Problem 35 - General Chemistry: Principles and Modern Applications

Question

A 0.418 g sample of gas has a volume of \(115 \mathrm{mL}\) at \(66.3^{\circ} \mathrm{C}\) and \(743 \mathrm{mm} \mathrm{Hg} .\) What is the molar mass of this gas?

Step-by-Step Solution

Answer

The molar mass of the gas is approximately 103.58 g/mol.

Step 1: Convert the pressure from mmHg to atm

Knowing that \(1 \: atm \approx 760 \: mmHg \), the pressure is converted to atm by dividing 743 by 760, approximately equal to 0.977 atm.

Step 2: Convert the volume from mL to L

Since 1 L contains 1000 mL, the volume is 115 mL, which is equivalent to 0.115 L.

Step 3: Convert the temperature from Celsius to Kelvin

The Kelvin scale is used in gas law calculations. The temperature in Kelvin can be found by adding 273.15 to the Celsius temperature, resulting in a temperature of \(339.45 \: K\).

Step 4: Use the Ideal Gas Law equation to solve for the number of moles, n

Next, rearrange the ideal gas law equation to solve for the number of moles \(n = P*V/(R*T)\). Substituting the previous values and using \(R = 0.0821 \: L*atm/(K*mol)\), the number of moles n is approximately 0.004036 mol.

Step 5: Calculate molar mass given mass and number of moles

Finally, to find the molar mass, you divide the given mass of the sample by the number of moles calculated in step 4. Given mass is 0.418 g. Thus, molar mass is approximately 103.58 g/mol.

Key Concepts

Understanding Molar Mass

Molar mass is a fundamental concept in chemistry that helps us understand the mass of one mole of a particular substance. It is expressed in grams per mole (g/mol). To find the molar mass of a gas, you need to know both the mass of the gas sample and the number of moles present.

• First, weigh your sample to determine its mass in grams.
• Use the ideal gas law to calculate the number of moles (\( n \)), which is derived from dividing the sample mass by the molar mass.
• This allows you to rearrange the ideal gas equation and solve for molar mass: \( \text{molar mass} = \text{mass} / \text{number of moles} \).

In our example, the mass was 0.418 g, and by applying the ideal gas law, the number of moles was found to be 0.004036 mol. Therefore, the molar mass of the gas calculated was approximately 103.58 g/mol.

Gas Pressure Conversion Simplified

In chemical equations, pressure is often measured in different units such as mmHg (millimeters of mercury) and atm (atmospheres). Converting between these units is necessary to keep your gas law calculations accurate and consistent.

• The conversion factor between mmHg and atm is crucial: \( 1 \, \text{atm} = 760 \, \text{mmHg} \).
• To convert from mmHg to atm, divide the pressure value in mmHg by 760.

For example, if you have a pressure of 743 mmHg, converting this involves computing \( 743 \, \text{mmHg} / 760 \approx 0.977 \, \text{atm} \). This standardization is vital for plugging into the ideal gas law formula effectively.

Gas Volume Conversion Techniques

Volume can also appear in different units and usually requires conversion for precise usage in chemical calculations. Typically, volume is measured in milliliters (mL) or liters (L).

• One liter contains 1000 milliliters, so to convert from mL to L, divide by 1000.
• This conversion ensures uniformity across computations, especially when using the gas constant.

In the given exercise, a sample had a volume of 115 mL. Converting this volume to liters involves simply dividing 115 by 1000, resulting in 0.115 L. Handling volume conversions correctly enables you to apply the ideal gas law accurately and obtain valid results in your calculations.