Question

Several small molecules (besides water) are important in biochemical systems: \(\mathrm{O}_{2}, \mathrm{CO}, \mathrm{CO}_{2}\) and NO. You have isolated one of these and to identify it you determine its molar mass. You release 0.37 g of the gas into a flask with a volume of \(732 \mathrm{mL}\) at \(21^{\circ} \mathrm{C}\). The gas pressure in the flask is \(209 \mathrm{mm}\) Hg. What is the unknown gas?

Short answer

The unknown gas is carbon monoxide (CO).

Step-by-step solution

Convert Given Values to SI Units

Start by converting all given measurements to the appropriate SI units. The volume is given in milliliters, so convert it to liters by dividing by 1000: \[ 732 \text{ mL} = 0.732 \text{ L} \] The pressure is given in mm Hg; convert it to atmospheres using the conversion: 1 atm = 760 mm Hg: \[ 209 \text{ mm Hg} = \frac{209}{760} \text{ atm} = 0.275 \text{ atm} \] The temperature is given in Celsius; convert it to Kelvin by adding 273.15: \[ 21^\circ\text{C} = 21 + 273.15 = 294.15 \text{ K} \]

Use Ideal Gas Law to Find Moles

Use the ideal gas equation \( PV = nRT \), where \( P \) is the pressure in atm, \( V \) is the volume in liters, \( n \) is the number of moles, \( R \) is the ideal gas constant \( 0.0821 \, \text{L atm} / \text{mol K} \), and \( T \) is the temperature in Kelvin.Rearrange the ideal gas law to solve for \( n \): \[ n = \frac{PV}{RT} \] Substitute the known values into the equation: \[ n = \frac{(0.275 \, \text{atm})(0.732 \, \text{L})}{(0.0821 \, \text{L atm} / \text{mol K})(294.15 \, \text{K})} \] Calculate the result to find \( n \).

Calculate Molar Mass

Using the number of moles from Step 2 and the mass of the gas, calculate its molar mass. The molar mass \( M \) is given by the formula: \[ M = \frac{m}{n} \] where \( m \) is the mass of the gas (0.37 g). After finding \( n \) from Step 2, substitute \( n \) into the formula to find \( M \).

Determine the Unknown Gas

Compare the calculated molar mass from Step 3 with the known molar masses of the gases: - Oxygen (\( \mathrm{O}_2 \)) has a molar mass of 32.00 g/mol.- Carbon monoxide (\( \mathrm{CO} \)) has a molar mass of 28.01 g/mol.- Carbon dioxide (\( \mathrm{CO}_2 \)) has a molar mass of 44.01 g/mol.- Nitric oxide (\( \mathrm{NO} \)) has a molar mass of 30.01 g/mol.Identify the gas by matching the closest molar mass from the list.

Key concepts

Molar Mass

Molar mass is a crucial concept in chemistry, especially for identifying and comparing substances. It is defined as the mass of one mole of a substance, typically measured in grams per mole (g/mol). The molar mass of a compound is determined by summing the molar masses of its constituent elements. Molar mass helps chemists understand how much of each element is present in a compound. For example, in this exercise, we are determining the molar mass of an unknown gas to identify it.

Once we calculate moles (using the ideal gas law), we can divide the mass of the sample by the moles of gas we've found. This provides the molar mass of the gas. Knowing the molar masses of potential gases—oxygen (\(\mathrm{O}_{2}\)), carbon monoxide (\(\mathrm{CO}\)), carbon dioxide (\(\mathrm{CO}_{2}\)), and nitric oxide (\(\mathrm{NO}\))—helps in matching our calculated value with these known molar masses, thus identifying the unknown gas.

When learning about molar mass, quickly recall:

• Molar mass is mass per mole.
• Used to convert between grams and moles.
• Essential for identifying substances by comparing known molar masses.

Gas Identification

Identifying a gas involves determining its properties, one of which is its molar mass. By isolating a sample and calculating its molar mass, we can compare this to known values for various gases. This is done by using mathematical tools such as the ideal gas law.

The calculated molar mass acts like a fingerprint, helping us to identify which gas we are dealing with. In this exercise, after computing the molar mass, we compare it to known molar masses of common gases like oxygen, carbon monoxide, carbon dioxide, and nitric oxide. This comparison is crucial because each gas has a unique molar mass that distinguishes it.

Gas identification is vital in science and industry to:

• Ensure safety by identifying hazardous gases.
• Ensure the correct gases are used in processes.
• Assist in environmental monitoring and compliance.

Unit Conversion

Unit conversion is a fundamental skill in chemistry and science, allowing us to standardize measurements and perform calculations accurately. In the scenario of determining a gas's molar mass using the ideal gas law, proper unit conversion is essential. Here are the critical conversions used:

- **Volume**: Often given in milliliters (mL), it's converted to liters (L) because the ideal gas law uses liters. This is done by dividing by 1000 (e.g., 732 mL to 0.732 L).
- **Pressure**: Given in millimeters of mercury (mm Hg), it's converted to atmospheres (atm), using the conversion factor 1 atm = 760 mm Hg. Thus, 209 mm Hg converts to 0.275 atm.
- **Temperature**: Measured in degrees Celsius, converted to Kelvin by adding 273.15, because the ideal gas law works with Kelvin scale to make calculations reliable.

Correct unit conversion ensures that all measurements align with the ideal gas law parameters, providing accurate results. It is a necessary step to:

• Facilitate accurate calculations.
• Ensure consistency across different measurement systems.
• Enable clear communication of scientific data.