To find the ratio of the total surface energies before and after the change, we can use the formula for the surface energy of a drop of mercury, which is given by:

E = 2πrT

where E is the surface energy, r is the radius of the drop, and T is the surface tension of mercury.

Let's assume the radius of each small drop is R. So, the surface energy of each small drop before the change is:

E1 = 2πR*T

The total surface energy before the change is the sum of the surface energies of the two small drops:

E_total_before = E1 + E1 = 2E1

Now, let's consider the single large drop formed after the coalescence. The radius of the large drop can be calculated using the formula for the volume of a sphere:

V = (4/3)πR^3

Since the volume is conserved during the coalescence, we can equate the volumes of the two small drops to the volume of the large drop:

2*(4/3)πR^3 = (4/3)πR_large^3

Simplifying this equation, we find:

R_large = (2^(1/3))*R

Now, we can calculate the surface energy of the large drop after the change:

E_large = 2πR_large*T

Substituting the value of R_large, we get:

E_large = 2π*(2^(1/3))*R*T

The ratio of the total surface energies before and after the change is:

(E_total_before) / (E_large) = (2E1) / (2π*(2^(1/3))*R*T)

Simplifying this expression, we find:

(E_total_before) / (E_large) = 1 / (π*(2^(1/3)))

Therefore, the ratio of the total surface energies before and after the change is 1 : (π*(2^(1/3))).

Question

To find the ratio of the total surface energies before and after the change, we can use the formula for the surface energy of a drop of mercury, which is given by:

E = 2πrT

where E is the surface energy, r is the radius of the drop, and T is the surface tension of mercury.

Let's assume the radius of each small drop is R. So, the surface energy of each small drop before the change is:

E1 = 2πR*T

The total surface energy before the change is the sum of the surface energies of the two small drops:

E_total_before = E1 + E1 = 2E1

Now, let's consider the single large drop formed after the coalescence. The radius of the large drop can be calculated using the formula for the volume of a sphere:

V = (4/3)πR^3

Since the volume is conserved during the coalescence, we can equate the volumes of the two small drops to the volume of the large drop:

2*(4/3)πR^3 = (4/3)πR_large^3

Simplifying this equation, we find:

R_large = (2^(1/3))*R

Now, we can calculate the surface energy of the large drop after the change:

E_large = 2πR_large*T

Substituting the value of R_large, we get:

E_large = 2π*(2^(1/3))*R*T

The ratio of the total surface energies before and after the change is:

(E_total_before) / (E_large) = (2E1) / (2π*(2^(1/3))*R*T)

Simplifying this expression, we find:

(E_total_before) / (E_large) = 1 / (π*(2^(1/3)))

Therefore, the ratio of the total surface energies before and after the change is 1 : (π*(2^(1/3))).

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