To determine the mass of neon in the sign, we'll use the Ideal Gas Law, which is given by the equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in liters)
- \( n \) = number of moles of gas
- \( R \) = ideal gas constant (0.0821 L·atm/(K·mol))
- \( T \) = temperature (in Kelvin)
Step 1: Convert Pressure from Torr to Atmospheres
First, we need to convert the pressure from torr to atmospheres. The conversion factor is:
\[ 1 \text{ atm} = 760 \text{ torr} \]
So, we find:
\[ P = \frac{1.78 \text{ torr}}{760 \text{ torr/atm}} \approx 0.0023421 \text{ atm} \]
Step 2: Convert Temperature from Celsius to Kelvin
Next, we convert the temperature from Celsius to Kelvin:
\[ T(K) = T(°C) + 273.15 = 33 + 273.15 = 306.15 \text{ K} \]
Step 3: Solve for Moles (n) using the Ideal Gas Law
We are missing the volume \( V \). To proceed further, we typically need a specific volume. For this example, let’s assume we have a volume \( V \) of 1 liter (you can replace it with another specific volume if you have it).
Using the Ideal Gas Law with \( V = 1 \text{ L} \):
\[ n = \frac{PV}{RT} \]
Substituting in our known values:
\[ n = \frac{(0.0023421 \text{ atm})(1 \text{ L})}{(0.0821 \text{ L·atm/(K·mol)})(306.15 \text{ K})} \]
Calculating the denominator:
\[ n = \frac{0.0023421}{0.025192} \]
\[ n \approx 0.093\text{ moles} \]
Step 4: Calculate the Mass of Neon
Now we find the mass of neon (Ne) using the molar mass of neon, approximately \( 20.18 \text{ g/mol} \):
\[ \text{mass} = n \times \text{molar mass} = 0.093 \text{ moles} \times 20.18 \text{ g/mol} \]
Calculating the mass:
\[ \text{mass} \approx 1.88 \text{ g} \]
Final Answer:
Thus, the mass of neon in the sign, rounded to two significant figures, is approximately:
\[ \boxed{1.9 \text{ g}} \]
Make sure to adjust the volume \( V \) if you have a different value for it to get the correct mass accordingly!