To find the mass of neon gas contained in the sign, we can use the Ideal Gas Law, which is represented as:
\[ PV = nRT \]
where:
- \( P \) is the pressure in atmospheres,
- \( V \) is the volume in liters,
- \( n \) is the number of moles of gas,
- \( R \) is the ideal gas constant (0.0821 L·atm/(K·mol)),
- \( T \) is the temperature in Kelvin.
Step 1: Convert Pressure and Temperature
First, we need to convert the pressure from torr to atm and temperature from Celsius to Kelvin.
- Pressure in atm:
\[ P = 1.78 , \text{torr} \times \left(\frac{1 , \text{atm}}{760 , \text{torr}}\right) = 0.0023421 , \text{atm} \]
- Temperature in Kelvin:
\[ T = 33 , \text{°C} + 273.15 = 306.15 , \text{K} \]
Step 2: Calculate Moles of Neon (n)
Assuming we know the volume of the sign or cylinder, denoted as \( V \), we can plug the values into the ideal gas law. However, since volume was not given, let's represent \( V \) in the calculations assuming you have it.
Rearranging the Ideal Gas Law gives:
\[ n = \frac{PV}{RT} \]
Step 3: Substitute Values
Plugging in the values for \( P, R, \) and \( T \):
\[ n = \frac{(0.0023421 , \text{atm}) \times V}{(0.0821 , \text{L·atm/(K·mol)}) \times (306.15 , \text{K})} \]
Calculating the denominator:
\[ RT = 0.0821 \times 306.15 \approx 25.1872 \]
Now we calculate \( n \):
\[ n = \frac{0.0023421 \times V}{25.1872} \]
Step 4: Convert Moles to Grams
To find the mass of neon, we use the molar mass of neon:
- Molar mass of neon (Ne) = 20.18 g/mol.
The mass \( m \) in grams can be found using:
\[ m = n \times \text{molar mass} \]
Thus:
\[ m = \left( \frac{(0.0023421 \times V)}{25.1872} \right) \times 20.18 \]
\[ m \approx \frac{0.0472398 , V}{25.1872} \]
Final Step: Presenting the Result
At this point without a specific volume \( V \), we cannot obtain a numeric solution. However, the final relationship expressing the mass of neon contained in the sign is found as follows:
\[ m \approx 0.0018746 , V , \text{grams} \]
To have a numeric answer, please provide the volume of the cylinder/signed space (in liters) to calculate the mass in grams.
If you assume a certain volume, plug it into (\( V \)) to get your grams of neon in the sign. Don’t forget to round your answer according to significant figures once you have \( V \).