The level of toluene (a flammable hydrocarbon) in a storage tank may fluctuate between 10 and 400 cm from the top of the tank. since it is impossible to see inside the tank, an open-end manometer with water or mercury as the manometer fluid is to be used to determine the toluene level. one leg of the manometer is attached to the tank 500 cm from the top. a nitrogen blanket at atmospheric pressure is maintained over the tank contents. felder, richard m.; rousseau, ronald w.; bullard, lisa g.. elementary principles of chemical processes, 4th edition (page 81). wiley. kindle edition.

The toluene level where in the tank where the height of the manometer fluid level in the open arm is equal to the height where the manometer is connected to the tank is h =150 cm

The manometer reading is R

Generally at the point where the height of the open arm is equal to the height of the of the point connected to the tank ,

The pressure at the height of the both arms of the manometer corresponding to the base of the tank are equal

i.e [tex]P_1 = P_2[/tex]

Here [tex]P_1[/tex] is the pressure of the manometer at the point corresponding to the base of the tank and this is mathematically represented as

[tex] P_{atm} + P_1 = P_{atm} + P_t[/tex]

Here [tex]P_t[/tex] is the pressure due to the toluene level in the tank and in the arm of the manometer connected to the tank and this is mathematically represented as

[tex]P_t = \rho_t * g * h_i[/tex]

Here

[tex]\rho_t [/tex] is the density of toluene with value [tex]\rho_t = 867 kg/m^3 [/tex]

[tex]h_i[/tex] is the height of the connected arm above the point equivalent to the base of the tank , this mathematically represented as

[tex]h_i = d - h + R[/tex]

and [tex] P_2 [/tex] is the the pressure at the open arm of the manometer at the point equivalent to the base of the base of the tank and this is mathematically represented as

[tex] P_2 = \rho_f * g * h_f [/tex]

Here

[tex]\rho_f[/tex] is the density of the fluid in use , if it is water the density is

[tex]\rho_w = 1000 \ kg /m^3 [/tex]

and if it is mercury the density is

[tex]\rho_m = 13600 \ kg /m^3 [/tex]

[tex]h_f[/tex] is the height of the fluid in the open arm of the manometer from the point equivalent to the base of the tank which is equivalent the manometer reading R

So when the fluid is water we have

[tex] P_{atm} + \rho_t* g *(d - h + R) = P_{atm} + \rho_f * g * h_f[/tex]

=> [tex] \rho_t* (d - h + R) = \rho_w * h_f[/tex]

=> [tex] 867 (500 - 150 + R) = 1000 * R [/tex]

=> [tex] R = 2281.6 \ cm [/tex]

So when the fluid is mercury we have

[tex] \rho_t* (d - h + R) = \rho_m * h_f[/tex]

=> [tex] 867 (500 - 150 + R) = 13600 * R [/tex]

=> [tex] R = 23.83 \ cm [/tex]

The difference in the mercury reading for mercury due to the fact that they have different densities as we have seen in this calculation

So the best fluid to use is mercury because for water a slight change in toluene level will cause a large change in height .