Question

Given is a container with a volume of 100 cubic meters. The content (CO2) is under a relative pressure of 100 mbar and has a constant temperature of 35 °C.

A constant flow of 60 cubic meters per hour (relative to 50 mbar) is exiting the container.

When will a pressure of 30 mbar be reached in the container?

I would use the ideal gas law:

PV=nRT

T=35+273.15=308.15 K

R=8.314 J/(molK)

Amount of substance: Container volume at 100 mbar:

V1=100 m3

P1=100 mbar+1 bar=1.1 bar=110,000 Pa

n1=P1⋅V/R⋅T

n1=1.1 bar⋅100 m3/0.08206 L⋅bar/(mol⋅K)⋅308.15 K

n1=4,293.6 mol

Amount of substance: Withdrawal flow Q (per hour) relative to 50 mbar:

V2=60 m3

P2=50 mbar+1 bar=105,000 Pa

n2=P2⋅V2/R⋅T

n2=105,000 Pa⋅60 m3/8.314 J/(molK)⋅308.15 K

n2=2,459.1 mol

Amount of substance: Container volume at 30 mbar:

V3=V1

P3=30 mbar+1 bar=1.03 bar=103,000 Pa

n3=P3⋅V3/R⋅T

n3=103,000 Pa⋅100 m/38.314 J/(molK)⋅308.15 K

n3=4,020.4 mol

The sought time is then

1 h⋅(n1−n2)/n3 = 1 h⋅(4,293.6−4,020.4) mol / 2,459.1 mol=0.11 h=6.7 min

Is this correct?

Answer 1

The student's calculation to determine when the pressure in the container would reach 30 mbar contained a mistake in the calculation of the final number of moles and the misuse of n2. The ideal gas law needs to be applied correctly, taking into account the constant temperature and the hourly flow rate of the gas out of the container using the correct pressures for each stage of the calculation.

Step-by-step explanation:

To determine when a pressure of 30 mbar will be reached within the container, we apply the ideal gas law in the form of PV=nRT, where:

• P is the pressure,
• V is the volume,
• n is the number of moles,
• R is the ideal gas constant,
• T is the temperature in Kelvin.

Initially, the volume (V1) of the gas is 100 m3 at an absolute pressure of P1 = 100 mbar + 1 bar = 1.1 bar = 110,000 Pa. The temperature remains constant at T = 35°C + 273.15 = 308.15 K. We need to convert bars to Pascals, keeping in mind that 1 bar = 105 Pa.

Using the ideal gas equation, we calculate the initial number of moles (n1) as follows:

n1 = P1 × V1 / (R × T)

Similarly, the flow of gas exiting per hour (n2) at P2 = 60 m3 relative to 50 mbar + 1 bar = 105,000 Pa is:

n2 = P2 × V2 / (R × T)

We also calculate the remaining moles (n3) when the pressure in the container reaches P3 = 30 mbar + 1 bar = 103,000 Pa at the same volume (V3 = V1).

To find the time to reach 30 mbar, we take the difference between the initial and final moles and divide by the rate of change per hour:

Time = (n1 - n3) / (n2 - n3)

However, there seems to be a mistake in the original calculation provided by the student, primarily due to an incorrect calculation of n3 and the misuse of n2. The actual time should consider the constant flow rate (n2) and how it reduces the number of moles inside the container (n1) until n3 is reached.