Question

100 grams of boiling water (temperature 100

Answer 1

This question requires the heat capacity equation , with Q being the amount of heat energy that is transferred in a material of mass m and heat capacity c when it undergoes a change in temperature T.

Part (a)

Step 1) Since the heat lost by the water is transferred to the pan, the change in heat energy of the water is equal to the change for the aluminum pan:

Step 2) Expand that using the heat capacity equation:

The expressions involving the initial temperature of the water T_wand the initial temperature of the pan T_p are written like because the water is cooling off, but the pan is heating up.

Step 3) The final temperature T_f of the water and pan needs to be isolated, so next distribute to get rid of the parentheses:

Step 4) Get the terms with the final temperature alone on the same side:

Step 5) Factor out the common final temperature on the right:

Step 6) Divide both sides to get the T_f alone on the right:

Step 7) Then insert the values, with the mass of the water m_w being the 100 g, the mass of the pan m_p being 950 g, the initial temperature of the water T_w being 100^o C, the initial temperature of the pan T_p being 26^o C, the heat capacity of the water c_w being 4.2 J/g/K, and the heat capacity of the aluminum pan c_p being 0.9 J/g/K:

Step 8) Calculate that:

So the final temperature of the water and pan combination is about 50.4^o C.

Part (b)

Step 1) Heat Q is added to the water and pan in the form of work W due to stirring, and heat energy Q_H from the stove. Write that as an equation:

The goal is to find the heat added from the stove, Q_H.

Step 2) Expand the left side by using the heat capacity equation, using that both the water and the pan will be heating up from the same initial temperature To:

Step 3) Combine like terms on the left:

Step 4) Fill in the same values for the water and pan, though using 50.4^o C as the initial temperature, and 78.4^o C as the final temperature, as well as the 22,510 J of work for W:

Step 5) Simplify that:

Step 6) Find Q_H:

So about 15,000 J (or 15 kJ) worth of heat energy was added from the stove to help heat the water and pan to the 78.4^o C.