To determine the air pressure at the top of the mountain, we can use the combined gas law formula which relates pressure (P), volume (V), and temperature (T).
The formula is:
[tex]\[
\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}
\][/tex]
Given data:
- At the valley floor:
- [tex]\(P_1 = 1.0 \, \text{atm}\)[/tex]
- [tex]\(V_1 = 0.985 \, \text{L}\)[/tex]
- [tex]\(T_1 = 25^{\circ} \text{C} + 273.15 = 298.15 \, \text{K}\)[/tex]
- At the top of the mountain:
- [tex]\(V_2 = 1.030 \, \text{L}\)[/tex]
- [tex]\(T_2 = 22^{\circ} \text{C} + 273.15 = 295.15 \, \text{K}\)[/tex]
- We need to find [tex]\(P_2\)[/tex].
First, rearrange the formula to solve for [tex]\(P_2\)[/tex]:
[tex]\[
P_2 = \frac{P_1 V_1 T_2}{T_1 V_2}
\][/tex]
Plugging in the values:
[tex]\[
P_2 = \frac{(1.0 \, \text{atm}) (0.985 \, \text{L}) (295.15 \, \text{K})}{(298.15 \, \text{K}) (1.030 \, \text{L})}
\][/tex]
Performing the calculation step-by-step:
1. Multiply the values in the numerator:
[tex]\[
1.0 \times 0.985 \times 295.15 = 290.02275
\][/tex]
2. Multiply the values in the denominator:
[tex]\[
298.15 \times 1.030 = 307.0955
\][/tex]
3. Divide the results:
[tex]\[
P_2 = \frac{290.02275}{307.0955} \approx 0.9467 \, \text{atm}
\][/tex]
So, the air pressure at the top of the mountain is approximately [tex]\(0.9467 \, \text{atm}\)[/tex]. Considering significant figures based on the given data, the most appropriate answer would be:
[tex]\[
P_2 = 0.947 \, \text{atm}
\][/tex]
This ensures the precision of the result matches the precision of the given values.
