An air bubble at the bottom of a lake 35.0 m deep has a volume of 1.88 cm^3. If the temperature at the bottom is 5.4 C and at the top 19.5 C, what is the volume of the bubble just before it reaches the surface?
You have a temp change, and a pressure change. Figure the pressure at surface as one atm(10.3m water), and the pressure at 35 m is then 45.3m water total.
P1Vi/T1=P2V2/T2
V2=P1V2T2/T1P2 where condition 1 is at the depth, and condition 2 is at the surface. Be certain to change temps to Kelvins
PV/T = constant (remember T in Kelvin)
pressure = 1 atmosphere at surface, about 10^5 pascals
add to that rho g h for pressure at the bottom where rho is density of water (10^3 kg/m^3) and g is 9.8 m/s^2 and h is 35 m
To solve this problem, we can use Boyle's Law and the Ideal Gas Law.
1. Boyle's Law states that the pressure and volume of a gas are inversely proportional at constant temperature. Mathematically, it can be expressed as:
P1 * V1 = P2 * V2
Where P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
2. The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of gas. Mathematically, it is written as:
PV = nRT
Where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.
Now let's calculate step by step:
Step 1: Convert the temperatures to Kelvin.
T1 = 5.4 + 273.15 = 278.55 K
T2 = 19.5 + 273.15 = 292.65 K
Step 2: Convert the initial volume to liters.
V1 = 1.88 cm^3 = 1.88 * 10^(-3) L
Step 3: Calculate the initial pressure using the Ideal Gas Law.
Using the equation PV = nRT, we can rearrange it to solve for P.
P1 = nRT1 / V1
Since we don't have the number of moles (n), we can assume that the number of moles remains constant. Therefore, the initial pressure can be expressed as:
P1 = (nR / V1) * T1
Step 4: Calculate the final pressure.
The final pressure (P2) is equal to the pressure at the top of the lake, which is the atmospheric pressure. We can estimate the atmospheric pressure to be around 101.3 kPa.
Step 5: Calculate the final volume using Boyle's Law.
Using the equation P1 * V1 = P2 * V2, we can rearrange it to solve for V2.
V2 = (P1 * V1) / P2
Let's substitute the values we have:
V2 = (P1 * V1) / P2
Step 6: Convert the final volume to cm^3, if necessary.
If the question specifically asks for the volume in cm^3, you can convert it by multiplying by 1000:
V2_cm^3 = V2 * 1000
Now you can calculate the final volume using these steps and the given values.