Question

Suppose you are given two flasks at the same temperature, one of volume $2\mathrm{L}$ and the other of volume $3\mathrm{L}$. The 2-L flask contains $4.8\mathrm{g}$ of gas, and the gas pressure is $X$ atm. The 3-L flask contains $0.36\mathrm{g}$ of gas, and the gas pressure is $0.1\mathrm{X}$. Do the two gases have the same molar mass? If not, which contains the gas of higher molar mass?

Short answer

The two gases do not have the same molar mass. The gas in Flask 1 has a higher molar mass than the gas in Flask 2.

Step-by-step solution

Set up the ideal gas law for each flask

First, we will set up the ideal gas law for each flask: 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. For Flask 1: P1 = x kPa V1 = 2 L m1 = 4.8 g For Flask 2: P2 = 0.1x kPa V2 = 3 L m2 = 0.36 g Assume that both flasks are at the same temperature T.

Solve for the number of moles in each flask

We can rearrange the ideal gas law to solve for n: n = PV/RT We will convert the pressures to Pa and use the ideal gas constant R = 8.314 J/mol·K. For Flask 1: n1 = (x*1000 Pa)(2 L) / (8.314 J/mol·K)(T) For Flask 2: n2 = (0.1x*1000 Pa)(3 L) / (8.314 J/mol·K)(T)

Calculate the molar masses for each flask

Now we can find the molar mass (M) for each flask using the formula M = m/n: For Flask 1: M1 = m1/n1 = (4.8 g) / ((x*1000)(2 L) / (8.314 J/mol·K)(T)) For Flask 2: M2 = m2/n2 = (0.36 g) / ((0.1x*1000)(3 L) / (8.314 J/mol·K)(T))

Compare the molar masses and identify which gas has the higher molar mass

To determine if the two gases have the same molar mass, we can compare M1 and M2. Note that the temperature T will cancel out in both denominators, so it is not necessary to know the actual value. If M1 = M2, the gases have the same molar mass. If M1 > M2, the gas in Flask 1 has a higher molar mass, otherwise, the gas in Flask 2 has a higher molar mass. Let's simplify the expressions for M1 and M2: M1 = (4.8 g) / (2x*1000 / 8.314) = 4.8 * 8.314 / 2000x M2 = (0.36 g) / (0.1*3x*1000 / 8.314) = 0.36 * 8.314 / 300x Now, compare M1 and M2: 4.8 * 8.314 / 2000x = 0.36 * 8.314 / 300x Divide both sides by 8.314 and multiply both sides by 300*2000: 4.8 * 2000 = 0.36 * 300 9600 = 108 Since 9600 > 108, it means that M1 > M2, and the gas in Flask 1 has a higher molar mass than the gas in Flask 2. So, the two gases do not have the same molar mass.

Key concepts

Molar Mass

Understanding the molar mass of a substance is a fundamental aspect of chemistry and crucial in various calculations. Molar mass, represented by the symbol 'M', is the mass of one mole of a chemical element or chemical compound. In simple terms, it's the mass in grams of one mole of particles, such as atoms or molecules. The molar mass is typically expressed in units of grams per mole (g/mol).

For example, the molar mass of water (H₂O) is approximately 18.015 g/mol, because it contains two hydrogen atoms (2 x 1.008 g/mol) and one oxygen atom (16.00 g/mol).

In the context of the exercise, where we're comparing gases in flasks, molar mass allows us to determine the type of gas by relating the mass of a sample to the number of moles. This is done using the formula M = m/n, where 'm' is the mass of the gas sample and 'n' is the number of moles. By doing so for both flasks and comparing their molar masses, we can deduce whether they contain the same gas or different ones and which one has a higher molar mass.

Gas Laws

The behavior of gases and their interactions with changes in pressure, volume, and temperature can be described by gas laws. One of the pivotal equations summarizing these relationships is the ideal gas law, which states that PV = nRT. Here, 'P' stands for the pressure of the gas, 'V' is the volume it occupies, 'n' is the number of moles, 'R' is the ideal gas constant, and 'T' is the absolute temperature in Kelvin.

For the gases in our exercise, we apply the ideal gas law to each flask separately, with the same constant 'R' and 'T', following the steps provided in the solution. This law underpins the calculations required to connect the empirical data from our experiment (pressure, volume, and temperature) to the molar amount of gas, which will then be used to find the molar mass.

Real Gas vs. Ideal Gas

It's important to note that the ideal gas law is an approximation that holds true under certain conditions, often at low pressure and high temperature where the gas molecules behave independently of each other. In reality, gas molecules can exert forces upon one another and occupy physical space, so the ideal gas law is adjusted for these real gas behaviors in more advanced chemistry applications.

Chemistry Calculations

Chemistry calculations are a vital tool for understanding and predicting the behavior of substances during chemical reactions and changes in states. These calculations involve quantifying the relationships between elements and compounds, such as determining reaction yields, concentration of solutions, molar mass, and more through stoichiometry, gas laws, and other chemical principles.

For example, when studying the behavior of gases, as in the exercise with the two flasks, we make precise calculations using the ideal gas law. These calculations often require unit conversions (like converting the pressure to Pascals) and algebraic manipulations, such as solving for the number of moles or rearranging formulas to isolate the desired variable. An underlying principle of these calculations is the law of conservation of mass, which ensures that the mass remains constant, irrespective of the transformations it undergoes.

Effective chemistry calculations are essential for accurate scientific research, quality control in manufacturing, environmental monitoring, and in medical settings for drug formulations. Mastery of these calculations provides a framework for students to grasp complex chemical concepts and to execute laboratory experiments with predictability and precision.