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One 63-g ice cube at 0.0 °C is dropped into 5.00 x 10^2 mL of tea to make iced tea. The tea was initially at 20.0 °C. When thermal equilibrium is reached, all of the ice will have melted, and the temperature of the mixture will be somewhere between 20.0 °C and 0.0 °C. Calculate the final temperature of the beverage.

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Answer:

To calculate the final temperature of the beverage when the ice has melted and thermal equilibrium is reached, you can use the principle of conservation of energy. The heat lost by the tea as it cools down is equal to the heat gained by the ice as it melts.

The equation for the heat exchange is:

Q_lost = Q_gained

The heat lost by the tea can be calculated using the formula:

Q_lost = m * c * ΔT

Where:

m = mass of the tea (in grams)

c = specific heat capacity of the tea

ΔT = change in temperature (final temperature - initial temperature)

The heat gained by the ice can be calculated as the heat needed to melt the ice and raise its temperature to the final temperature. The total heat gained is the sum of the heat needed to melt the ice and the heat needed to raise the temperature of the resulting water to the final temperature.

Q_gained = (heat needed to melt the ice) + (heat needed to raise the temperature of the water)

Now, let's calculate each part:

1. Heat needed to melt the ice:

Q_melt = m_ice * L_f

Where:

m_ice = mass of the ice (63 g)

L_f = heat of fusion for ice (336 J/g)

2. Heat needed to raise the temperature of the resulting water:

Q_raise = m_water * c_water * ΔT

Where:

m_water = mass of the water produced when the ice melts (equal to the mass of the ice)

c_water = specific heat capacity of water

ΔT = final temperature - 0°C

Now, set Q_lost equal to Q_gained:

m * c * ΔT = (m_ice * L_f) + (m_water * c_water * ΔT)

You know the values for m_ice, L_f, c_water, and the initial and final temperatures. You need to find m, which is the mass of the tea.

Rearrange the equation and solve for m:

m = ((m_ice * L_f) + (m_water * c_water * ΔT)) / (c * ΔT)

Now, plug in the values and calculate m:

m = ((63 g * 336 J/g) + (63 g * 4.18 J/(g°C) * ΔT)) / (4.18 J/(g°C) * ΔT)

Once you find the mass of the tea (m), you can use it to calculate the final temperature (ΔT) by rearranging the equation:

ΔT = ((m_ice * L_f) + (m_water * c_water * ΔT)) / (m * c)

Now you can solve for ΔT and find the final temperature.

User Daniel Kurz
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