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The problem presented in the image involves determining the **maximum safe operating temperature** of a gas reaction vessel containing **sulfur hexafluoride (SF₆)**. Here's a breakdown of the provided information and how we might approach the problem:

### Given Data:
1. **Vessel dimensions**: 
   - Diameter: 39.0 cm 
   - Height: 46.8 cm
2. **Maximum safe pressure**: 5.70 MPa
3. **Mass of sulfur hexafluoride**: 9.24 kg
4. The temperature is requested in **degrees Celsius**, and the answer must include the correct number of significant figures.

### Approach:
To solve this, we'll likely use the **ideal gas law** \( PV = nRT \), where:
- \( P \) is pressure (in pascals),
- \( V \) is volume (in cubic meters),
- \( n \) is the number of moles of gas,
- \( R \) is the gas constant (for SF₆, \( R = 8.314 \, \text{J/mol·K} \)),
- \( T \) is temperature (in kelvins).

We'll need to:
1. **Convert dimensions** of the vessel to meters and compute the **volume**.
2. **Convert pressure** from MPa to Pascals (since \( 1 \, \text{MPa} = 10^6 \, \text{Pa} \)).
3. Calculate the number of **moles** of SF₆ using the molar mass (146.06 g/mol).
4. Rearrange the ideal gas law to solve for temperature \( T \).

Let's start solving step by step:

#### 1. **Volume of the vessel (cylinder)**
The volume of a cylinder is given by:
\[
V = \pi r^2 h
\]
Where:
- \( r = \frac{39.0}{2} \, \text{cm} = 19.5 \, \text{cm} = 0.195 \, \text{m} \)
- \( h = 46.8 \, \text{cm} = 0.468 \, \text{m} \)

Substituting the values:
\[
V = \pi (0.195)^2 \times 0.468 \, \text{m}^3
\]
\[
V \approx 0.0561 \, \text{m}^3
\]

#### 2. **Convert pressure** to Pascals:
\[
P = 5.70 \, \text{MPa} = 5.70 \times 10^6 \, \text{Pa}
\]

#### 3. **Number of moles of SF₆**:
The molar mass of SF₆ is 146.06 g/mol, so the number of moles \( n \) is:
\[
n = \frac{9.24 \, \text{kg} \times 1000}{146.06 \, \text{g/mol}} \approx 63.26 \, \text{mol}
\]

#### 4. **Using the ideal gas law to solve for temperature**:
Rearrange the ideal gas law for temperature \( T \):
\[
T = \frac{PV}{nR}
\]
Substitute the known values:
\[
T = \frac{(5.70 \times 10^6) \times 0.0561}{63.26 \times 8.314}
\]
\[
T \approx 608.92 \, \text{K}
\]

#### 5. **Convert to Celsius**:
\[
T_{\text{Celsius}} = T - 273.15
\]
\[
T_{\text{Celsius}} = 608.92 - 273.15 \approx 335.77^\circ C
\]

Therefore, the maximum safe operating temperature is approximately **335.8°C** (rounded to 4 significant figures).

---

Would you like more details on any of the steps or further clarifications?

### Related Questions:
1. What is the relationship between pressure, temperature, and volume in gases?
2. How do you calculate the number of moles of a gas from its mass?
3. What are the limitations of using the ideal gas law for real gases?
4. How would the solution change if the gas was not ideal?
5. What units of pressure, volume, and temperature are typically used in chemical engineering?

#### Tip:
Always ensure that your units are consistent when applying gas laws—converting to SI units (e.g., meters, Pascals, Kelvin) ensures accuracy in calculations.

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. Be sure your answer has the correct number of significant digits.

Learn how to calculate the maximum safe operating temperature of a high-pressure gas reaction vessel. Using the ideal gas law, this problem involves key concepts such as unit conversion, volume calculation, and mole calculation. The vessel contains sulfur hexafluoride gas with specific dimensions and pressure limits.

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