A 10.0 g sample of liquid gallium metal, at its melting point, is added to 50.0g of water. The initital temperature of the water is 24.0 degrees C. If the molar enthalpy of freezing gallium is -5.54 kj/mol, what is the final temperature of the water?
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To find the final temperature of the water after adding gallium at its melting point, we need to use the concept of heat transfer. This involves calculating the heat gained by the water and equating it to the heat lost by the gallium during the phase change.
First, let's calculate the heat gained by the water using the formula:
q = m * c * ΔT
where q is the heat gained, m is the mass of water, c is the specific heat capacity of water, and ΔT is the change in temperature.
Given:
Mass of water (m) = 50.0 g
Initial temperature (T1) = 24.0 °C
The specific heat capacity of water (c) is approximately 4.18 J/g·°C.
So, the heat gained by the water can be calculated as:
q_water = m * c * ΔT_water
To find ΔT_water, we need to know the final temperature (Tf) of the water. Let's assume that the final temperature of the water is Tf.
ΔT_water = Tf - T1
Now, let's calculate the heat lost by the gallium during freezing using the concept of enthalpy change:
q_gallium = n * ΔH
where q_gallium is the heat lost by the gallium, n is the number of moles of gallium, and ΔH is the molar enthalpy of freezing gallium.
The number of moles of gallium (n) can be calculated using the molar mass of gallium (approximately 69.72 g/mol) and the given mass of gallium.
First, convert the mass of gallium from grams to moles:
n = mass / molar mass
Given:
Mass of gallium = 10.0 g
Molar mass of gallium ≈ 69.72 g/mol
Now, calculate the number of moles of gallium:
n = 10.0 g / 69.72 g/mol
Next, calculate the heat lost by the gallium:
q_gallium = n * ΔH
Given:
Molar enthalpy of freezing gallium (ΔH) = -5.54 kJ/mol
Note: Convert kJ to J by multiplying by 1000.
Now, calculate the heat lost by the gallium:
q_gallium = n * ΔH * 1000
Finally, according to the principle of conservation of energy, the heat lost by gallium should be equal to the heat gained by water:
q_gallium = q_water
n * ΔH * 1000 = m * c * ΔT_water
Substituting the known values, we can solve for ΔT_water:
(10.0 g / 69.72 g/mol) * (-5.54 kJ/mol) * 1000 = (50.0 g) * (4.18 J/g·°C) * (Tf - 24.0 °C)
Simplifying the equation, we can solve for Tf:
Tf = [(10.0 g / 69.72 g/mol) * (-5.54 kJ/mol) * 1000] / [(50.0 g) * (4.18 J/g·°C)] + 24.0 °C
Calculating the above expression will give you the final temperature (Tf) of the water after adding the gallium.