Gas Laws: Boyle's, Charles's, and the Ideal Gas Law
Apply the ideal gas law and combined gas law to solve multi-step problems involving pressure, volume, and temperature changes.
Problem Scenario
A rigid steel tank contains 5.00 L of nitrogen gas at 25.0°C and 2.50 atm. The tank is heated until the pressure reaches 4.00 atm. A valve is then opened, releasing gas into a flexible balloon until the tank pressure drops to 1.00 atm at 25.0°C. Calculate the final temperature inside the tank after heating, the moles of N₂ originally in the tank, and the volume of the balloon at 25.0°C and 1.00 atm.
Given Data
Requirements
- Find the temperature after heating using Gay-Lussac's law
- Calculate moles of N₂ originally in the tank using the ideal gas law
- Determine moles remaining in the tank after gas release
- Calculate the volume of gas in the balloon
Solution
Step 1:
Find temperature after heating. Since the tank is rigid (constant volume) with no gas escaping: P₁/T₁ = P₂/T₂. So T₂ = T₁ × (P₂/P₁) = 298.15 K × (4.00/2.50) = 477.0 K = 203.9°C.
Step 2:
Calculate initial moles using PV = nRT. n = PV/(RT) = (2.50 atm × 5.00 L)/(0.08206 L·atm/(mol·K) × 298.15 K) = 12.50/24.47 = 0.511 mol N₂.
Step 3:
Find moles remaining in tank after valve opens. The tank cools back to 25.0°C with pressure at 1.00 atm: n_remaining = PV/(RT) = (1.00 × 5.00)/(0.08206 × 298.15) = 5.00/24.47 = 0.204 mol.
Step 4:
Calculate moles released: n_released = 0.511 − 0.204 = 0.307 mol N₂.
Step 5:
Find balloon volume at 25.0°C and 1.00 atm: V = nRT/P = (0.307 × 0.08206 × 298.15)/1.00 = 7.51 L.
Final Answer
After heating: T = 477 K (204°C). Original tank contains 0.511 mol N₂. After release, the balloon holds 0.307 mol of N₂ at a volume of 7.51 L (at 25°C and 1 atm).
Key Takeaways
- ✓Gay-Lussac's law: at constant volume, P/T is constant
- ✓The ideal gas law PV = nRT connects all four gas variables
- ✓Always convert temperatures to Kelvin for gas law calculations
- ✓Conservation of matter: moles released = moles initial − moles remaining
Common Errors to Avoid
- ✗Using Celsius instead of Kelvin in gas law equations
- ✗Applying Boyle's law when temperature is not constant
- ✗Forgetting that a rigid container means constant volume, not constant pressure
- ✗Not accounting for the temperature change when calculating released gas
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Common questions about this problem type
Use PV = nRT when you need to find a specific variable (n, P, V, or T) at a single set of conditions. Use the combined gas law (P₁V₁/T₁ = P₂V₂/T₂) when the same gas sample changes conditions and you need to relate before and after.
Gas laws are derived from the kinetic molecular theory where temperature is proportional to average kinetic energy. Zero Kelvin means zero molecular motion. Celsius has an arbitrary zero point, so ratios in Celsius give incorrect results.