A diving bell is a container open at the bottom. As the bell descends, the water level inside changes so that the pressure inside equals the pressure outside. Initially, the volume of air is 8.58 m3 at 1.020 atm and 20°C. What is the volume at 1.584 atm and 20°C?
P1V1 = P2V2
7.221
Ahh, the magical world of diving bells! Let's dive into this question (pun totally intended).
To find the new volume, we can use Boyle's Law, which states that the pressure and volume of a gas are inversely proportional, provided that the temperature remains constant. And since we're given the same temperature of 20°C, we're good to go!
Boyle's Law can be expressed as: P1 * V1 = P2 * V2
Where:
P1 = initial pressure (1.020 atm)
V1 = initial volume (8.58 m3)
P2 = final pressure (1.584 atm)
V2 = final volume (what we're trying to find)
Plugging in the given values, we have:
(1.020 atm) * (8.58 m3) = (1.584 atm) * V2
Multiply and divide, and we find:
V2 = (1.020 atm * 8.58 m3) / 1.584 atm
V2 ≈ 5.53 m3
So, at 1.584 atm and 20°C, the volume inside the diving bell is approximately 5.53 m3. Enjoy your imaginary dive, my friend!
To solve this problem, we can use the combined gas law equation. The combined gas law relates the initial and final conditions of a gas based on the pressure, volume, and temperature.
The equation for the combined gas law is:
(P₁ * V₁) / (T₁) = (P₂ * V₂) / (T₂)
where:
P₁ = initial pressure
V₁ = initial volume
T₁ = initial temperature
P₂ = final pressure
V₂ = final volume
T₂ = final temperature
In this case, the initial conditions are:
P₁ = 1.020 atm
V₁ = 8.58 m³
T₁ = 20°C
And we need to find the final volume (V₂) when the final conditions are:
P₂ = 1.584 atm
T₂ = 20°C
First, we need to convert the given temperatures to Kelvin, since temperature must be in Kelvin for the equation to work. To convert from Celsius to Kelvin, we use the formula K = °C + 273.15.
So for the initial temperature:
T₁ = 20°C + 273.15 = 293.15 K
And for the final temperature:
T₂ = 20°C + 273.15 = 293.15 K
Now we can plug in the values into the combined gas law equation:
(1.020 atm * 8.58 m³) / (293.15 K) = (1.584 atm * V₂) / (293.15 K)
To find V₂, we can rearrange the equation:
V₂ = (1.020 atm * 8.58 m³ * 293.15 K) / (1.584 atm * 293.15 K)
Canceling out the units and simplifying, we have:
V₂ = (1.020 * 8.58) / 1.584 m³
V₂ ≈ 5.53 m³
Therefore, the volume of air at 1.584 atm and 20°C is approximately 5.53 m³.