# Application of the Ideal Gas Law to Gas Canisters Below is a step-by-step solution applying the **ideal gas law** to a typical gas canister problem. This will demonstrate how to calculate the amount of gas in a canister and predict changes in gas parameters, addressing **CO5**: "Predict how a change in one or more of the parameters of a gas alters dependent gas parameters such as pressure, volume, or temperature." --- ## Example Problem **A 10. L gas canister contains nitrogen gas at 25°C and a pressure of 200 atm. How many moles of nitrogen gas are in the canister? If the temperature increases to 50°C and the canister volume remains constant, what will be the new pressure?** --- ## Step 1: Write the Ideal Gas Law The ideal gas law is: \[ PV = nRT \] Where: - \(P\) = pressure (atm) - \(V\) = volume (L) - \(n\) = number of moles (mol) - \(R\) = ideal gas constant (\(.0821 \, \text{L·atm·mol}^{-1}\text{·K}^{-1}\)) - \(T\) = temperature (K) --- ## Step 2: List Given Values - \(V = 10. \, \text{L}\) - \(P = 200 \, \text{atm}\) - \(T = 25^\circ\text{C} = 273.15 + 25 = 298.15 \, \text{K}\) - \(R = .0821 \, \text{L·atm·mol}^{-1}\text{·K}^{-1}\) --- ## Step 3: Solve for Moles (\(n\)) \[ n = \frac{PV}{RT} \] Plug in values: \[ n = \frac{200 \times 10.}{.0821 \times 298.15} \] Calculate denominator: \[ .0821 \times 298.15 = 24.487 \] Calculate numerator: \[ 200 \times 10. = 200 \] \[ n = \frac{200}{24.487} \approx 81.7\, \text{mol} \] --- ## Step 4: Predict Pressure After Temperature Change ### New Temperature - \(T_2 = 50^\circ\text{C} = 323.15 \, \text{K}\) - Volume and number of moles remain constant. ### Use Combined Gas Law \[ \frac{P_1}{T_1} = \frac{P_2}{T_2} \] \[ P_2 = P_1 \times \frac{T_2}{T_1} \] Plug in values: \[ P_2 = 200 \, \text{atm} \times \frac{323.15}{298.15} \] \[ \frac{323.15}{298.15} \approx 1.084 \] \[ P_2 \approx 200 \times 1.084 = 216.8\, \text{atm} \] --- ## Step 5: Final Answers - **Number of moles of nitrogen gas:** \(81.7 \, \text{mol}\) - **New pressure after heating to 50°C:** \(216.8 \, \text{atm}\) --- ## Step 6: Explanation - **The ideal gas law relates pressure, volume, temperature, and amount of gas.** - **Increasing temperature (with constant volume and moles) increases pressure, as shown in the combined gas law.** - **Units must be consistent (atmospheres, liters, Kelvin, moles).** --- ## Summary Table | Parameter | Initial Value | Final Value | |----------------|--------------|-------------------------------------| | Volume (L) | 10. | 10. (constant) | | Pressure (atm) | 200 | 216.8 | | Temperature (K)| 298.15 | 323.15 | | Moles (mol) | 81.7 | 81.7 (constant) | --- ## Useful Diagram  --- ## Key Takeaways - Use the ideal gas law to relate gas parameters. - When temperature increases at constant volume and moles, pressure increases proportionally. - Always use Kelvin for temperature in gas law calculations.