Final answer:
To find the maximum safe operating temperature for a reaction vessel containing boron trifluoride gas, the ideal gas law is used with a given pressure and the vessel's dimensions to calculate the temperature in Kelvin, which is then converted to degrees Celsius. The maximum safe temperature is found to be approximately 63166°C.
Step-by-step explanation:
To calculate the maximum safe operating temperature for the reaction vessel containing boron trifluoride gas, we can make use of the ideal gas law, which is PV = nRT, where P is the pressure, V is the volume, n is the amount of substance in moles, R is the ideal gas constant, and T is the temperature in Kelvin. First, we need to determine the volume of the gas cylinder and convert the mass of boron trifluoride to moles.
The volume (V) of the cylinder is πr²h, where r is the radius (16.0 cm) and h is the height (38.4 cm). Remember to convert centimeters to meters:
V = π(0.16 m)²(0.384 m) = π(0.0256 m²)(0.384 m) = 0.0309 m³.
Next, we calculate the number of moles (n) of boron trifluoride using its molar mass (approximately 67.80 g/mol):
n = 1.27 kg / (67.80 g/mol) = 1270 g / (67.80 g/mol) = 18.73 mol.
Now, we can rearrange the ideal gas law to solve for temperature (T = rac{PV}{nR}) and substitute the given pressure (P = 3.20 MPa), using 8.314 J/(mol·K) for R, the ideal gas constant:
T = rac{(3.20 × 10¶ Pa)(0.0309 m³)}{(18.73 mol)(8.314 J/(mol·K))} = rac{9.888 × 10¶ Pa·m³}{155.8387 J/K} = 63439.54 K.
Finally, convert the temperature from Kelvin to degrees Celsius:
T = 63439.54 K - 273.15 = 63166.39°C.
Therefore, the maximum safe operating temperature the engineer should recommend for this reaction is rounded to 63166°C.