Avg PO4^3- []'s in the deep and surface ocn's are 2.11 and 0.71umolkg^-1. Calc the residence time of PO4^3- in the deep ocn considering the carbon budget(Figure below) Write the general equation, calculation and units.

------ represents a Box outline

Surface---------------------------------
[P]
------------/\--------------==----------
---------\/-----------------==----------
(arrows moving \/(sinking)
up and down through Jc = 5.0Gt/y
surface and deep)

v =4.6 m/y

deep------------==----------------------
----------------==----------------------
\/ (sinking)
Bc = 0.15Gt/y

sediments-------------------------------

To calculate the residence time of PO4^3- in the deep ocean, we need to find the amount of PO4^3- in the deep ocean and the rate at which it is being replenished or removed.

First, we can calculate the amount of PO4^3- in the deep ocean using the average concentration given:
Amount of PO4^3- in the deep ocean = Average concentration * volume of the deep ocean
Given: Average concentration = 2.11 umol/kg and volume of the deep ocean = Bc = 0.15 Gt/y (1 Gt = 10^15 g)

Converting the volume to units of umol/kg:
0.15 Gt/y * 10^15 g/Gt * 10^6 umol/g = 1.5 x 10^23 umol

Next, we need to determine the rate at which the PO4^3- is being replenished or removed from the deep ocean.

From the figure provided, we can see that there is a sinking flux (arrows moving down through the surface and deep) and a sinking rate (v = 4.6 m/y).

To calculate the sinking flux, we can multiply the sinking rate by the average concentration in the surface ocean:
Sinking flux = Sinking rate * average concentration in the surface ocean
Given: Sinking rate = v = 4.6 m/y and average concentration in the surface ocean = [P] = 0.71 umol/kg

Sinking flux = 4.6 m/y * 0.71 umol/kg = 3.266 umol/m^2/y

Now, we can calculate the residence time using the carbon budget:
Residence time = Amount of PO4^3- in the deep ocean / Sinking flux

Residence time = (1.5 x 10^23 umol) / (3.266 umol/m^2/y)

To simplify the calculation, let's convert the volume of the deep ocean to area using the sinking rate:
Volume of the deep ocean = Area of the deep ocean * sinking rate

Area of the deep ocean = Volume of the deep ocean / sinking rate
Area of the deep ocean = (1.5 x 10^23 umol) / (4.6 m/y) = 3.26 x 10^22 m^2

Now, we calculate the residence time:
Residence time = (1.5 x 10^23 umol) / (3.266 umol/m^2/y)
Residence time = 4.6 x 10^22 y

Therefore, the residence time of PO4^3- in the deep ocean is approximately 4.6 x 10^22 years.