How do i calculate the final tempurature of two items?

Question

200g of water at 20 degrees celsius and 100g of iron nails at 80 degrees celsius when mixed what will be the final tempurature

Answer 1

To answer this question you need to apply conservation of energy. That is to say that in an isolated system (no heat/work input or output) the total energy level stays the same. Think about there being an invisible box around the water at 20C and the nails at 80C. Once you combine the two together the total amount of energy cannot change. Mathematically we can write this as:

E1 = E2

This means the energy at state 1, when the two are separated, is the same as the energy at state 2, when they are mixed together. Now since we are dealing with solids I will assume that Cp is equal to Cv as they are relatively incompressibe. We will also use the state of 0 degrees kelvin as the baseline as any object at 0K has a thermal energy of 0J. So we can rewrite the energy equation as:

mw * Cpm * Tw1 + mi * Cpi * Ti1 = mw * Cpm * Tw2 + mi * Cpi * Ti2

This is from the equation:

E = m * Cp * T

Which is simply a measure of the internal thermal energy. Now we can get the values of Cpw and Cpi from a table. These are:

Cpw = 4186 J/(kg*K)

and

Cpi = 450 J/(kg*K)

Now we know everything on the left hand side of the equation. We can caculate the seperate components as (remember to add 273C to the temeperatures to have them in terms of Kelvin for the reason above):

mw * Cpm * Tw1 = 0.2kg * 4186 J/(kg*K) * (20 + 273)K

mw * Cpm * Tw1 = 254299 J

mi * Cpi * Ti1 = 0.1kg * 450 J/(kg*K) * (80 + 273)K

mi * Cpi * Ti1 = 15885 J

Now we have the left hand side as the sum of these two energy's. Therefore:

E1 = 270184 J

This is the total thermal energy at state 1, before everything mixes. Now there is one other equation that we need to solve. If everything is well mixed and allowed to reach equiibrium, the temperature of the iron and of the water should be the same in state two. This equation can be writted as:

Tw2 = Ti2

We will call this second state temperature simply T2. So we can write:

E1 = (mw * Cpw + mi * Cpi) * T2

Notice there is only one unknown in this, which is T2. We can rearrange to solve for T2:

T2 = E1 / (mw * Cpw + mi * Cpi)

T2 = 270184J / (0.2*4186 + 0.1*450)

T2 = 306.26 K

We can convert this to celsius by subtracting 273 K.

T2 = (306.26 - 273) C

T2 = 33.26C

Notice that this is between the two initial temperatures. Intuitively this makes sense as when you put something hot into something cold, the final temperature is inbetween the two.

Best of Luck

Let me know if you need any more information.