Problem 36 - Chemistry: The Central Science in SI Units, Global Edition

Question

A neon sign is made of glass tubing whose inside diameter is \(3.0 \mathrm{~cm}\) and length is \(10.0 \mathrm{~m}\). If the sign contains neon at a pressure of \(265 \mathrm{~Pa}\) at \(30^{\circ} \mathrm{C}\), how many grams of neon are in the sign? (The volume of a cylinder is \(\pi r^{2} h\).)

Step-by-Step Solution

Answer

There are approximately 0.159 grams of neon gas in the sign.

Step 1: Find the volume of the cylinder

To find the volume of the cylinder, we will use the formula provided: \(V = \pi r^{2} h\) Since the inside diameter of the sign is 3.0 cm, we can find the radius (r) by dividing it by 2: \(r = \frac{3.0 cm}{2} = 1.5 cm\) Now, convert the radius to meters by dividing by 100, and then use the given length (h) which is already in meters: \(r = \frac{1.5 cm}{100} = 0.015 m\) \(h = 10.0 m\) Now, plug the values of r and h into the formula and compute the volume: \(V = \pi (0.015 m)^{2} (10.0 m)\) \(V = \pi (0.000225 m^2) (10.0 m)\) \(V = \pi (0.00225 m^3)\) \(V \approx 0.00707 m^3\)

Step 2: Use the ideal gas law to find the number of moles of neon gas

The ideal gas law is given by: \(PV = nRT\) Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. We are given the pressure (P) and temperature (T) and have calculated the volume (V). The ideal gas constant R for calculation in SI units is 8.314 J/(mol·K). To find the number of moles (n), we can rearrange the formula and solve for n: \(n = \frac{PV}{RT}\) First, convert the temperature from Celsius to Kelvin by adding 273.15: \(T = 30^{\circ}C + 273.15 = 303.15 K\) Now, plug the values of P, V, and T into the equation: \(n = \frac{(265 Pa)(0.00707 m^{3})}{(8.314 J/(mol·K))(303.15 K)}\) \(n \approx 0.00791 mol\)

Step 3: Convert the number of moles to grams

To find the mass of neon gas in grams, we can use the molar mass of neon, which is 20.18 g/mol. Multiply the number of moles by the molar mass to find the mass in grams: \(mass = n \times molar\ mass\) \(mass = (0.00791 mol)(20.18 g/mol)\) \(mass \approx 0.159 g\) So, there are approximately 0.159 grams of neon gas in the sign.

Key Concepts

Cylinder Volume

To understand how to calculate the volume of a cylinder, think of a can of soup or any tube-shaped object. The volume is how much space is inside this shape. For cylinders, the volume is found with the formula: \[ V = \pi r^2 h \] Here, \( V \) is the volume, \( r \) is the radius of the cylinder's base, and \( h \) is its height. In this case, the base of the neon sign is a circle. To find \( r \), simply divide the given diameter by 2. In this problem, the inside diameter is 3.0 cm, so the radius \( r \) becomes \( 1.5 \) cm.

• Important: Convert all units to meters for consistency in calculations. Therefore, \( 1.5 \) cm converts to \( 0.015 \) m.
• The height in this scenario is given directly as \( 10.0 \) m.

With these values, the volume of the cylinder can be calculated using the formula to get approximately \( 0.00707 \) cubic meters (\( m^3 \)). This volume tells us how much space the neon gas occupies inside the tube.

Moles Calculation

The ideal gas law is a powerful equation used to relate pressure, volume, and temperature with the amount of gas in moles. The formula is: \[ PV = nRT \] Where:

• \( P \) is the pressure (in pascals, Pa),
• \( V \) is the volume (in cubic meters, \( m^3 \)),
• \( n \) is the number of moles (our variable of interest),
• \( R \) is the ideal gas constant (8.314 J/(mol·K)),
• \( T \) is the temperature (in Kelvin).

To solve for \( n \) (number of moles), rearrange the formula: \[ n = \frac{PV}{RT} \] In this lesson:

• Pressure \( P \) is given as \( 265 \) Pa.
• Temperature must be converted from Celsius to Kelvin by adding \( 273.15 \). The given temperature of 30°C becomes \( 303.15 \) K.
• Volume has been found as \( 0.00707 \ m^3\).

By plugging these values into the rearranged equation, you get approx \( 0.00791 \) moles of neon gas inside the cylinder. The calculation demonstrates how gases behave under different physical conditions.

Molar Mass

Molar mass is an important concept in chemistry. It refers to the mass of one mole of a substance, providing the bridge to convert between moles and grams. The units of molar mass are typically grams per mole (g/mol). The periodic table shows that the molar mass of neon (Ne) is 20.18 g/mol. This is useful for converting moles of neon gas into grams, which is often required in solving problems involving substances like gases. Conversion to grams is straightforward with the formula: \[ \text{mass} = n \times \text{molar mass} \] Where \( n \) is the number of moles. Insert the known values for neon in this scenario:

• The calculated \( n \) (0.00791 mol).
• The molar mass of neon (20.18 g/mol).

The resulting mass is \( \approx 0.159 \) grams of neon gas. This tells us that the total amount of neon in the tube is around \( 0.159 \) grams, highlighting the step-by-step conversion from moles to a more tangible measure, such as grams.