The "steam" above a freshly made cup of instant coffee is really water vapor droplets condensing after evaporating from the hot coffee. What is the final temperature of 200 g of hot coffee initially at 90.0°C if 2.00 g evaporates from it? The coffee is in a Styrofoam cup, and so other methods of heat transfer can be neglected. Assume that coffee has the same physical properties as water.
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To determine the final temperature of the coffee after 2.00 g evaporates, we can use the principle of conservation of energy.
The energy lost due to the evaporation of 2.00 g of coffee can be calculated using the equation:
Q = m * ΔHvap
where Q is the energy, m is the mass, and ΔHvap is the enthalpy of vaporization.
For water, the enthalpy of vaporization is approximately 40.7 kJ/mol. However, we need to convert grams to moles to use this value.
The molar mass of water is approximately 18 g/mol, so the number of moles of coffee that evaporated is:
n = m / M
where n is the number of moles, m is the mass evaporated (2.00 g), and M is the molar mass of water (18 g/mol).
n = 2.00 g / 18 g/mol = 0.1111 mol
Now, we can calculate the energy lost:
Q = n * ΔHvap = 0.1111 mol * 40.7 kJ/mol = 4.52 kJ
The energy lost due to evaporation is equal to the energy gained by the remaining coffee in order to maintain energy conservation. Since the coffee is in a Styrofoam cup, which is a good insulator, we can neglect heat loss to the surroundings. Therefore, the energy gained by the coffee is equal to the heat lost due to evaporation.
The specific heat capacity of coffee and water is approximately 4.18 J/g°C. We can use the principle of energy conservation to relate the change in temperature to the energy gained:
Q = mcΔT
where Q is the energy gained (4.52 kJ), m is the mass of the remaining coffee (200 g - 2.00 g = 198 g), c is the specific heat capacity, and ΔT is the change in temperature.
Rearranging the equation to solve for ΔT:
ΔT = Q / (mc)
Substituting the known values:
ΔT = 4.52 kJ / (198 g * 4.18 J/g°C) = 0.054°C
The final temperature can be obtained by subtracting ΔT from the initial temperature:
Final temperature = Initial temperature - ΔT = 90.0°C - 0.054°C = 89.946°C (rounded to three decimal places)
Therefore, the final temperature of the coffee after 2.00 g evaporates is approximately 89.946°C.