Equilibrium Temperature of Cup and Water When Adding Ice - Homework Solution

[map7s]

Homework Statement

A 35 g ice cube at 0.0°C is added to 110 g of water in a 62 g iron cup. The cup and the water have an initial temperature of 36°C. Find the equilibrium temperature of the cup and its contents.

Homework Equations

heat lost by water+cup Q(w)=m(water)c(w)deltaT+m(cup)c(c)deltaT
amt. of heat needed to melt ice Q(ice)=m(ice)L(f)
amt. of heat left Q=Q(w)-Q(ice)
deltaT=Q/((m(w)+m(ice))c(w)+(m(cup))c(cup))

The Attempt at a Solution

I tried plugging in my numbers to the equations above but I didn't get the right answer. I'm fairly certain that my equations are correct but the only one that I am uncertain about is the last one.

 

[map7s]

Never mind...I found out that it was simply a miscalculation error in converting units. :)

 

[kyle8921]

I'm not sure if the deltaT should be positive or negative in this case.

I would like to first clarify that the provided equations are not entirely accurate and may lead to incorrect results. The correct equations for this problem would be:

Heat lost by water and cup: Q = (mwater * cwater + mcup * ccup) * ΔT

Heat needed to melt ice: Q = mice * Lf

Equilibrium temperature: Q = 0, which means that the heat lost by the water and cup must be equal to the heat needed to melt the ice.

Now, plugging in the given values, we get:

Heat lost by water and cup: Q = (110 g * 4.18 J/g°C + 62 g * 0.45 J/g°C) * (36°C - Tf)

Heat needed to melt ice: Q = 35 g * 334 J/g

Equilibrium temperature: (110 g * 4.18 J/g°C + 62 g * 0.45 J/g°C) * (36°C - Tf) = 35 g * 334 J/g

Solving for Tf, we get Tf = -0.46°C. This means that the equilibrium temperature of the cup and its contents will be slightly below 0°C, which is the melting point of ice. This makes sense because the ice will absorb heat from the water and cup, causing them to cool down and reach an equilibrium temperature below the initial temperature of 36°C.

Therefore, it is important to use the correct equations and units when solving scientific problems to ensure accurate results. Additionally, it is always a good practice to double check your calculations and results to avoid any errors.