Answer:
Approximately 121.35 grams of ice cubes must be added to the hot tea to bring the final temperature of the mixture to 20°C.
Step-by-step explanation:
To calculate the number of grams of ice cubes needed, we can use the principle of conservation of energy, which states that the heat lost by the hot tea will be equal to the heat gained by the ice cubes when they melt.
The heat lost by the hot tea can be calculated using the formula:
Q_hottea = m_hottea * C_hottea * ΔT_hottea
where:
m_hottea = mass of hot tea (in grams)
C_hottea = specific heat capacity of hot tea (approximately 4.18 J/g°C)
ΔT_hottea = change in temperature of hot tea (final temperature - initial temperature)
The heat gained by the ice cubes can be calculated using the formula:
Q_ice = m_ice * L_f
where:
m_ice = mass of ice cubes (in grams)
L_f = latent heat of fusion of ice (approximately 334 J/g)
Since the heat lost by the hot tea is equal to the heat gained by the ice cubes, we can set up the equation:
m_hottea * C_hottea * ΔT_hottea = m_ice * L_f
Now, we can plug in the given values:
C_hottea = 4.18 J/g°C
ΔT_hottea = 20°C - 85°C = -65°C (negative because the hot tea is losing heat)
L_f = 334 J/g
Now, let's solve for m_ice (mass of ice cubes):
m_ice = (m_hottea * C_hottea * ΔT_hottea) / L_f
Given that the initial temperature of the hot tea is 85°C and the density of water is approximately 1.0 g/mL, the mass of 1.0 L of hot tea can be calculated as:
m_hottea = 1.0 L * 1.0 g/mL = 1000 grams
Now, substitute the values into the equation:
m_ice = (1000 g * 4.18 J/g°C * -65°C) / 334 J/g
m_ice = -1000 * 4.18 * -65 / 334 ≈ 121.35 grams
Therefore, approximately 121.35 grams of ice cubes must be added to the hot tea to bring the final temperature of the mixture to 20°C. Since mass cannot be negative in this context, we interpret the negative sign in the calculation as an indication that heat is transferred from the hot tea to the ice, causing it to melt. So, in practice, you would need to add about 121.35 grams of ice cubes.