A person makes a quantity of iced tea by mixing 668 g of hot tea (essentially water) with an equal mass of ice at its melting point. If the initial hot tea is at a temperature of 86∘C, what are the (a) final temperature of the ice tea (in ∘C) and (b) mass of the remaining ice? what are the (c) final temperature of the ice tea (in ∘C) and (d) mass of the remaining ice if the initial hot tea is at a temperature of 68∘C? The specific heat of water is 4186 J/kg⋅K) and the latent heat of fusion is 333 kJ/kg.
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To solve this problem, we need to consider the heat gained or lost by each component of the mixture: the hot tea and the ice.
(a) To find the final temperature of the iced tea, we can start by calculating the heat lost by the hot tea and the heat gained by the ice. Since there is no heat loss or gain to the surroundings, the heat lost by the hot tea is equal to the heat gained by the ice.
The equation for heat is:
Q = mcΔT
Where:
Q = heat energy (Joules)
m = mass (kg)
c = specific heat capacity (J/kg⋅K)
ΔT = change in temperature (K)
The heat lost by the hot tea can be calculated using this equation as:
Q_hot_tea = m_hot_tea * c_water * ΔT_hot_tea
Where:
m_hot_tea = mass of hot tea (kg)
c_water = specific heat capacity of water (J/kg⋅K)
ΔT_hot_tea = initial temperature of the hot tea - final temperature of the iced tea
Since the hot tea is essentially water, its specific heat capacity is the same as that of water, which is 4186 J/kg⋅K.
Now, let's work on finding the heat gained by the ice:
Q_ice = m_ice * L_fusion + m_ice * c_water * ΔT_ice
Where:
m_ice = mass of ice (kg)
L_fusion = latent heat of fusion (J/kg)
c_water = specific heat capacity of water (J/kg⋅K)
ΔT_ice = final temperature of the iced tea - melting point temperature of ice (0°C)
Since the mass of hot tea is equal to the mass of ice, we can represent it as m_hot_tea = m_ice. Substituting this into the equations above, we get:
Q_hot_tea = m_ice * c_water * ΔT_hot_tea
Q_ice = m_ice * L_fusion + m_ice * c_water * ΔT_ice
Since Q_hot_tea = Q_ice, we can set them equal to each other and solve for the final temperature of the iced tea:
m_ice * c_water * ΔT_hot_tea = m_ice * L_fusion + m_ice * c_water * ΔT_ice
Now we can solve for the final temperature of the iced tea (ΔT_ice):
ΔT_ice = (m_ice * c_water * ΔT_hot_tea) / (m_ice * c_water - m_ice * L_fusion)
Substituting the given values:
m_ice = 668 g = 0.668 kg
ΔT_hot_tea = 86°C - final temperature of the iced tea
L_fusion = 333 kJ/kg = 333000 J/kg
c_water = 4186 J/kg⋅K
We can now calculate the final temperature of the iced tea.
(b) To find the mass of the remaining ice, we know that the initial mass of ice is equal to the mass of hot tea, which is 668 g. Subtracting the previously calculated mass of ice that melted (m_ice) from the initial mass of ice will give us the mass of the remaining ice.
Now, let's calculate the final temperature of the iced tea and the mass of the remaining ice using the given information.
(c) To repeat the calculation for a different initial temperature (68°C), we can use the same equations and substitute the new values:
ΔT_hot_tea = 68°C - final temperature of the iced tea
We can now calculate the final temperature of the iced tea and the mass of the remaining ice for this scenario as well.