Heat of mixture of two substances (no state changes)

[Esoremada]

Homework Statement

A 100-g sample of Cu(s), initially at 100oC, is added to 140 g of water, initially at 25oC. What is the final temperature of the mixture?

Specific heat
(in J g-1 oC-1)
Cu(s) 0.385
H2O(l ) 4.184

Homework Equations

Q = m * ΔT * c

heat lost = heat gained
m₁ * ΔT₁ * c₁ = m₂ * ΔT₂ * c₂

The Attempt at a Solution

100 * (T₂ - 100) * 0.385 = 140 * (T₂ - 25) * 4.185
38.5* (T₂ - 100) = 585.9 * (T₂ - 25)
38.5T₂ - 3850 = 585.9T₂ - 14647.5
T₂ = 19.725°CBut the temperature can't be lower than both their starting temperatures, what did I do wrong? The answer is supposed to be 29.6

 

[gneill]

Take a look at the expressions you're using for ΔT for the two substances. One should be cooling down while the other is warming up, but you want both to yield positive numbers.

 

[TSny]

Hello, Esoremada.

You've been snagged by a common sign error for this type of problem. The expression Q = mcΔT represents the heat gained by an object. So, you have to be careful when writing heat lost by an object.

The system as a whole is isolated. So, the total heat gained by the system is zero:

Q₁+Q₂=0

m₁c₁ΔT₁ + m₂c₂ΔT₂ = 0

Rearrange to

-m₁c₁ΔT₁ = m₂c₂ΔT₂

Note the negative sign on the left (the heat lost term).

So, if you want to set it up as Heat Lost = Heat Gained, then you'll need to handle the sign of the heat lost.

[ASIDE: Did we just set a record for the number of posts within 1 minute?]

 

[Chestermiller]

100 - t₂

 

[Dick]

Homework Statement

A 100-g sample of Cu(s), initially at 100oC, is added to 140 g of water, initially at 25oC. What is the final temperature of the mixture?

Specific heat
(in J g-1 oC-1)
Cu(s) 0.385
H2O(l ) 4.184

Homework Equations

Q = m * ΔT * c

heat lost = heat gained
m₁ * ΔT₁ * c₁ = m₂ * ΔT₂ * c₂

The Attempt at a Solution

100 * (T₂ - 100) * 0.385 = 140 * (T₂ - 25) * 4.185
38.5* (T₂ - 100) = 585.9 * (T₂ - 25)
38.5T₂ - 3850 = 585.9T₂ - 14647.5
T₂ = 19.725°C


But the temperature can't be lower than both their starting temperatures, what did I do wrong? The answer is supposed to be 29.6

Heat lost and heat gained should both be positive. You know T2 is between 100 and 25. You have a sign problem. Change (T2-100) to (100-T2).