A compartment measures 3 meters by 5 meters and is 2.8 meters high. A fire raises the temperature from 20oC to 1000 oC. If the starting pressure was 1 atmosphere, what pressure is present at the elevated temperature assuming the compartment remains closed?
To calculate the final pressure in the compartment, we need to use the ideal gas law equation, which is:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature (in Kelvin)
We will need to convert the temperatures from Celsius to Kelvin by adding 273.15. The starting temperature is 20°C + 273.15 = 293.15K, and the final temperature is 1000°C + 273.15 = 1273.15K.
First, let's find the initial number of moles in the compartment using the ideal gas law equation:
P1V1 = nRT1
Since the compartment is closed, the volume remains the same, so we can rewrite the equation as:
P1 = nRT1 / V
Given:
P1 = 1 atm (initial pressure)
V = 3m x 5m x 2.8m = 42m^3 (volume of the compartment)
R = 0.0821 atm·L/(mol·K) (ideal gas constant)
T1 = 293.15K (initial temperature)
Now we can calculate the initial number of moles (n1):
n1 = P1V / (R * T1)
n1 = (1 atm * 42m^3) / (0.0821 atm·L/(mol·K) * 293.15K)
n1 ≈ 5.62 moles
Next, we can use the same formula to find the final pressure (P2):
P2 = n2RT2 / V
Given:
T2 = 1273.15K (final temperature)
To calculate the final number of moles (n2), we can use the equation:
n1T1 = n2T2
n2 = (n1T1) / T2
n2 = (5.62 moles * 293.15K) / 1273.15K
n2 ≈ 1.292 moles
Now we can substitute the values into the equation for P2:
P2 = (1.292 moles * 0.0821 atm·L/(mol·K) * 1273.15K) / 42m^3
P2 ≈ 47.99 atm
Therefore, the pressure present at the elevated temperature of 1000°C in the closed compartment is approximately 47.99 atmospheres.