# Describe what would happen to the temperature of water if a person poured 2 liters of water at 25 degrees C into a container that has 2 liters of water at 75 degrees C. What temperature would the mixture become?

##### 1 Answer

Conceptually, we should be able to figure this out in less than

*The final temperature for two equal masses of water combined will be halfway between the high and low temperatures.*

Mathematically, we assume **conservation of thermal energy** *out* from the hot body of water *into* the cold body of water:

#q_(cold) = mC_PDeltaT_(cold)#

#= -q_(hot) = -mC_PDeltaT_(hot)# ,where

#q# is heat flow,#q_(hot) < 0# , and#q_(cold) > 0# .#C_P = "4.184 J/g"^@ "C"# is the specific heat capacity of water at constant pressure (lab bench conditions), and#m# is the mass of the water in#"g"# .

Both bodies of water have the same volume and hence *approximately the same mass* (more or less...), assuming similar densities at these different temperatures.

If we assume we don't know what the **final temperature** is, except for that it will be **the same** for both bodies of water (as required to reach thermal equilibrium), then:

#cancel(mC_P)DeltaT_(cold) = -cancel(mC_P)DeltaT_(hot)#

#=> T_f - T_(i,cold) = -(T_f - T_(i,hot))#

#=> T_f - 25^@ "C" = -(T_f - 75^@ "C")#

#=> 25^@ "C" - T_f = T_f - 75^@ "C"#

#=> 100^@ "C" = 2T_f#

#=># #color(blue)(T_f ~~ 50^@ "C")#

Indeed,