Three liquids are at temperatures of 4 ◦C, 24◦C, and 29◦C, respectively. Equal masses of the first two liquids are mixed, and the equi- librium temperature is 21◦C. Equal masses of the second and third are then mixed, and the equilibrium temperature is 26.1 ◦C. Find the equilibrium temperature when equal masses of the first and third are mixed. Answer in units of ◦C.

T₁₃ = 24.1°C

Explanation:

Given

m = mass of each liquids (all masses are equal)

C₁  = specific heat of the first liquid

C₂ = specific heat of the second  liquid

C₃ = specific heat of the third liquid

T₁ = 4°C (temperature of the first liquid)

T₂ = 24°C  (temperature of the second liquid)

T₃ = 29°C  (temperature of the third liquid)

Temperature of 1+2 liquids mix: T₁₂ = 21°C

Temperature of 2+3 liquids mix: T₂₃ = 26.1°C  

Temperature of 1+3 liquids mix: T₁₃ = ?

We can apply the relation ∑Q = 0

We assume the system is isolated and the process is adiabatic.

Mix 1:

Q₁ + Q₂ = 0

where

Q₁ = m*C₁*(T₁-T₁₂)

and

Q₂ = m*C₂*(T₂-T₁₂)

then

m*C₁*(T₁-T₁₂) + m*C₂*(T₂-T₁₂) = 0

⇒ C₁*(T₁-T₁₂) + C₂*(T₂-T₁₂) = 0

⇒ (4°C-21°C)*C₁ + (24°C-21°C)*C₂ = 0

⇒ -17°C*C₁ + 3°C*C₂ = 0

⇒ C₁ = (3/17)*C₂ = 0.176*C₂     (i)

Mix 2:

Q₂ + Q₃ = 0

where

Q₂ = m*C₂*(T₂-T₂₃)

and

Q₃ = m*C₃*(T₃-T₂₃)

then

m*C₂*(T₂-T₂₃) + m*C₃*(T₃-T₂₃) = 0

⇒ C₂*(T₂-T₂₃) + C₃*(T₃-T₂₃) = 0

⇒ (24°C-26.1°C)*C₂  + (29°C-26.1°C)*C₃ = 0

⇒ -2.1°C*C₂ + 2.9°C*C₃ = 0

⇒ C₃ = 0.724*C₂      (ii)

Mix 3:

Q₁ + Q₃ = 0

where

Q₁ = m*C₁*(T₁-T₁₃)

and

Q₃ = m*C₃*(T₃-T₁₃)

then

m*C₁*(T₁-T₁₃) + m*C₃*(T₃-T₁₃) = 0

⇒ C₁*(T₁-T₁₃) + C₃*(T₃-T₁₃) = 0

⇒ (4°C-T₁₃)*C₁ + (29°C-T₁₃)*C₃ = 0

If we use the following relations  C₁ = 0.176*C₂ and C₃ = 0.724*C₂  we obtain

(4°C-T₁₃)*0.176*C₂ + (29°C-T₁₃)*0.724*C₂ = 0

⇒ (4°C-T₁₃)*0.176 + (29°C-T₁₃)*0.724 = 0

⇒ 0.706°C - 0.176*T₁₃ + 21°C - 0.724*T₁₃ = 0

⇒ 0.9*T₁₃ = 21.706°C

⇒ T₁₃ = 24.1°C

the order for strong to weak forces such as

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force

the fundamental forces are those forces that act between to objects or element to make interactions. there are four types of fundamental forces such as strong and weak nuclear forces (due to protons and electrons), electromagnetic force (due to the motion of electrons) and gravitational force (due to orbital movement of the planet). if these forces are arranged on the basis of their maximum strength then the order is

strong nuclear force → electromegnetic force → weak nuclear force → gravitational force