Nitrogen gas is introduced into a large deflated plastic bag. No gas is allowed to escape, but as more and more nitrogen is added, the bag inflates to accommodate it. The pressure of the gas within the bag remains at 1 \rm atm and its temperature remains at room temperature (20^\circ {\rm C}). How many moles n have been introduced into the bag by the time its volume reaches 22.4 \rm L?

To determine the number of moles of nitrogen gas introduced into the bag, we can use the ideal gas law equation:

PV = nRT

where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of gas
R is the ideal gas constant
T is the temperature of the gas

Given:
Pressure (P) = 1 atm
Volume (V) = 22.4 L
Temperature (T) = 20°C = 20 + 273.15 = 293.15 K (since temperature must be in Kelvin for the ideal gas law)

By substituting these values into the ideal gas law equation, we can solve for the number of moles (n):

n = (PV) / (RT)

Substituting the given values:

n = (1 atm * 22.4 L) / (0.0821 L·atm/(mol·K) * 293.15 K)

Calculating this expression:

n ≈ 0.967 moles

Therefore, approximately 0.967 moles of nitrogen gas have been introduced into the bag by the time its volume reaches 22.4 L.