using the mass of the gas sample.

Gas Pressure and Volume

Understanding gas pressure and volume is crucial when dealing with gases. Pressure is the force applied by the gas molecules against the walls of its container, and it is typically measured in atmospheres (atm). Volume refers to the space that the gas occupies, usually expressed in liters (L).

According to Boyle's Law, if temperature remains constant, there is an inverse relationship between the pressure and volume of a gas: when one increases, the other decreases. However, when using the Ideal Gas Law, both pressure and volume are considered together in relation to temperature and moles of the gas involved. Knowing the correct values for pressure and volume is essential when applying the Ideal Gas Law, as they directly affect the calculation of moles, thus influencing the final determination of properties like the molar mass.

Temperature Conversion

Converting temperature from Celsius to Kelvin is an essential step when using the Ideal Gas Law because the law requires the absolute temperature scale. The difference between these two scales is that Kelvin starts from absolute zero, the temperature at which all molecular motion ceases.

The conversion formula is simple:

• \(T(K) = T(°C) + 273.15\)

For example, when the temperature given is \(20^{\circ}C\), converting it to Kelvin, we add \(273.15\) to get \(293.15 K\). This step is crucial because Kelvin ensures that you are working with absolute temperatures, thus aligning with the requirements of the Ideal Gas Law. Proper temperature conversion allows for accurate calculations, ultimately leading to precise determinations of physical properties like molar mass.