in a cylinder that was 9.0 mm in diameter and 215 mm in length.
I want to know the temperature (in Kelvin) and pressure (in atm) within this tube after combustion assuming that the cork did not “shoot” out of the end, i.e., the pressure was contained within the tube after ignition.
Balanced Reaction: 2 C6H14 + 19 O2 12 CO2 + 14 H2O
We’re going to simplify the problem slightly by assuming that there is no heat/energy loss to the walls of the cylinder (i.e., all the heat produced goes to heating the product gases), the combustion reaction goes to completion, no PV work was performed and the heat capacities (given below) are independent of temperature.
The system before ignition: 1 atm O2 gas and hexane vapor. (O2 is limiting reagent for the combustion reaction which produced H2O(g) and CO2(g) as products.
The heat of combustion is -3891 kJ/mole of hexane
The constant volume heat capacities are…
33.6 J/K-mol for H2O(g)
37.1 J/K-mol for CO2(g)
These are the steps to follow:
(1) compute the cylindrical chamber volume, V
(2) compute the number of moles n in volume initially using n = PV/RT
(3) 2/21 of the total moles initially present are C6H14. Compute that number.
(4) Compute the heat release, assuming the reaction goes to completion. Get that from the number of moiles of hexane and the heat of reaction per mole.
(5) Compute the final temperature such that the heat release equals the heat transferred to the reaction products H2O and CO2.
To answer this question and calculate the temperature and pressure within the cylinder after combustion, we need to follow the given steps:
Step 1: Compute the cylindrical chamber volume, V
To calculate the volume of the cylinder, we can use the formula for the volume of a cylinder: V = πr^2h, where r is the radius (half of the diameter) and h is the length of the cylinder.
Given diameter: 9.0 mm
Radius (r) = 9.0 mm / 2 = 4.5 mm = 0.0045 m
Length (h) = 215 mm = 0.215 m
Volume (V) = π(0.0045^2)(0.215) = 4.57 x 10^-6 m^3
Step 2: Compute the number of moles (n) initially present in the volume using n = PV/RT
We can use the ideal gas equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant (8.314 J/(mol*K)), and T is the temperature in Kelvin.
Given pressure: 1 atm = 101325 Pa
Temperature: Unknown (let's call it T)
n = (PV) / (RT) = (101325 * 4.57 x 10^-6) / (8.314 * T)
Step 3: Compute the number of moles of C6H14
The balanced reaction shows that 2 moles of C6H14 react with 19 moles of O2. So, in the initial mixture, the ratio of C6H14 to total moles is 2/21.
Total moles initially present = n (from Step 2)
Moles of C6H14 = (2/21) * Total moles
Step 4: Compute the heat release assuming the reaction goes to completion
The heat of combustion is given as -3891 kJ/mole of hexane. To calculate the heat release, we multiply the moles of C6H14 (from Step 3) by the heat of combustion.
Heat release = Moles of C6H14 * Heat of combustion
Step 5: Compute the final temperature such that the heat release equals the heat transferred to the reaction products H2O and CO2
Since we assume no heat loss to the walls of the cylinder, the heat release (from Step 4) should equal the heat transferred to the reaction products H2O and CO2. We can calculate this using the heat capacities and the change in temperature.
Heat transferred = (Moles of H2O * Heat capacity of H2O) + (Moles of CO2 * Heat capacity of CO2) * Change in temperature
Let's solve this step-by-step to find the final temperature (T).