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QUESTION

# 10.0g of a metal, initially at 25 Celsius, are placed into 10.0 g of water, initially at 100 Celsius. Which metal will have the lowest final temperature? A) Aluminium (0.902 J/g*Celsius) B) Gold (0.129 J/g* Celsius) C) Copper (0.385 J/g*Cesius)

Aluminum will produce the lowest final temperature at "86.7"^("o")"C".

Q=mcDeltaT, where Q is the amount of heat energy lost or gained, m is the mass in grams, and DeltaT is the change in temperature, T_"final" - T_"initial".

The final temperature occurs when the metal and water temperatures are the same, meaning they are at equilibrium. At equilibrium, the magnitude of the heat energy gained by the metal is equal to the magnitude of the heat energy lost by the water. The of water is "4.184 J/g"^("o")"C".

In order to find the final temperature, or the temperature at equilibrium, set the heat capacity for the metal and the water equal, except Q for water is negative because it loses heat energy.

Q_"metal" = -Q_"water" (The negative sign indicates that the water loses heat energy.)

mc(T_f - T_i) = -mc(T_f - T_i)

A) Aluminum

10.0g*0.902 "J/g^oC"*(T_f - 25^oC) = -10.0g*4.184 "J/g*"^oC"*(T_f - 100^oC)

9.02(T_f - 25^oC) = -41.84(T_f - 100^oC)

Apply the distributive property.

9.02T_f - 225.5^oC = -41.84T_f + 4184^oC

Add 41.84T_f to both sides, and add 225.5^oC to both sides.

50.86T_f = 4410^oC

Divide both sides by 50.86.

T_f = 86.7^oC

B) Gold

10.0g*0.129 "J/g^oC"*(T_f - 25^oC) = -10.0g*4.184 "J/g*"^oC"*(T_f - 100^oC)

1.29(T_f - 25^oC) = -41.84(T_f - 100^oC)

Apply the distributive property.

9.02T_f - 32.25^oC = -41.84T_f + 4184^oC

Add 41.84T_f to both sides, and add 32.25^oC to both sides.

43.13T_f = 4216^oC

Divide both sides by 43.13.

T_f = 97.8^oC

C) Copper

10.0g*0.385 "J/g^oC"*(T_f - 25^oC) = -10.0g*4.184 "J/g*"^oC"*(T_f - 100^oC)

3.85(T_f - 25^oC) = -41.84(T_f - 100^oC)

Apply the distributive property.

3.85T_f - 96.25^oC = -41.84T_f + 4184^oC

Add 41.84T_f to both sides, and add 96.25^oC to both sides.

45.69T_f = 4280^oC

Divide both sides by 45.69.

T_f = 93.7^oC