Question

A neon sign is made of glass tubing whose inside diameter is \(2.5 \mathrm{~cm}\) and whose length is \(5.5 \mathrm{~m}\). If the sign contains neon at a pressure of \(1.78\) torr at \(35^{\circ} \mathrm{C}\), how many grams of neon are in the sign? (The volume of a cylinder is \(\pi r^{2} h\).)

Short answer

There are approximately 0.00624 grams of neon in the sign.

Step-by-step solution

Identify the given values

We are given the following information: - Inside diameter of the glass tubing = 2.5 cm = 0.025 m (converted to meters) - Length of the glass tubing = 5.5 m - Pressure, P = 1.78 torr - Temperature, T = 35°C Additionally, we are provided with the molar gas constant, R = 8.314 J/(mol·K).

Convert given values to appropriate units

We first need to convert the temperature from Celsius to Kelvin and the pressure from torr to Pa (Pascals). T = 35°C = 35 + 273.15 = 308.15 K P = 1.78 torr × 101325 Pa / 760 torr = 239.43 Pa

Calculate the volume of the tubing using the formula for the volume of a cylinder

The inside diameter of the tubing is given as 2.5 cm, so the radius is half of that: r = 1.25 cm = 0.0125 m. We can now calculate the volume, V, using the formula for the volume of a cylinder: \(V = \pi r^2h\). V = \(\pi × (0.0125)^2 × 5.5\) V ≈ 0.00267 m³

Use the Ideal Gas Law to find the number of moles (n) of the neon gas

Now that we have all our given values in appropriate units, we can find the number of moles of neon gas using the Ideal Gas Law formula: \(PV = nRT\). Rearrange the formula to solve for n: \( n = \frac{PV}{RT}\). n = \(\frac{(239.43\,\text{Pa})(0.00267\,\text{m}^3)}{(8.314\,\text{J/mol·K})(308.15\,\text{K})}\) n ≈ 0.0003091 mol

Convert moles of neon gas to grams

Finally, we convert the number of moles (n) of neon gas to grams using the molar mass of neon, which is 20.18 g/mol. mass = n × molar mass of neon mass = 0.0003091 mol × 20.18 g/mol ≈ 0.00624 g So, there are approximately 0.00624 grams of neon in the sign.

Key concepts

Neon Gas

Neon gas, a noble gas, is one of the lightest elements in its group. It is non-reactive, odorless, colorless, and known for producing a bright glow when electricity passes through it. Neon is commonly used in signage and lighting because of its striking luminescence. Its chemical symbol is Ne, and it exists in a gaseous state under standard conditions. Since it's a noble gas, it does not easily form compounds with other elements. To use neon in lighting, it is contained within sealed glass tubes and subjected to a strong electrical current.
This current excites the neon atoms, causing them to emit light. The color of the light can vary depending on the gas pressure and the type of glass used in the tubing, but neon itself typically glows bright orange-red. Because of these properties, neon has practical applications in advertising and decoration, providing a vivid visual effect for both commercial and artistic purposes.

Cylinder Volume Calculation

Calculating the volume of a cylinder is an important step when dealing with gases in tubing, such as in neon signs. A cylinder's volume can be determined using the formula: \[ V = \pi r^2 h \]where \( r \) is the radius of the cylinder and \( h \) is the height (or length).
In this exercise, the inside diameter of the glass tubing is given as 2.5 cm. To find the radius, simply divide the diameter by 2, which provides a radius of 1.25 cm, or 0.0125 m after conversion to meters. Then, the volume of the tubing with a length of 5.5 m is calculated.
This formula helps in determining the container's capacity to hold gases at a certain pressure and temperature, playing a key role in executing the Ideal Gas Law for practical calculations, such as determining amounts of gases for industrial uses.

Pressure Conversion

In many chemical problems, pressure is not provided in the SI unit of Pascals (Pa), so a conversion is necessary. In this exercise, the initial pressure of neon gas is given in torr, a unit commonly used in vacuum and low-pressure measurements. To employ the Ideal Gas Law properly, we convert this pressure to Pascals. To convert from torr to Pa: - Multiply the pressure in torr by 101325 Pa / 760 torr. - This conversion factor arises because 1 atm is defined as 101325 Pa and is equivalent to 760 torr.
Following this conversion allows the values to be consistent with the units of the Ideal Gas Law constant, R, which is used as 8.314 J/(mol·K) here, ensuring accurate calculation of number of moles. Correct pressure conversion is crucial in achieving the right outcomes when performing such thermodynamic calculations.

Molar Mass of Neon

The molar mass of neon is a fixed constant used to convert between moles and grams. The molar mass, also known as molecular weight, for neon is set at 20.18 grams per mole. This means that one mole of neon atoms has a mass of 20.18 grams.In calculations involving gases, such as in the ideal gas law, it is often necessary to determine how many grams of a gas are present given a certain pressure, volume, and temperature. Here, after finding the number of moles \( n \) using the Ideal Gas Law equation:
\[ n = \frac{PV}{RT} \]the mass in grams can be easily found by multiplying the number of moles by the molar mass. This conversion is pivotal in chemical processes and applications, translating theoretical moles into practical, measurable quantities important for experimentation and industrial applications.