- Convert temperatures to Kelvin: \(T_1 = 298.15\text{ K}\) and \(T_2 = 295.15\text{ K}\).
- Use the combined gas law to express the final pressure: \(P_2 = \frac{P_1 V_1 T_2}{V_2 T_1}\).
- Substitute the values to find \(P_2 \approx 0.9467\text{ atm}\).
- Round the final result to two significant figures to arrive at \(\boxed{0.95\text{ atm}}\).
Explanation
- Problem Analysis and Setup
We are given a sealed bag of potato chips with initial conditions at the valley floor: pressure \(P_1 = 1.0\text{ atm}\), temperature \(T_1 = 25^{\circ}\text{C}\), and volume \(V_1 = 0.985\text{ L}\). At the mountain top, the conditions change to: temperature \(T_2 = 22^{\circ}\text{C}\) and volume \(V_2 = 1.030\text{ L}\). We need to determine the air pressure \(P_2\) at the mountain top using the combined gas law: \(\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}\).
- Temperature Conversion
First, we convert the temperatures from Celsius to Kelvin, as required for the gas law, using the conversion \(K = ^{\circ}\text{C} + 273.15\).
For the initial temperature: \(T_1 = 25 + 273.15 = 298.15\text{ K}\).
For the final temperature: \(T_2 = 22 + 273.15 = 295.15\text{ K}\).
- Calculating the Final Pressure
Rearranging the combined gas law to solve for the final pressure \(P_2\), we obtain:
\[P_2 = \frac{P_1 \times V_1 \times T_2}{V_2 \times T_1}\]
Substituting the known values into the equation:
\[P_2 = \frac{1.0\text{ atm} \times 0.985\text{ L} \times 295.15\text{ K}}{1.030\text{ L} \times 298.15\text{ K}}\]
Performing the calculation:
\[P_2 \approx \frac{290.72275}{307.0945} \approx 0.946688\text{ atm}\]
- Final Answer Determination
Finally, we consider the significant figures. The values \(1.0\text{ atm}\), \(25^{\circ}\text{C}\), and \(22^{\circ}\text{C}\) each have two significant figures. Therefore, our result should be reported with two significant figures. Rounding \(0.946688\) to two significant figures gives \(0.95\). The air pressure at the top of the mountain is \(\boxed{0.95\text{ atm}}\).
Examples
In aviation or meteorology, understanding how air pressure changes with altitude is critical for safety and weather forecasting. As pilots fly at higher altitudes, the ambient pressure decreases, causing sealed containers (like pressurized cabins or even snack bags) to expand. By using the combined gas law, professionals can predict these changes in volume or pressure, ensuring that aircraft systems are correctly pressurized and that scientific instruments operating in the atmosphere provide accurate data.