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

The neon sign contains approximately 0.482 grams of neon.

Step-by-step solution

Calculate the volume of the glass tubing

To find the volume of the glass tubing, we'll use the formula for the volume of a cylinder: \(V = \pi r^{2}h\). First, we need to find the radius (r) of the cylinder. Since we're given the diameter (2.5 cm), we can divide it by 2 to find the radius: \(r = 1.25 cm\) (or 0.0125 m, since we need the value in meters). Then using the radius and the length (h) of 5.5 m, we can calculate the volume as follows: \[ V = \pi (0.0125)^{2}(5.5) \approx 0.00269 \, m^3 \]

Convert the pressure from torr to atm

We're given the pressure in torr (1.78 torr), so we need to convert it to atm using the conversion factor: 1 atm = 760 torr. So, the pressure in atm is: \[ P = \frac{1.78 \, \text{torr}}{760 \, \text{torr/atm}} \approx 0.00234 \, \text{atm} \]

Convert the temperature from Celsius to Kelvin

We're given the temperature in degrees Celsius (35°C), and we need to convert it to Kelvin using the conversion formula: \(T(K) = T(°C) + 273.15\). So, the temperature in Kelvin is: \[ T = 35 + 273.15 = 308.15 \, K \]

Use the Ideal Gas Law to solve for number of moles

The Ideal Gas Law is given by the equation: \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (0.08206 L atm/mol K), and T is the temperature. We'll need to plug in the values and solve for n: \[ n = \frac{PV}{RT} = \frac{(0.00234 \, atm)(0.00269 \, m^3)}{(0.08206 \, L\,atm/mol\,K)(308.15 \,K)} \] First, we'll need to convert the volume from m³ to L: \[ 0.00269 \,m^3 \times \frac{1000 \, L}{1\, m^3} = 2.69 \, L \] Now, we can determine the moles (n): \[ n = \frac{(0.00234 \, atm)(2.69 \, L)}{(0.08206 \, L \, atm/mol \, K)(308.15 \, K)} \approx 0.0239 \, mol \]

Convert moles of neon to grams

To convert moles of neon to grams, we need the molar mass of neon, which is approximately 20.18 g/mol. Using the moles calculated above, we can find the mass of neon as follows: \[ m = n \times \text{molar mass} = (0.0239 \, mol)(20.18 \, g/mol) \approx 0.482 \, g \] Thus, there are approximately 0.482 grams of neon in the sign.

Key concepts

Moles to Grams Conversion

Understanding how to convert moles to grams is crucial in chemistry, especially when dealing with the quantification of substances in a reaction or contained within a system like our neon sign. To begin, we need to know the molar mass of the substance in question, which is the mass of one mole of that substance. The molar mass of an element can be found on the periodic table and is expressed in grams per mole (g/mol).

In the example given, to convert the number of moles of neon gas to grams, we multiply the moles of neon by its molar mass. The calculation is straightforward: if we have the number of moles (n) of neon and the molar mass (Mm) of neon, the mass (m) in grams is simply calculated by the formula:
\[ m = n \times Mm \.\] For instance, with neon having a molar mass of 20.18 g/mol, your calculation will determine the mass in the neon sign efficiently. This step is vital for translating the abstract concept of moles into tangible grams that can be measured on a scale.

Cylinder Volume Calculation

When we talk about the volume of a cylinder in the context of the ideal gas law, it’s important to understand it as the space inside the cylinder that can be occupied by a gas. The volume of the cylinder is found using the formula \[ V = \pi r^{2} h \.\]

In this formula, \( V \) represents the volume, \( r \) is the radius of the base of the cylinder, \( h \) is the height or length of the cylinder, and \( \pi \) is a constant approximately equal to 3.14159. To ensure proper measurements, we need to convert all our dimensions to the same unit system; in many scientific contexts, metric units such as meters (m) or liters (L) are used.

For the problem at hand, the precise calculation of the cylinder volume was key in determining the amount of neon gas. The volume found was then used in the ideal gas law to find the moles of neon, which we ultimately converted to grams. The cylinder volume calculation is not just a fundamental geometric exercise but a building block in solving more complex chemistry problems.

Pressure and Temperature Conversions

In working with the ideal gas law, both pressure and temperature conversions are critical to achieving accurate results. For pressure, standard units are often atmospheres (atm), but measurements might be taken in other units such as torr or mmHg. To convert between these units, we use the relationship \[ 1 \, \text{atm} = 760 \, \text{torr} \.\] The given problem required converting torr to atm to maintain consistency with the gas constant used in the ideal gas law equation.

Regarding temperature, scientific calculations typically require temperatures to be in Kelvin (K) rather than Celsius (°C) or Fahrenheit (°F). Kelvin is the unit of the thermodynamic temperature scale and is crucial in gas laws because it ensures proportionality in the relationship between temperature and volume or pressure. We convert Celsius to Kelvin with the formula: \[ T(K) = T(°C) + 273.15 \.\]

In our neon sign problem, proper conversion of pressure and temperature was indispensable for the calculation. It allowed us to use the ideal gas law correctly, helping us to solve for the moles of neon and subsequently convert this value to grams. Grasping these conversion concepts provides a strong foundation for working with gases in a variety of chemical contexts.