Problem 93 - Chemistry Matter and Change

Question

Television Determine the pressure inside a television picture tube with a volume of 3.50 L that contains \(2.00 \times 10^{-5}\) g of nitrogen gas at \(22.0^{\circ} \mathrm{C}\) .

Step-by-Step Solution

Answer

The pressure inside the tube is approximately 4.94 x 10^-6 atm.

Step 1: Convert Grams to Moles

First, we need to find the number of moles of nitrogen gas present. The molecular weight of nitrogen (N2) is approximately 28 g/mol. So, we will convert the given mass of nitrogen gas into moles using the formula: \[ n = \frac{mass}{molecular\ weight} = \frac{2.00 \times 10^{-5}\ g}{28\ g/mol} \]Calculate this to get the number of moles.

Step 2: Apply the Ideal Gas Law

Now, we use the ideal gas law formula to find the pressure: \[ PV = nRT \] where \( P \) is the pressure, \( V \) is the volume (3.50 L), \( n \) is the number of moles calculated from Step 1, \( R \) is the ideal gas constant (0.0821 L atm/mol K), and \( T \) is the temperature in Kelvin. Convert the Celsius temperature to Kelvin by adding 273.15: \[ T = 22.0 + 273.15 = 295.15 \ K \] Rearrange for pressure: \[ P = \frac{nRT}{V} \] Substitute all the known values into this equation.

Step 3: Perform the Calculation

Substitute the values from previous steps into the pressure equation: \[ P = \frac{(7.14 \times 10^{-7}) \times 0.0821 \times 295.15}{3.50} \] Calculate the numerator and divide by the volume to find the pressure.

Step 4: Interpret the Result

The result from the calculation in Step 3 will give you the pressure in atmospheres. Make sure to verify the units and ensure the answer makes sense given the small amount of gas in a large volume.

Key Concepts

Conversion of Grams to Moles

In chemistry, the conversion of grams to moles is an essential step, especially when dealing with gases. It helps us understand how many molecules of a gas we have. The fundamental formula to use is:

• \( n = \frac{\text{mass of the substance}}{\text{molecular weight}} \)

To see an example, let's consider nitrogen gas. Nitrogen (\(N_2\)) has a molecular weight of about 28 g/mol. If you are given a tiny amount, say \(2.00 \times 10^{-5}\) g, you can find the number of moles:

• Plug the values into the formula: \( n = \frac{2.00 \times 10^{-5} \text{ g}}{28 \text{ g/mol}} \).

By performing this division, you convert the given mass (in grams) into moles. This is crucial since other calculations such as pressure often require the substance's quantity in moles. It's a straightforward process but very important to grasp.

Pressure Calculation

When dealing with gasses, pressure can be determined using the Ideal Gas Law. This law combines pressure, volume, the number of moles of gas, temperature, and a universal constant. The general formula is:

• \( PV = nRT \)

In this equation:

• \( P \) stands for pressure (in atmospheres)
• \( V \) is volume (in liters)
• \( n \) is the number of moles (calculated from the previous step)
• \( R \) is the ideal gas constant, roughly \(0.0821 \text{ L atm/mol K}\)
• \( T \) is the temperature in Kelvin

To find the pressure, rearrange the equation to solve for \( P \):

• \( P = \frac{nRT}{V} \)

Using this formula, one can knit together all the different parameters they've calculated or been given, discovering the internal pressure of a gaseous system. Understanding this allows you to predict how the gas will behave under different conditions.

Temperature Conversion to Kelvin

Temperature conversion to Kelvin is straightforward but crucial in gas calculations. Kelvin is the standard unit of temperature in the scientific world, particularly in gas laws. Converting from Celsius to Kelvin is necessary because the Ideal Gas Law requires temperature to be in Kelvin. Here's how you do it:

• Add 273.15 to the Celsius temperature.

For example, if the temperature is \(22.0^{\circ} \text{C}\):

• \( T = 22.0 + 273.15 = 295.15 \text{ K} \)

This temperature conversion ensures that calculations align with the constants used in scientific formulas. Using Kelvin makes it simpler to deal with temperature-related changes in a system, such as pressure or volume, as it starts at absolute zero, where all motion stops. It's a small adjustment but vital for precise scientific calculations.