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To solve this problem, we can apply the **Ideal Gas Law**, which is given by the equation:

\[
PV = nRT
\]

Where:
- \( P \) is the pressure (in Pascals),
- \( V \) is the volume of the gas (in cubic meters),
- \( n \) is the number of moles of gas,
- \( R \) is the ideal gas constant, \( 8.314 \, \text{J/(mol·K)} \),
- \( T \) is the temperature (in Kelvin).

### Step 1: Converting Given Values
1. **Pressure**: The pressure is given as \( 7.40 \, \text{MPa} \), which we need to convert to Pascals (Pa):
   \[
   7.40 \, \text{MPa} = 7.40 \times 10^6 \, \text{Pa}
   \]

2. **Volume of the Vessel**: The vessel is a cylinder with a diameter of \( 31.0 \, \text{cm} \) and height \( 37.2 \, \text{cm} \). We need to calculate the volume of the cylinder using the formula for the volume of a cylinder:
   \[
   V = \pi r^2 h
   \]
   First, convert the dimensions to meters:
   \[
   r = \frac{31.0 \, \text{cm}}{2} = 0.155 \, \text{m}, \quad h = 37.2 \, \text{cm} = 0.372 \, \text{m}
   \]
   Now calculate the volume \( V \):
   \[
   V = \pi (0.155)^2 (0.372) \, \text{m}^3
   \]

3. **Mass of Sulfur Hexafluoride**: The gas in the vessel is \( 4.48 \, \text{kg} \) of sulfur hexafluoride (\( \text{SF}_6 \)). To calculate the number of moles \( n \), we use the molar mass of sulfur hexafluoride, which is approximately \( 146.06 \, \text{g/mol} \). Thus, the number of moles is:
   \[
   n = \frac{4.48 \, \text{kg} \times 1000 \, \text{g/kg}}{146.06 \, \text{g/mol}}
   \]

### Step 2: Solving for Temperature
Rearrange the Ideal Gas Law to solve for temperature \( T \):
\[
T = \frac{PV}{nR}
\]
Substitute the known values to find \( T \).

Finally, convert \( T \) from Kelvin to degrees Celsius:
\[
T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15
\]

I'll compute the values next.The maximum safe operating temperature for the sulfur hexafluoride gas in the reaction vessel is approximately **541.6°C**.

Would you like further details on the calculations, or do you have any questions?

Here are five related questions you might find useful:
1. How does the ideal gas law apply to real gases under high pressure?
2. What safety factors are typically considered when calculating maximum safe operating temperatures?
3. How would the calculations change if the gas used were different, such as nitrogen?
4. What methods are used to ensure that pressure vessels do not exceed safe temperature limits?
5. How do variations in the volume of the vessel affect the maximum temperature?

**Tip:** Always check if the gas behaves ideally under given conditions, as the ideal gas law assumes no intermolecular forces, which may not hold at high pressures or low temperatures.

A chemical engineer must calculate the maximum safe operating temperature of a high-pressure gas reaction vessel. The vessel is a stainless-steel cylinder that measures 31.0 cm wide and 37.2 cm high. The maximum safe pressure inside the vessel has been measured to be 7.40 MPa. For a certain reaction, the vessel may contain up to 4.48 kg of sulfur hexafluoride gas. Calculate the maximum safe operating temperature the engineer should recommend for this reaction. Write your answer in degrees Celsius. Be sure your answer has the correct number of significant digits.

This problem involves calculating the maximum safe operating temperature for sulfur hexafluoride gas inside a high-pressure stainless-steel cylinder using the ideal gas law. Key steps include unit conversion, volume calculation of the cylinder, and applying the ideal gas equation to solve for temperature. The answer is provided in degrees Celsius with proper significant figures.

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