Problem 31 - Chemistry: The Central Science

Question

Suppose you are given two 1-L flasks and told that one contains a gas of molar mass 30 , the other a gas of molar mass 60 , both at the same temperature. The pressure in flask A is \(\mathrm{X}\) atm, and the mass of gas in the flask is \(1.2 \mathrm{~g}\). The pressure Which flask contains the gas of molar mass 30 , and which contains the gas of molar mass 60 ?

Step-by-Step Solution

Answer

Flask A contains the gas with a molar mass of 30 g/mol, and Flask B contains the gas with a molar mass of 60 g/mol.

Step 1: Recall the Ideal Gas Law

The Ideal Gas Law relates the pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas through the equation: \[ PV = nRT, \] where R is the ideal gas constant, with a value of \( 8.3145 \frac{J}{mol K} \) or \(0.08206 \frac{L atm}{mol K}\). Step 2: Convert the mass to number of moles

Step 2: Convert the mass to the number of moles

We'll need to relate the given mass of the gas (in grams) with a number of moles (n) to eventually use it in the Ideal Gas Law. For this, we'll use the molar mass (M) of the substance: \[ n = \frac{m}{M}, \] where n is the number of moles, m is the mass of the substance, and M is its molar mass. Step 3: Write an equation for each flask

Step 3: Write equations for each flask

Let's consider flask A containing the gas with molar mass 30 and flask B containing a gas with molar mass 60. Given the Pressure, we'll use the Ideal Gas Law to write an equation for each flask: \[ P_A V = \frac{m_A}{30} RT, \] \[ P_B V = \frac{m_B}{60} RT, \] where \(P_A\) is the pressure , \(m_A\) is the mass of gas in flask A, and \(P_B\) and \(m_B\) are flask B pressure and mass. Step 4: Use the given information to find the Pressure ratios

Step 4: Use the given information to find the Pressure ratios

We'll now use the given information about flask A and flask B: flask A's pressure is X atm and mass is 1.2 g. \[ P_A V = \frac{1.2}{30} RT, \] flask B's pressure is 0.5X atm and mass is 1.2 g. \[ P_B V = \frac{1.2}{60} RT, \] Step 5: Solve for the pressure ratios

Step 5: Solve for the pressure ratios

Now we'll compare both equations to find the pressure ratio: \[ \frac{P_B V}{P_A V} = \frac{\frac{1.2}{60} RT}{\frac{1.2}{30} RT}, \] Cancelling similar terms and solving for the pressure ratio, we get: \[ \frac{P_B}{P_A} = \frac{1}{2}, \] Step 6: Determine the flasks' content.

Step 6: Determine the flasks' content.

Since we already know that the pressure ratio between flask A and flask B is equal to 1/2, we can see that: Flask A: Pressure = X atm and Gas with Molar Mass = 30 g/mol Flask B: Pressure = 0.5 X atm and Gas with Molar Mass = 60 g/mol So, flask A contains gas of molar mass 30, and the flask B contains the gas of molar mass 60.

Key Concepts

Molar Mass

Molar mass is a key concept in chemistry that helps us understand the amount of matter in a given substance. It is defined as the mass of one mole of a given substance and is usually expressed in grams per mole (g/mol).
For example, if you have a substance with a molar mass of 30 g/mol, it means that each mole of this substance weighs 30 grams. Knowing the molar mass allows you to convert between mass and the number of moles using the formula:

• \( n = \frac{m}{M} \),

where \( n \) is the number of moles, \( m \) is the mass of the substance, and \( M \) is its molar mass.
This conversion is essential when you want to understand how much of a gas you have in a chemical reaction or a container. In our example, flask A with a molar mass of 30 g/mol will have a different number of moles at the same mass compared to flask B with a molar mass of 60 g/mol.

Pressure Ratio

When dealing with gases, the term pressure ratio is crucial, especially in comparative studies of different gases.
The pressure ratio is the comparison of the pressure values between two instances or flasks, measuring how much one pressure is in relation to another.
In the given exercise, by comparing the pressures in flasks A and B, we determine that the ratio \( \frac{P_B}{P_A} = \frac{1}{2} \), meaning that the pressure in flask B is half that in flask A.

• Flask A has a pressure of \( X \) atm.
• Flask B has a pressure of \( 0.5X \) atm.

Understanding these pressure ratios helps in identifying specific properties of gases within containers, such as understanding how different molar masses influence the behavior of gases under similar conditions.

Number of Moles

The number of moles is a fundamental measurement in chemistry, signifying the amount of substance present. It represents a specific quantity, Avogadro's number, approximately \( 6.022 \times 10^{23} \) entities (such as atoms or molecules).
In the context of the Ideal Gas Law, the number of moles is crucial because it allows you to relate physical quantities like pressure, volume, and temperature.

• Using the formula \( n = \frac{m}{M} \), you find the number of moles from mass \( m \) and molar mass \( M \).

In the exercise, converting the mass of the gas in each flask to moles was essential to apply the Ideal Gas Law. This conversion informed us about how much gas was present and allowed for comparison based on different molar masses, leading to the determination of which flask contained which gas.

Gas Properties

Gases have unique properties that distinguish them from liquids and solids, primarily their ability to expand and fill a container, regardless of the container's size.
A few of the key properties of gases include:

• Pressure: Gases exert pressure uniformly on the walls of their container.
• Temperature: Typically, the higher the temperature, the more kinetic energy the gas molecules have, influencing pressure and volume.
• Volume: A gas will expand to occupy the entire volume of its container.

These properties are interconnected through the Ideal Gas Law, \( PV = nRT \), which helps predict and measure gas behavior under various conditions.
In this particular exercise, understanding these properties was crucial, as it provided insight into how we could use the Ideal Gas Law to compare the behavior of gases with different molar masses at different pressures.