The volume of the air bubble when it reaches the surface of the lake is 5.26 x 10⁻⁶ m³.
How to calculate the volume of the air bubble at the surface?
The volume of the air bubble at the surface is calculated by applying general gas equation.
P₁V₁/T₁ = P₂V₂/T₂
where;
- P₁ is the pressure at the bottom of the lake
- P₂ is the pressure at the surface of the lake
- T₁ is the temperature at the bottom of the lake
- T₂ is the temperature at the surface of the lake.
- V₁ is the volume at the bottom of the lake
- V₂ is the volume at the surface of the lake
The pressure at the bottom of the lake is calculated as;
P₁ = 1 atm + ρgh
P₁ = 1.013 x 10⁵ Pa + (1000 x 9.8 x 40)
P₁ = 493,300 Pa
The pressure at the surface of the lake:
P₂ = 1 atm
P₂ = 1.013 x 10⁵ Pa
The temperature at the bottom of the lake:
T₁ = 12 ⁰C + 273
T₁ = 285 K
The temperature at the surface of the lake:
T₂ = 35 ⁰C + 273
T₂ = 308 K
The volume at the bottom of the lake:
V₁ = 1 cm³
V₁ = 1 x 10⁻⁶ m³
The volume of the air bubble at the surface is calculated as;
V₂ = (P₁V₁T₂) / (P₂T₁)
V₂ = (493,300 x 1 x 10⁻⁶ x 308) / (1.013 x 10⁵ x 285)
V₂ = 5.26 x 10⁻⁶ m³
The complete question is below:
An air bubble of volume 1 cm³ rises from the bottom of a lake 40m deep to the surface at a temperature of 12°C. Calculate the volume of the air bubble when it reaches the surface, which is at a temperature of 35 ⁰C. (Given atmospheric pressure, 1 atm = 1.013 x 10⁵ Pa , density of water = 1000 kg/m³, and gravitational acceleration = 9.8 m/s²).