1. Convert Temperature to Kelvin

* T(K) = T(°C) + 273.15

* T(K) = 27°C + 273.15 = 300.15 K

2. Calculate Total Moles of Gas

* Total moles = moles of hydrogen + moles of oxygen + moles of helium

* Total moles = 3 + 6 + 9 = 18 moles

3. Use the Ideal Gas Law

* PV = nRT

* P = pressure (what we want to find)

* V = volume (10 L)

* n = number of moles (18 moles)

* R = ideal gas constant (0.0821 L·atm/mol·K)

* T = temperature (300.15 K)

4. Solve for Pressure

* P = (nRT) / V

* P = (18 moles * 0.0821 L·atm/mol·K * 300.15 K) / 10 L

* P ≈ 44.5 atm

Therefore, the total pressure of the gas mixture in the vessel is approximately 44.5 atmospheres.