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 280 g of hot coffee initially at 90.0°C if 2.50 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.
cM[t(ini) –t(fin)] = mr
cMΔt = mr
Δt= mr/cM= 2.5•10⁻³•2260•10³/4183•0.28 = 4.8℃
T(fin) = 90-4.8 = 85.2℃
To determine the final temperature of the remaining coffee, you can use the principle of conservation of energy.
Here's how you can solve it step by step:
1. Start by calculating the heat lost by the hot coffee as it cools down. The amount of heat lost (Q) can be calculated using the formula Q = mcΔT, where:
- m is the mass of the coffee (280 g)
- c is the specific heat capacity of water (4.18 J/g°C)
- ΔT is the change in temperature (initial temperature - final temperature)
2. Since 2.50 g of coffee evaporates, the energy required for evaporation (Qevap) can be calculated using the formula Qevap = mL, where:
- m is the mass evaporated (2.50 g)
- L is the latent heat of vaporization of water (2260 J/g)
3. The total heat lost by the coffee is equal to the heat lost through cooling plus the heat lost through evaporation, so we have: Qtotal = Q + Qevap.
4. Let's assume the final temperature of the coffee is T. Therefore, the change in temperature ΔT = (90.0 - T). Substitute this value into the equations to get:
Q = mcΔT
Qevap = mL
Qtotal = Q + Qevap
Qtotal = mcΔT + mL
5. Rearrange the equation to solve for the final temperature (T):
mcΔT + mL = Qtotal
mc(90.0 - T) + 2.50 * 2260 = mcΔT
6. Simplify the equation by canceling out common terms:
90.0mc - mcT + 5650 = mcT
7. Rearrange the equation to isolate the final temperature (T):
T = (90.0mc + 5650) / 2mc
8. Substitute the given values into the equation and calculate the final temperature:
T = (90.0 * 280 + 5650) / (2 * 280)
T ≈ 60.4°C
Therefore, the final temperature of the coffee after 2.50 g evaporates is approximately 60.4°C.
To find the final temperature of the hot coffee after some of it evaporates, we can use the principle of conservation of energy. The amount of heat lost by the hot coffee will be equal to the amount of heat gained by the water vapor that evaporates.
First, let's calculate the amount of heat lost by the hot coffee:
Q_lost = m * c * ΔT
Where:
m = mass of the hot coffee
c = specific heat capacity of water (assumed to be the same for coffee)
ΔT = change in temperature of the hot coffee
The mass of the hot coffee that remains can be calculated by subtracting the mass of the evaporated water from the initial mass of the coffee:
m_hot_coffee = m_initial - m_evaporated
Given:
m_initial = 280 g
m_evaporated = 2.50 g
So,
m_hot_coffee = 280 g - 2.50 g
m_hot_coffee = 277.50 g
Next, let's calculate the amount of heat gained by the water vapor:
Q_gained = m_evaporated * L_v
Where:
m_evaporated = mass of water evaporated
L_v = latent heat of vaporization (for water)
The latent heat of vaporization for water is approximately 2.26 x 10^6 J/kg.
Q_gained = 2.50 g * (2.26 x 10^6 J/kg)
Q_gained = 5.65 x 10^6 J
Since Q_lost = Q_gained, we can equate the two equations:
m_hot_coffee * c * ΔT = m_evaporated * L_v
Rearranging the equation to solve for ΔT:
ΔT = (m_evaporated * L_v) / (m_hot_coffee * c)
Substituting the given values:
ΔT = (2.50 g * 2.26 x 10^6 J/kg) / (277.50 g * 4.18 J/g°C)
Simplifying,
ΔT ≈ 4.82°C
To find the final temperature, we subtract ΔT from the initial temperature of the hot coffee:
Final Temperature = Initial Temperature - ΔT
Final Temperature = 90.0°C - 4.82°C
Final Temperature ≈ 85.18°C
Therefore, the final temperature of the hot coffee, after 2.50 g evaporates, is approximately 85.18°C.