Understanding Air Pressure, Temperature, and Mass: A Comprehensive Guide

Understanding the properties of air, such as its pressure, temperature, and mass, is crucial in various scientific and engineering fields. This article delves into the problem of a tank containing 20.0 liters of air at 30°C and 5.01x105 N/m2 pressure. It explores how to calculate the mass of the air and its volume at one atmospheric pressure at 0°C, while also considering the impact of humidity on the air's density.

Problem Statement and Analysis

A given tank contains 20.0 liters of air at a temperature of 30°C and a pressure of 5.01x105 N/m2. This initial condition gives us a unique opportunity to explore the principles of ideal gas behavior under different conditions. First, we need to determine the mass of the air in the tank. To do this, we will employ the ideal gas law.

Calculating the Mass of Air

To find the mass of the air, we first need to find the number of moles of air in the tank using the ideal gas law, which is represented as:

PV nRT

Where:

P is the pressure (Pa) 5.01x105 N/m2 V is the volume (m3) 20 L 0.020 m3 n is the number of moles R is the universal gas constant (8.314 J/(mol·K)) T is the temperature (K) 30°C 273.15 303.15 K

Rearranging the equation to solve for n:

n PV / RT

Substituting the given values:

n (5.01x105 N/m2 x 0.020 m3) / (8.314 J/(mol·K) x 303.15 K)

n ≈ 3.21 moles

Now, we can calculate the mass of the air using the molar mass of dry air, which is given as 28.97 kg/kmol (or 28.97 g/mole). The mass, m, can be found using the formula:

m n x M

Where M is the molar mass (28.97 g/mole or 28.97 kg/kmol).

m 3.21 moles x 28.97 kg/mole

m ≈ 92.38 kg

Impact of Humidity on Air Density

It is important to understand that humid air is less dense than dry air at the same temperature and pressure. Humidity affects the density of air according to the Dalton's law of partial pressures. For dry air, we can use the ideal gas law to calculate the density. The density, ρ, of an ideal gas can be calculated using the formula:

ρ PM / (RT)

Where P is the pressure, M is the molar mass, R is the universal gas constant, and T is the temperature. For the given conditions:

ρ (5.01x105 N/m2 x 28.97 x 10-3 kg/mole) / (8.314 J/(mol·K) x 303.15 K)

ρ ≈ 1.97 kg/m3

Considering the impact of humidity, we must adjust the density to account for the fact that humid air is less dense than dry air.

Volume of Air at One Atmospheric Pressure at 0°C

Next, we need to calculate the volume of the air at one atmospheric pressure (101,325 N/m2) and a temperature of 0°C (273.15 K). Assuming the tank maintains the same number of moles of air, we can use the ideal gas law to find the new volume (V) at these conditions:

P1V1 / T1 P2V2 / T2

Where:

P1 5.01x105 N/m2 V1 0.020 m3 T1 303.15 K P2 101,325 N/m2 T2 273.15 K V2 is the volume needed to be calculated

Rearranging the equation to solve for V2:

V2 (P1V1 / T1) x (T2 / P2)

Substituting the values:

V2 (5.01x105 N/m2 x 0.020 m3 / 303.15 K) x (273.15 K / 101,325 N/m2)

V2 ≈ 0.040 m3 or 40 L

Conclusion

Understanding the properties of air allows us to make precise calculations in various scientific and engineering applications. In this case, we determined that a tank containing 20.0 liters of air at 30°C and 5.01x105 N/m2 pressure has a mass of approximately 92.38 kg. Furthermore, at one atmospheric pressure and 0°C, the volume of the air would be approximately 40 liters. The impact of humidity on air density cannot be overstated, as it affects the overall behavior of the gas in different atmospheric conditions.

Keywords

air pressure, temperature and volume, humidity, molar mass, atmospheric pressure