Problem 100 - Chemistry: The Central Science in SI Units, Global Edition

Question

A \(500 \mathrm{~mL}\) incandescent light bulb is filled with \(1.5 \times 10^{-5}\) mol of xenon to minimize the rate of evaporation of the tungsten filament. What is the pressure of xenon in the light bulb at \(25^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The pressure of xenon gas in the light bulb at \(25^{\circ}\mathrm{C}\) can be found using the Ideal Gas Law, \(P = \frac{nRT}{V}\). Convert the given temperature to Kelvin, \(T(K) = 25 + 273.15 = 298.15\,\text{K}\). Convert volume to liters, \(V (L) = 500\,\text{mL} / 1,000 = 0.5\,\text{L}\). Substitute the known values into the equation: \(P = \frac{(1.5 \times 10^{-5}\,\text{mol})(0.0821 \, \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}})(298.15\,\text{K})}{0.5\,\text{L}}\). Calculate the pressure: \(P \approx 7.25 \times 10^{-4}\,\text{atm}\).

Step 1: Convert temperature to Kelvin

Since the given temperature is in Celsius, we have to convert it to Kelvin. To do so, we simply add 273.15 to the temperature in Celsius. \( T(K) = T(°C) + 273.15 \)

Step 2: Convert volume to liters

The volume given is in milliliters (mL), and we need it to be in liters (L) because the Ideal Gas Law requires volume in liters. To convert the volume from mL to L, we simply divide by 1,000. \( V(L) = V(mL) / 1,000 \)

Step 3: Apply the Ideal Gas Law

Now, we plug the known values into the Ideal Gas Law equation, PV = nRT, to find the pressure of the xenon gas. R is the ideal gas constant, which is 0.0821 L atm / (mol K). We know the moles of xenon gas (1.5 x 10⁻⁵ mol), the converted temperature in Kelvin, and the converted volume in liters. We need to rearrange the Ideal Gas Law formula to solve for pressure P. \( P = \frac{nRT}{V} \)

Step 4: Calculate pressure

Now, plug in the known values and calculate the pressure of xenon gas. \( P = \frac{(1.5 \times 10^{-5} \,\text{mol})(0.0821 \, \frac{\text{L} \cdot \text{atm}}{\text{mol} \cdot \text{K}} )(25 + 273.15 \,\text{K})}{500\text{mL} / 1,000} \) Make sure the units are consistent, and then solve for pressure P in atmospheres.

Key Concepts

gas pressure calculation

To determine the pressure of xenon gas in a light bulb, we use the Ideal Gas Law, an important equation in chemistry. The Ideal Gas Law formula is given by \( PV = nRT \). Here, \( P \) stands for pressure, \( V \) for volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

The main objective is to find the pressure, \( P \). We rearrange the equation to make \( P \) the subject, which results in the equation: \( P = \frac{nRT}{V} \).

• Moles (n): This is the amount of xenon gas present, which is given as \( 1.5 \times 10^{-5} \) mol.
• Gas Constant (R): The ideal gas constant \( R \) is \( 0.0821 \frac{\text{L atm}}{\text{mol K}} \).
• Temperature (T): Converted into Kelvin, it's \( 298.15 \) K. More on this in the next section.
• Volume (V): After converting to liters, it is \( 0.500 \) L. See the volume conversion section for details.

By plugging these values into the rearranged Ideal Gas Law equation, you can calculate the pressure accurately.

temperature conversion

Real-life problems often provide temperature in degrees Celsius, which is a practical scale for everyday use. However, when dealing with the Ideal Gas Law, it's essential to convert it to Kelvin. The Kelvin scale is used because it starts at absolute zero, which is the point where particles have minimal thermal motion.

To convert Celsius to Kelvin, a simple addition is required: add 273.15 to the Celsius temperature:

• For example, for \( 25^{\circ} \text{C} \), the conversion is \( T(K) = 25 + 273.15 = 298.15 \text{ K} \).

This conversion is crucial because the Kelvin scale ensures that the values used in the gas law are absolute, preventing negative numbers that could lead to errors in calculations.

volume conversion

The Ideal Gas Law requires volumes to be in liters. However, sometimes data is given in milliliters (mL), so a conversion is necessary. The conversion is straightforward since there are 1,000 milliliters in a liter.

To convert from milliliters to liters, you divide the number of milliliters by 1,000:

• For a given volume of \( 500 \text{ mL} \), the conversion is \( V(L) = 500/1,000 = 0.500 \text{ L} \).

This simple conversion allows the use of the Ideal Gas Law without any unit-related errors, ensuring accuracy in calculating properties like pressure.

xenon gas properties

Xenon is a noble gas known for its lack of reactivity. In applications like filling light bulbs, xenon is chosen because of its unique properties.

Here's why xenon is ideal in such settings:

• Low Reactivity: As a noble gas, xenon doesn't react with the tungsten filament, maintaining the integrity of the bulb's components.
• Dense Atmosphere: Xenon is heavier than air, which provides better insulation for the filament. This helps in minimizing the evaporation of tungsten, extending the bulb's life.
• Thermal Management: The density of xenon also contributes to better thermal management, which is crucial in maintaining the right pressure and preventing overheating.

Understanding these properties not only highlights the reasons behind its use but also aids in comprehending how the Ideal Gas Law applies to real-world scenarios where xenon gas is involved.