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 0.00657 grams of neon in the sign.

Step-by-step solution

Identify the Ideal Gas Law equation

The Ideal Gas Law equation is written as \(PV = nRT\), where P is the pressure, V is the volume, n is the amount of gas in moles, R is the gas constant, and T is the temperature in Kelvin.

Set up equations to convert pressure and temperature

In the given problem, we have the pressure in torr and the temperature in Celsius. We need to convert these values to consistent units with the Ideal Gas Law. Convert the pressure to atm and temperature to Kelvin. Pressure: \(1 \text{ atm} = 760 \text{ torr}\) Temperature: \(T_\text{K} = T_\text{C} + 273.15\)

Convert pressure and temperature to their respective units

Pressure: \(P_\text{atm} = \frac{1.78 \text{ torr}}{760 \text{ torr/atm}} = 0.0023421 \text{ atm}\) Temperature: \(T_\text{K} = 35^\circ\text{C} + 273.15 = 308.15 \, \text{K}\)

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

The volume of a cylinder formula is given by \(V = \pi r^2h\), where r is the radius and h is the height. Using the given diameter: \(d = 2.5 \mathrm{~cm} \Rightarrow r = \frac{d}{2} = 1.25\mathrm{~cm}\) (convert cm to meters) \(r = 0.0125\mathrm{~m}\) Length of the tubing \(h = 5.5 \mathrm{~m}\) Now we'll compute the volume of the tubing: \(V = \pi (0.0125\mathrm{~m})^2 (5.5 \mathrm{~m}) = 0.0026945 \, \text{m}^3\)

Determine the number of moles of neon using the Ideal Gas Law and the known gas constant

We'll use the Ideal Gas Law equation, rearranging for moles and using the gas constant value for atm. \(n = \frac{PV}{RT}\) \(n = \frac{(0.0023421 \, \text{atm})(0.0026945 \, \text{m}^3)}{(0.0821 \, \text{L atm/mol K})(308.15 \, \text{K})} = 0.00032568 \, \text{mol}\)

Calculate the mass of neon using the molar mass of neon

Given that neon has a molar mass of 20.18 grams per mole, we can calculate the mass of the neon in the tubing: Mass of neon \(= n \times Molar \, mass \, of \, neon\) Mass of neon \(= (0.00032568 \, \text{mol})(20.18 \, \text{g/mol}) = 0.0065689 \, \text{g}\) So, there are approximately 0.00657 grams of neon in the sign.

Key concepts

Gas Pressure Conversion

Understanding how to convert gas pressure between different units is crucial for applying the Ideal Gas Law accurately. In the problem, neon gas pressure is given in "torr" but needs to be converted to "atmospheres" (atm) for use in the Ideal Gas Law.

• Atmospheric pressure is a standard measurement used in gas law calculations.
• Torr is a unit of pressure originating from the millimetre of mercury (mmHg).
• The conversion between torr and atm is essential for solving pressure-related problems in chemistry.

To convert, note that 1 atm = 760 torr. Therefore, to convert the pressure from torr to atm, you divide the given pressure (in torr) by 760. For example, if the pressure is 1.78 torr, the conversion to atm is:\[P_{\text{atm}} = \frac{1.78\, \text{torr}}{760\, \text{torr/atm}} = 0.0023421 \text{ atm}\]This conversion is an integral step to solve the problem using the Ideal Gas Law.

Volume of a Cylinder

The volume of a cylindrical object is a three-dimensional measurement that describes how much space the cylinder occupies. For geometric shaped objects like cylindrical glass tubes, it's easy to calculate the volume when you know the formula:

• The volume of a cylinder is calculated using the formula: \(V = \pi r^{2} h\).
• "\(\pi\)" (pi) is a constant approximately equal to 3.14159.
• "\(r\)" stands for the radius, which is half of the cylinder's diameter.
• "\(h\)" is the total length or height of the cylinder.

For our exercise, convert radius to meters as a standard unit and apply them to the formula. Given:\(\text{diameter} = 2.5\,\mathrm{cm} \), so \( r = \frac{2.5}{2} = 1.25\, \text{cm} = 0.0125\,\text{m}\) and\( h = 5.5\,\text{m} \). The volume calculation will be:\[V = \pi (0.0125\, \text{m})^2 (5.5\,\text{m}) = 0.0026945 \text{ m}^3\]This formula will allow you to find the volume necessary to compute the number of moles of neon in the tube using the Ideal Gas Law.

Temperature Conversion

Temperature conversion is necessary when dealing with gas laws because these formulas need temperature inputs in Kelvin to function properly. The Kelvin scale is an absolute temperature scale starting at absolute zero, which is where all molecular motion stops.

• The Celsius to Kelvin conversion formula is: \(T_{\text{K}} = T_{\text{C}} + 273.15\).
• "Kelvin (K)" is the SI unit of thermodynamic temperature.
• This conversion handles temperatures above absolute zero, ensuring calculations are correct with the gas constant used in the Ideal Gas Law.

In our problem, we convert \(35^{\circ} \text{C}\) to Kelvin:\[T_{\text{K}} = 35^{\circ}\text{C} + 273.15 = 308.15 \text{K}\]Using Kelvin ensures precise calculations when applying the Ideal Gas Law to find the amount of neon in the sign.

Molar Mass Calculation

Molar mass, also known as molecular weight, is a measure of the mass of one mole of a substance. It serves as the link between the amount of substance (in moles) and the mass (in grams), which is critical for converting the results from the Ideal Gas Law into meaningful quantities. For neon:

• The molar mass of neon is 20.18 g/mol.
• This value is crucial for finding out how much 0.00032568 moles of neon weigh.
• You multiply the number of moles by the molar mass to find the mass in grams.

Using the moles we calculated earlier, the mass of neon is found by:\[\text{Mass of neon} = (0.00032568\, \text{mol}) \times (20.18\, \text{g/mol}) = 0.0065689 \text{ g}\]This tells you the total grams of neon gas present in the sign's tubing, completing the conversion from moles via the molar mass.