Question

A gas bubble with a volume of $1.0{\mathrm{mm}}^3$ originates at the bottom of a lake where the pressure is $3.0$ atm. Calculate its volume when the bubble reaches the surface of the lake where the pressure is 695 torr, assuming that the temperature doesn't change.

Short answer

The volume of the gas bubble when it reaches the surface of the lake is $3.28{\mathrm{mm}}^3$.

Step-by-step solution

List the given values

Initial Volume (Vi) = 1.0 mm³ Initial Pressure (Pi) = 3.0 atm Final Pressure (Pf) = 695 torr

Convert units

We need to convert the pressures to the same unit. Let's convert atm to torr: 1 atm = 760 torr 3.0 atm × (760 torr/1 atm) = 2280 torr Initial Pressure (Pi) = 2280 torr Additionally, convert the volume to a larger unit like liters: 1.0 mm³ × (1 cm³ / 1000 mm³) × (1 L / 1000 cm³) = 1.0 × 10⁻⁶ L Initial Volume (Vi) = $1.0\times {10}^{-6}$ L

Apply Boyle's Law

Boyle's Law states that for an ideal gas at constant temperature, the product of the initial pressure and volume is equal to the product of the final pressure and volume: Pi × Vi = Pf × Vf Now, solve for the final volume (Vf): Vf = (Pi × Vi) / Pf

Calculate the final volume

Plug in the given values and solve for Vf: Vf = ($2280\mathrm{torr}$ × $1.0\times {10}^{-6}\mathrm{L}$) / $695\mathrm{torr}$ Vf = $2.28\times {10}^{-3}\mathrm{L}$ / $695\mathrm{torr}$ Vf = $3.28\times {10}^{-6}$ L

Convert the final volume to mm³

To convert the final volume back to mm³, use the following conversion: Vf = $3.28\times {10}^{-6}$ L × (1000 cm³ / 1 L) × (1000 mm³ / 1 cm³) = 3.28 mm³ The volume of the gas bubble when it reaches the surface of the lake is 3.28 mm³.

Key concepts

Pressure Conversion

One of the critical steps in gas calculations is converting pressure units. Pressure can be measured in various units such as atmospheres (atm) and torr. However, to perform accurate calculations, these units need to be consistent.
To convert pressure from atmospheres to torr or vice versa, remember that:

• 1 atm is equivalent to 760 torr.
• To convert from atm to torr, multiply by 760.
• To convert from torr to atm, divide by 760.

In our original problem, the pressure of the gas bubble at the bottom of the lake is given as 3.0 atm. Using the conversion factor, you calculate this as 2280 torr. By ensuring that both initial and final pressures are in the same units (torr in this case), you can apply Boyle's Law more effectively.

Gas Laws

Gas laws provide vital relationships between pressure, volume, and temperature of gases. One of the famous gas laws is Boyle's Law. This principle describes how the pressure of a gas tends to increase as the volume of the container decreases, provided the temperature remains constant.
Boyle's Law is mathematically expressed as:

• $$P_i\times V_i=P_f\times V_f$$

Where:

• $P_i$ = Initial pressure
• $V_i$ = Initial volume
• $P_f$ = Final pressure
• $V_f$ = Final volume

This equation states that the product of the initial pressure and volume (on one side of the equation) is equal to the product of the final pressure and volume (on the other side). In the initial problem, you used Boyle's Law to calculate how the volume of the gas bubble changes as it ascends to the lake surface with reduced pressure.

Volume Calculation

Volume calculation is another critical aspect of gas problems. Sometimes, you need to convert between various volume units, such as milliliters, liters, or cubic millimeters.
In the original problem, the initial volume of the gas bubble is given as 1.0 mm³. Converting small units like mm³ into more standard or bigger units like liters makes handling numbers in calculations easier:

• A cube with 10 mm on each side has a volume of 1 cm³.
• 1000 cm³ equals 1 liter.
• Thus, 1 mm³ is equivalent to $1\times {10}^{-6}$ liters.

After calculating in liters, don't forget to convert the final volume back to the original units if needed. In this instance, the volume was eventually converted back to mm³ for the solution, providing a clear and comprehensible result. This step ensures that your answer matches the problem's original context.