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To calculate the maximum safe operating temperature of the high-pressure gas reaction vessel, we can use the **Ideal Gas Law** in its rearranged form, which is commonly used in situations like this one:

\[
PV = nRT
\]

Where:
- \( P \) is the pressure in pascals (Pa),
- \( V \) is the volume in cubic meters (m³),
- \( n \) is the number of moles of gas,
- \( R \) is the ideal gas constant \((R = 8.314 \, \text{J/mol·K})\),
- \( T \) is the temperature in kelvins (K).

We are given:
- Pressure \( P = 5.70 \, \text{MPa} = 5.70 \times 10^6 \, \text{Pa} \),
- Dimensions of the cylinder: diameter = 39.0 cm, height = 46.8 cm,
- Mass of sulfur hexafluoride gas (SF₆) = 9.24 kg.

### Step 1: Volume of the vessel
The volume of the vessel, assuming it's a cylinder, can be calculated using the formula:

\[
V = \pi r^2 h
\]

Where:
- \( r = \frac{39.0 \, \text{cm}}{2} = 19.5 \, \text{cm} = 0.195 \, \text{m} \),
- \( h = 46.8 \, \text{cm} = 0.468 \, \text{m} \).

Substitute these values into the volume formula:

\[
V = \pi (0.195)^2 \times 0.468 = 0.0557 \, \text{m}^3
\]

### Step 2: Moles of SF₆ gas
To find the number of moles \( n \), we use the molar mass of sulfur hexafluoride (SF₆), which is approximately:

\[
M_{\text{SF}_6} = 146.06 \, \text{g/mol} = 0.14606 \, \text{kg/mol}
\]

Now calculate the number of moles \( n \):

\[
n = \frac{9.24 \, \text{kg}}{0.14606 \, \text{kg/mol}} = 63.26 \, \text{mol}
\]

### Step 3: Solve for temperature \( T \)
Now rearrange the ideal gas law to solve for temperature \( T \):

\[
T = \frac{PV}{nR}
\]

Substitute the known values:

\[
T = \frac{(5.70 \times 10^6) \times 0.0557}{63.26 \times 8.314}
\]

First, calculate the numerator:

\[
(5.70 \times 10^6) \times 0.0557 = 317490 \, \text{Pa·m}^3
\]

Now calculate the denominator:

\[
63.26 \times 8.314 = 526.15 \, \text{mol·J/K}
\]

Finally, solve for \( T \):

\[
T = \frac{317490}{526.15} = 603.46 \, \text{K}
\]

### Step 4: Convert to Celsius
To convert kelvins to Celsius:

\[
T_{\text{Celsius}} = T_{\text{Kelvin}} - 273.15 = 603.46 - 273.15 = 330.31^\circ \text{C}
\]

### Final Answer:
The maximum safe operating temperature is approximately **330°C**.

---

Would you like more details or have any questions? Here are some related questions:

1. How is the Ideal Gas Law derived, and what are its limitations?
2. What are the assumptions when using the Ideal Gas Law in high-pressure scenarios?
3. How does the molar mass of a gas affect its behavior under pressure?
4. How would changes in the vessel volume affect the operating temperature?
5. What safety measures are typically considered when designing pressure vessels?

**Tip:** Always remember to check if the gas in question behaves ideally, especially at high pressures where deviations might occur.

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 39.0 cm wide and 46.8 cm high. The maximum safe pressure inside the vessel has been measured to be 5.70 MPa. For a certain reaction, the vessel may contain up to 9.24 kg of sulfur hexafluoride gas. Calculate the maximum safe operating temperature the engineer should recommend for this reaction. Write your answer in degrees Celsius.

This problem requires calculating the maximum safe operating temperature of a high-pressure vessel using the Ideal Gas Law. With a vessel containing 9.24 kg of sulfur hexafluoride gas, a diameter of 39.0 cm, and height of 46.8 cm, the maximum pressure is 5.70 MPa. This solution involves cylinder volume calculation, unit conversions, and application of the Ideal Gas Law.

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