To determine the amount of nitrogen gas in the tank after it has been opened and some gas has escaped, we can use the ideal gas law, which is expressed as PV = nRT. Here, P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin. Let's break this down step by step.
Step 1: Convert Temperature to Kelvin
The temperature given is 20 degrees Celsius. To convert this to Kelvin, we add 273.15:
- T(K) = 20 + 273.15 = 293.15 K
Step 2: Use the Ideal Gas Law
Initially, we know the pressure (P) is 3 atm, the volume (V) is 30 liters, and the temperature (T) is 293.15 K. We can rearrange the ideal gas law to solve for n (the number of moles of nitrogen gas):
Now, we need the value of R. The ideal gas constant R can be used in different units, but for our case, we will use:
Step 3: Calculate Initial Moles of Nitrogen Gas
Substituting the known values into the equation:
- n = (3 atm) * (30 L) / (0.0821 L·atm/(K·mol) * 293.15 K)
- n = 90 / (24.207) ≈ 3.71 moles
Step 4: Calculate Moles After Gas Escapes
After the gas escapes, the pressure in the tank drops to 2.4 atm while the temperature returns to 20 degrees Celsius (293.15 K). We can again use the ideal gas law to find the new number of moles:
- n' = (2.4 atm) * (30 L) / (0.0821 L·atm/(K·mol) * 293.15 K)
- n' = 72 / (24.207) ≈ 2.97 moles
Step 5: Determine the Amount of Nitrogen Gas That Escaped
To find out how much nitrogen gas escaped, we subtract the final number of moles from the initial number of moles:
- Amount escaped = n - n' = 3.71 moles - 2.97 moles = 0.74 moles
Final Thoughts
In summary, after the tank was opened and some nitrogen gas escaped, the amount of nitrogen gas remaining in the tank is approximately 2.97 moles, and about 0.74 moles of gas escaped. This example illustrates how the ideal gas law can be applied to real-world scenarios involving gas behavior under changing conditions.
