To find the volume of gas that a climber's oxygen tank would occupy if released at the top of Mt. Everest, we can use Boyle's Law, which states that the pressure and volume of a gas are inversely related (assuming temperature remains constant). The law can be expressed mathematically as:
\[ P_1 V_1 = P_2 V_2 \]
Where:
- \(P_1\) is the initial pressure (inside the tank)
- \(V_1\) is the initial volume (volume of the gas in the tank)
- \(P_2\) is the final pressure (atmospheric pressure at the summit)
- \(V_2\) is the final volume (volume of the gas after being released)
Let's set up the equation. We know that:
- \(P_1 = 35,000 , \text{mm Hg}\) (pressure in the tank)
- \(P_2 = 150 , \text{mm Hg}\) (pressure at the summit)
- \(V_1\) is the volume inside the tank that we do not know, and we will call it \(V_t\).
- \(V_2\) is the volume we are trying to find after the gas is released.
Rearranging Boyle's Law gives us:
\[ V_2 = \frac{P_1 V_1}{P_2} \]
Substituting the known values into the equation (and noting that \(V_1\) is just some constant value, so let's leave it as \(V_t\)), we get:
\[ V_2 = \frac{35000 , \text{mm Hg} \cdot V_t}{150 , \text{mm Hg}} \]
Now, simplifying this:
\[ V_2 = \frac{35000}{150} V_t \]
Calculating \( \frac{35000}{150} \):
\[ \frac{35000}{150} = 233.33 \]
Thus, the formula becomes:
\[ V_2 = 233.33 V_t \]
This result shows that the gas released at the summit of Mt. Everest will occupy approximately 233.33 times the initial volume of the gas when it was compressed in the tank, assuming ideal gas behavior and constant temperature.
To find the actual volume, you would need the actual volume of the tank \(V_t\). If you know that, simply multiply it by 233.33 to get the final volume at the top of Mt. Everest.
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