A neon sign consists of glass tubing with an inside diameter of 2.5 cm and a length of 5.5 m. If the sign contains neon at a pressure of 1.76 torr at 38°C, how many grams of neon does it hold? (The volume of a cylinder is πr²h.)

To determine the mass of neon in the sign, let's break down the process into a series of steps.

Step 1: Find the volume of the neon gas. Given the inside diameter of the tubing is 2.5 cm, we can find the radius, r, which is half the diameter. r = diameter / 2 = 2.5 cm / 2 = 1.25 cm

Convert the radius into meters for convenience, as standard units of volume are cubic meters (m³). 1 cm = 0.01 m, so r = 1.25 cm * 0.01 m/cm = 0.0125 m

Next, we calculate the volume, V, using the formula for the volume of a cylinder: V = πr²h h is the length of the tubing, which is equal to 5.5 m. V = π * (0.0125 m)² * 5.5 m

Calculating the volume, we get: V = 3.1416 * (0.00015625 m²) * 5.5 m ≈ 0.00269 m³

Step 2: Convert the pressure to atmospheres. 1 atm = 760 torr, so to convert 1.76 torr to atmospheres, you do the following: Pressure = 1.76 torr * (1 atm / 760 torr) ≈ 0.00232 atm

Step 3: Convert the temperature to Kelvin. K = °C + 273.15 K = 38°C + 273.15 ≈ 311.15 K

Step 4: Use the Ideal Gas Law to find the number of moles of neon, n. PV = nRT Where P is the pressure in atmospheres, V is the volume in m³, n is the number of moles, R is the gas constant (0.0821 L·atm/(mol·K) or 8.314 m³·Pa/(mol·K)), and T is the temperature in Kelvin.

First, we need to ensure the units match. Let's use the gas constant in standard units of m³·Pa/(mol·K) because the volume is in m³ and pressure in atm needs to be converted to Pascals. We know that 1 atm = 101325 Pascals.

So our pressure in Pascals is: Pressure = 0.00232 atm * 101325 Pa/atm ≈ 235.075 Pa

Now the Ideal Gas Law looks like this: n = PV / RT R = 8.314 m³·Pa/(mol·K), substituting we get: n ≈ (235.075 Pa * 0.00269 m³) / (8.314 m³·Pa/(mol·K) * 311.15 K)

Solving for n gives us: n ≈ (0.00063245725 Pa·m³) / (2588.5519 Pa·m³/(mol·K)) n ≈ 0.000244381 moles

Step 5: Convert moles of neon to grams. The molar mass of neon (Ne) is approximately 20.18 g/mol. Therefore, mass is: mass = n * molar mass mass = 0.000244381 moles * 20.18 g/mol

Finally, the mass of the neon is: mass ≈ 0.000244381 moles * 20.18 g/mol ≈ 0.00493 g

So, the neon sign contains approximately 0.00493 grams of neon.