Problem 5 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

Incandescent light bulbs, which have been phased out in favour of less energy- consuming lighting, are filled with an inert gas to prevent the filament from burning. Find the mass of argon needed to fill a \(75.0 \mathrm{cm}^{3}\) light bulb to a pressure of 1.05 atm at \(25.0^{\circ} \mathrm{C} .(\text { Section } 8.2)\)

Step-by-Step Solution

Answer

To fill the light bulb, approximately 0.129 g of argon is needed.

Step 1: Convert Volume to Liters

The volume of the light bulb is given as \(75.0 \text{ cm}^3\). To use the ideal gas law, we need to convert this to liters: \(1 \text{ L} = 1000 \text{ cm}^3\), so: \[ V = \frac{75.0}{1000} = 0.075 \text{ L} \]

Step 2: Convert Temperature to Kelvin

The temperature is given as \(25.0^{\circ} \text{C}\). To convert it to Kelvin, use the formula: \(T(K) = T(^{\circ} \text{C}) + 273.15\): \[ T = 25.0 + 273.15 = 298.15 \text{ K} \]

Step 3: Use Ideal Gas Law to Find Moles

Use the ideal gas law \(PV = nRT\) to find the number of moles \(n\). Where: - \(P = 1.05 \text{ atm}\) - \(V = 0.075 \text{ L}\) - \(R = 0.0821 \text{ L atm} / \text{mol K}\) - \(T = 298.15 \text{ K}\)Substituting in these values: \[ n = \frac{PV}{RT} = \frac{1.05 \times 0.075}{0.0821 \times 298.15} \approx 0.00322 \text{ moles} \]

Step 4: Calculate Mass of Argon

We know the number of moles of argon and need to find its mass. The molar mass of argon \(M\) is \(39.95 \text{ g/mol}\). Using the formula \(\text{mass} = n \times M\): \[ \text{mass} = 0.00322 \times 39.95 \approx 0.129 \text{ g} \]

Key Concepts

Inert Gas

Inert gases, also known as noble gases, are a group of elements characterized by their very low reactivity. This low reactivity is due to their complete valence electron shell, making them stable and unlikely to engage in chemical reactions. This group includes elements like helium, neon, argon, krypton, xenon, and radon. To illuminate the function of inert gases, consider their use in incandescent light bulbs. Here, an inert gas is used to surround the filament, which prevents the filament from burning out too quickly. The gas does this by not reacting with the filament at high temperatures. Hence, an inert gas extends the lifespan of the bulb, a property highly valued before more energy-efficient lighting options became available. This makes inert gases exceptionally useful in various applications where stability is essential.

Argon

Argon is one of the noble gases, falling under the category of inert gases because of its non-reactive nature. It is colorless, odorless, and is found in our atmosphere, comprising about 0.93% of it. Argon’s inertness makes it an excellent choice for use in situations where materials need to be protected from the reactive environmental gases like oxygen. For instance, in the light bulb example given, argon fills the bulb to prevent the filament from oxidizing and burning out. Argon's stable character is crucial because it does not corrode or reacts under high temperatures, offering a safe environment that enhances the bulb's longevity. The myriad uses of argon stretch beyond lighting—it is also utilized in welding and other industrial processes where inert atmospheres are required.

Molar Mass

Molar mass is a fundamental concept in chemistry referring to the mass of a given substance (chemical element or chemical compound) divided by the amount of substance, measured in moles. It is expressed in units of grams per mole ( ext{g/mol}). For instance, the molar mass of argon is 39.95 ext{g/mol}. The value stem from the atomic weight of argon in the periodic table. To calculate the mass of a sample when the number of moles is known, you multiply the number of moles by the molar mass. For the light bulb problem, by knowing the molar amount of argon, we can easily determine the exact mass used in the bulb using this method. This relationship between moles and mass is vital in chemical calculations and is governed by the ideal gas law and other stoichiometric equations.

Temperature Conversion

Temperature conversion is essential in scientific calculations, especially when dealing with gases. Most gas law calculations require temperatures in Kelvin, because Kelvin is the absolute temperature scale aligning with the properties of gases described in equations like the ideal gas law. Convert Celsius temperatures to Kelvin with the formula: \[ T(K) = T(°C) + 273.15 \] This conversion is vital because Kelvin starts at absolute zero, where theoretically, molecular motion stops. Therefore, in the light bulb problem, converting 25°C to Kelvin equals 298.15 K, which then allows us to accurately use the ideal gas law to determine moles. For anyone dealing with any thermodynamic calculations, understanding how to convert between these temperature scales ensures precision and consistency in findings, a keystone in scientific endeavors.