Problem 52 - Physics

Question

A tank contains \(0.85 \mathrm{~mol}\) of molecular nitrogen \(\left(\mathrm{N}_{2}\right)\). Determine the mass (in grams) of nitrogen that must be removed from the tank in order to lower the pressure from 38 to 25 atm. Assume that the volume and temperature of the nitrogen in the tank do not change.

Step-by-Step Solution

Answer

Remove approximately 8.15 grams of nitrogen.

Step 1: Use Ideal Gas Law Relationships

Since the problem states that the volume and temperature of the gas do not change, we can use the relationship from the ideal gas law, \( PV = nRT \), specifically the relationship \( P_1/n_1 = P_2/n_2 \). Here, \( P_1 = 38 \) atm and \( n_1 = 0.85 \) mol. The desired pressure \( P_2 \) is 25 atm.

Step 2: Calculate Moles Needed for New Pressure

Rearrange the equation \( P_1/n_1 = P_2/n_2 \) to solve for the moles at new pressure, \( n_2 \):\[ n_2 = \frac{P_2 \times n_1}{P_1} \]Substituting the values:\[ n_2 = \frac{25 \times 0.85}{38} \approx 0.5592 \text{ mol} \]

Step 3: Determine Moles to be Removed

Subtract the moles needed at the new pressure from the initial moles to find out how many moles must be removed:\[ n_{removed} = n_1 - n_2 = 0.85 - 0.5592 \approx 0.2908 \text{ mol} \]

Step 4: Convert Moles to Grams

Use the molecular weight of nitrogen, \( N_2 \), which is approximately 28.02 g/mol, to convert moles to grams:\[ \text{Mass to be removed} = 0.2908 \times 28.02 \approx 8.15 \text{ grams} \]

Key Concepts

Moles Calculation

In this exercise, one of the key concepts is calculating the number of moles when changes occur in a gas system. Moles are a way of expressing quantities of chemical entities — in this case, the molecules of nitrogen gas, \(\mathrm{N}_2\). When using the ideal gas law \(PV = nRT\), we focus on the variables:

• \(P\) is the pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant
• \(T\) is the temperature.

The relationship \(\frac{P_1}{n_1} = \frac{P_2}{n_2}\) derived from the ideal gas law illustrates how pressure and moles are related when temperature and volume remain constant. This allows us to find \(n_2\), the number of moles required at the new pressure, by rearranging and solving the equation. The moles that must be removed can then be determined by simply subtracting \(n_2\) from \(n_1\). Each step in this calculation is crucial for accurately determining how much gas needs adjustment in such scenarios.

Pressure Reduction

Understanding pressure reduction is an integral part of gas calculations using the ideal gas law. Pressure is the force exerted by gas molecules as they collide with the walls of their container. In this problem, we start at a high pressure \(38\) atm, and aim to reduce it to \(25\) atm. Pressure reduction is achieved by altering the amount of gas in the container, while keeping volume and temperature constant. The ideal gas law relationship \(PV = nRT\) tells us that as pressure decreases, the number of moles \(n\) must also decrease if \(V\) and \(T\) are consistent. This mathematical link enables us to deduce the moles that need to be removed.Through careful manipulation of this relationship, one can calculate the exact number of moles or mass that need to be extracted to achieve the desired pressure.

Molecular Weight

Molecular weight, also referred to as molar mass, is the mass of one mole of a substance. For nitrogen gas \(\mathrm{N}_2\), this is approximately \(28.02\) g/mol. This property is fundamental when converting between moles and grams. Once we know the number of moles to remove (\(n_{\text{removed}}\)), we use the molecular weight to convert this into a measurable mass:\[ \text{Mass to be removed} = n_{\text{removed}} \times \text{molecular weight of} \ \mathrm{N}_2 \]Using consistent units (mol and g/mol) ensures accurate conversion. Here, knowing the molecular weight of \(\mathrm{N}_2\) allows us to determine that \(0.2908\) mol corresponds to approximately \(8.15\) grams that need removal. Understanding molecular weight is crucial for practical applications, like adjusting gas quantities in tanks, which directly relate to chemical reactions or processes.