Cool Coffee with Melting Ice Cube: Solve for T

[aub]

Homework Statement

An insulated Thermos contains 190 cm3 of hot coffee at 83.0°C. You put in a 18.0 g ice cube at its melting point to cool the coffee. By how many degrees (in Celsius) has your coffee cooled once the ice has melted and equilibrium is reached? Treat the coffee as though it were pure water and neglect energy exchanges with the environment. The specific heat of water is 4186 J/kg·K. The latent heat of fusion is 333 kJ/kg. The density of water is 1.00 g/cm3.

(the ice is initially at 0 C)

The Attempt at a Solution

i tried this but didnt work :
190 * 4.186 * (83.0 - T) = 18.0 * 333 + 18.0 * 4.186 * (T -0) and solved for T.

can someone help please?

 

[Borek]

Why 3333?

Pay attention to units.

 

[aub]

Why 3333?

Pay attention to units.

i edited my previous post because i wrote 3333 here by mistake..

btw i tried both 333 and .333 and didnt work

 

[Borek]

Other than that equation looks OK to me.

 

[rwisz]

Based on the given information, we can use the principle of conservation of energy to solve for the final temperature (T) of the coffee once the ice has melted. This can be done by equating the energy gained by the ice (due to the latent heat of fusion) to the energy lost by the coffee (due to its decrease in temperature).

Let's first calculate the energy gained by the ice cube:

Q_ice = m_ice * L_fusion (where Q is the energy gained, m is the mass of the ice, and L_fusion is the latent heat of fusion)

Q_ice = 18.0 g * 333 kJ/kg = 5.994 kJ

Next, let's calculate the energy lost by the coffee:

Q_coffee = m_coffee * c * (T_final - T_initial) (where c is the specific heat of water, and T_final and T_initial are the final and initial temperatures of the coffee, respectively)

We can rearrange this equation to solve for T_final:

T_final = T_initial - (Q_coffee / (m_coffee * c))

Plugging in the values given in the problem, we get:

T_final = 83.0°C - (5.994 kJ / (190 cm3 * 1.00 g/cm3 * 4.186 J/g·K))

T_final = 83.0°C - (0.00794 K) = 82.992°C

Therefore, the coffee has cooled by approximately 0.008°C once the ice has melted and equilibrium is reached.