Problem 1 - University Physics with Modern Physics

Question

A \(20.0-\mathrm{L}\) tank contains 0.225 \(\mathrm{kg}\) of helium at \(18.0^{\circ} \mathrm{C}\) . The molar mass of helium is 4.00 \(\mathrm{g} / \mathrm{inol}\) (a) How many moles of helium are in the tank? (b) What is the pressure in the tank, in pascals and in atmospheres?

Step-by-Step Solution

Answer

There are 56.25 moles of helium in the tank. The pressure is approximately 6.82 MPa or 67.27 atm.

Step 1: Convert Mass to Moles

To convert mass to moles, use the formula: \[ n = \frac{m}{M} \]where \( m = 0.225 \text{ kg} \) and \( M = 4.00 \text{ g/mol} \) (converted to kg: \( 0.004 \text{ kg/mol} \)). \[ n = \frac{0.225}{0.004} = 56.25 \text{ moles} \]

Step 2: Use Ideal Gas Law to Find Pressure

We use the Ideal Gas Law \( PV = nRT \) to find the pressure. Given: \( V = 20.0 \text{ L} = 0.020 \text{ m}^3 \), \( n = 56.25 \text{ moles} \), \( R = 8.314 \text{ J/(mol} \cdot \text{K)} \), \( T = 18.0 + 273.15 = 291.15 \text{ K} \). Find \( P \): \[ P = \frac{nRT}{V} = \frac{56.25 \times 8.314 \times 291.15}{0.020} \]

Step 3: Calculate Pressure in Pascals

Perform the calculation to find \( P \) in pascals: \[ P = \frac{56.25 \times 8.314 \times 291.15}{0.020} = 6816122.4375 \, \text{Pa} \]Therefore, the pressure is approximately \( 6.82 \times 10^6 \text{ Pa} \).

Step 4: Convert Pressure to Atmospheres

To convert pressure from pascals to atmospheres, use:\[ 1 \text{ atm} = 101325 \text{ Pa} \]\[ P_{\text{atm}} = \frac{6816122.4375}{101325} \approx 67.27 \text{ atm} \]Thus, the pressure is approximately \( 67.27 \text{ atm} \).

Key Concepts

Moles Calculation

To determine the number of moles in a given mass of a substance, we use a simple formula from chemistry. Moles tell us how many atoms or molecules are present in a sample. The formula to calculate moles from mass is:

• \[ n = \frac{m}{M} \]

Where:

• \( n \) is the number of moles.


• \( m \) is the mass of the substance in kilograms.


• \( M \) is the molar mass of the substance in kilograms per mole.

For the specific problem involving helium gas, we start with 0.225 kg of helium. The molar mass of helium is known to be 4.00 g/mol. We need to convert this molar mass into kilograms per mole (\( 0.004 \text{ kg/mol} \) ). Now, we use the formula for calculating moles:

• \[ n = \frac{0.225}{0.004} = 56.25 \text{ moles} \]

This means there are 56.25 moles of helium in the tank.

Pressure Conversion

Converting pressure from one unit to another can be crucial in understanding the behavior of gases. Often, pressure is measured in Pascals (Pa) or atmospheres (atm). To fully grasp pressure conversion, let's see how this is done.
Start by calculating the pressure in Pascals using the Ideal Gas Law, given by:

• \[ PV = nRT \]

Once you've calculated the pressure (\( P \) ) from the Ideal Gas Law, you might need to convert it into a more convenient or familiar unit such as atmospheres. One atmosphere is equal to 101325 Pascals.

• The formula for converting Pascals to atmospheres is:


• \[ P_{\text{atm}} = \frac{P_{\text{Pa}}}{101325} \]

In this example, once we find the pressure to be approximately 6816122.4375 Pa, converting to atmospheres involves dividing by 101325:

• \[ P_{\text{atm}} = \frac{6816122.4375}{101325} \approx 67.27 \text{ atm} \]

Pressure conversion helps make sense of high or low-pressure scenarios depending on your context or needs.

Molar Mass

Molar mass is a crucial concept in chemistry that links mass, moles, and molecules. It is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For calculating chemical quantities, it's vital to know the molar mass of the substances involved.
Understanding molar mass allows us to convert a substance's mass to moles, enabling different chemical calculations, such as those involving gas laws. To perform these conversions, one must remember:

• The molar mass of a substance is typically found on the periodic table.


• For our case, helium has a molar mass of 4.00 g/mol, which is equivalent to 0.004 kg/mol.

Knowing this simple conversion enables us to tackle problems involving gases like helium in terms of mass and moles.

• For example, converting the total mass of helium in a tank to moles involves dividing by this constant molar mass.

Thus, molar mass is integral to correctly applying equations in chemistry, especially those involving the Ideal Gas Law, and grasping the basic principles of chemical conversions.