Problem 57 - Chemistry A Molecular Approach

Question

A 248-mL gas sample has a mass of \(0.433 \mathrm{~g}\) at a pressure of \(745 \mathrm{mmHg}\) and a temperature of \(28^{\circ} \mathrm{C}\). What is the molar mass of the gas?

Step-by-Step Solution

Answer

The molar mass of the gas is approximately 44.14 g/mol.

Step 1: Convert Temperature to Kelvin

Since the ideal gas law requires that the temperature be in Kelvin, first convert the Celsius temperature to Kelvin by using the formula: Kelvin = Celsius + 273.15. So, the temperature in Kelvin is: 28°C + 273.15 = 301.15 K.

Step 2: Convert Pressure to Atmospheres

The ideal gas law uses pressure in atmospheres, so convert the pressure from mmHg to atm. The conversion factor is 1 atm = 760 mmHg. The pressure in atmospheres is: 745 mmHg * (1 atm / 760 mmHg) = 0.98026 atm.

Step 3: Use the Ideal Gas Law to Find Moles of Gas

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 (0.0821 L*atm/mol*K), and T is the temperature in Kelvin. Rearrange the formula to solve for n: n = PV / (RT). With P = 0.98026 atm, V = 248 mL (0.248 L), R = 0.0821 L*atm/mol*K, and T = 301.15 K, we get n = (0.98026 atm * 0.248 L) / (0.0821 L*atm/mol*K * 301.15 K) = 0.00981 mol.

Step 4: Calculate the Molar Mass of the Gas

The molar mass is the mass of one mole of a substance. You can find it by dividing the mass of the gas sample by the number of moles of gas: Molar Mass = Mass of the Gas / Number of Moles. Using the mass of 0.433 g and the moles calculated as 0.00981 mol, we get Molar Mass = 0.433 g / 0.00981 mol = 44.14 g/mol.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical ideal gas. It is a good approximation to the behaviour of many gases under a variety of conditions, although it has several limitations. It’s written as PV = nRT.

Here’s what each symbol represents:

• P stands for pressure of the gas.
• V is the volume the gas occupies.
• n indicates the number of moles of gas.
• R is the ideal gas constant, which is 0.0821 L*atm/mol*K in most cases.
• T is the temperature in Kelvin.

Applying the Ideal Gas Law involves rearranging the formula to solve for the unknown variable in question. In our exercise, it is used to calculate the number of moles of gas when the pressure, volume, and temperature are given. Once the number of moles is found, it provides a crucial part of the puzzle in determining the molar mass of the gas.

Convert Celsius to Kelvin

Temperature conversions are fundamental to gas law calculations. The Kelvin scale is the standard unit of temperature in the physical sciences, so we often need to convert from Celsius to Kelvin before using the Ideal Gas Law.

The formula for this conversion is straightforward:
Kelvin = Celsius + 273.15.

This conversion is based on the fact that 0 Kelvin, also known as absolute zero, is the lowest possible temperature where all molecular motion stops, and it is exactly 273.15 degrees lower than 0°C, the freezing point of water. By adding 273.15 to any Celsius temperature, you bridge the gap between the two scales. As seen in our exercise, the temperature of the gas in Celsius was converted to Kelvin before using the formula for the Ideal Gas Law.

Gas Law Calculations

Gas law calculations are procedures that involve manipulating the variables of the Ideal Gas Law to find unknowns such as pressure, volume, temperature, or number of moles of the gas. To perform these calculations:

• Ensure all units are compatible with the gas constant (R), often requiring conversion of pressure into atmospheres (atm) or volume into liters (L), and temperature to Kelvin.
• Rearrange the Ideal Gas Law equation to isolate and solve for the unknown variable.
• In our textbook exercise, for instance, the focus was to find the molar mass of the gas, which is the mass of one mole of a substance. We first calculated the number of moles (n) using the given pressure, volume, and temperature.
• Next, molar mass is calculated by dividing the mass of the gas sample by the number of moles obtained from the initial calculation.

To prevent calculation errors, it is crucial to double-check that each unit is correctly converted and that each step of the rearrangement is done properly. The right molar mass helps in identifying the unknown gas or in determining its molecular formula which are important aspects in chemistry.