Question

(a) show that the pressure exerted by a fluid P (in pascals) is given by P= hdg, where h is the column of the fluid in metres, d is density in kg/m3, and g is the acceleration due to gravity (9.81 m/s2). (Hint: see appendix 2.). (b) The volume of an air bubble that starts at the bottom of a lake at 5.24 degree celsius increases by a factor of 6 as it rises to the surface of water where the temperature is 18.73 degree celsius and the air pressure is 0.973 atm. The density of the lake water is 1.02 g/cm3. Use the equation in (a) to determine the depth of the lake in metres.

Answer 1

56.4 m

Step-by-step explanation:

volume increases by factor of 6, i.e
`(V2)/(V1)` = 6

Initial temperature T1 at bottom of lake = 5.24°C = 278.24 K

Final temperature T2 at top of lake = 18.73°C = 291.73 K

NB to change temperature from °C to K we add 273

Final pressure P2 at the top of the lake = 0.973 atm

Initial pressure P1 at bottom of lake = ?

Using the equation of an ideal gas

`(P1V1)/(T1)` =
`(P2V2)/(T2)`

P1 =
`(P2V2T1)/(V1T2)` =
`(0.973*6*278.24)/(291.73)`

P1 = 5.57 atm

5.57 atm = 5.57 x 101325 = 564380.25 Pa

Density Ρ of lake = 1.02 g/
`cm^(3)` = 1020 kg/
`m^(3)`

acceleration due to gravity g = 9.81
`m/s^(2)`

Pressure at lake bottom = pgd

where d is the depth of the lake

564380.25 = 1020 x 9.81 x d

d =
`(564380.25)/(10006.2)` = 56.4 m