"You want to cook 250g of pasta (the pasta is at 20 degrees C originally). The instructions say to boil 2L of water, then place the dry pasta into the water. a) If the specific heat of the pasta is 1800J.kg-1.K-1, and the specific heat of water is 4186j.kg-1.k-1, what is the temperature of the water immediately after the pasta is added to the vigorously boiling water?"
So, I didn't know how to include time into this so I made the assumption that the whole system was in equilibrium immediately after the pasta was added. I took the equation
Q=cmDeltaT and set it to equal to 0, and came with this equation
c(pasta)*m(pasta)*Ti(pasta)-c(pasta)*m(pasta)*Tf(pasta) = - c(water)*m(water)*Ti(water)-c(water)*m(water)*(Tf(water).
After plugging in my numbers (at some point, I equated Tf(pasta) to Tf(water) because I was assuming they were in equilibrium) and got -98.4 degrees Celsius. Is there any other way to do this?
For b), The question asks how long the water will take to return to a boil if the burner is supplying 1 kW to the pot. I'm not even sure where to start with this one. Even an equation with a time component would be appreciated.
Thank you in advance :)
To solve part a) of the question, it seems like you made some assumptions and set up an equation using the principle of conservation of energy. However, there is an error in your equation. Let's go through the correct approach together.
The heat gained by the water and the pasta must be equal to the heat lost by the pasta, as the system reaches equilibrium. The equation can be set up as follows:
(c_pasta * m_pasta * (Tf_pasta - Ti_pasta)) + (c_water * m_water * (Tf_water - Ti_water)) = 0
Where:
c_pasta: specific heat of pasta (1800 J/kg.°C)
m_pasta: mass of pasta (250 g = 0.25 kg)
Tf_pasta: final temperature of pasta (unknown)
Ti_pasta: initial temperature of pasta (20 °C)
c_water: specific heat of water (4186 J/kg.°C)
m_water: mass of water (2 L = 2 kg)
Tf_water: final temperature of water (unknown)
Ti_water: initial temperature of water (100 °C, assuming boiling)
We can rearrange this equation to solve for Tf_water:
Tf_water = [c_pasta * m_pasta * (Ti_pasta - Tf_pasta)] / (c_water * m_water) + Ti_water
Substituting the given values:
Tf_water = [1800 * 0.25 * (20 - Tf_pasta)] / (4186 * 2) + 100
Now, let's solve for Tf_pasta by assuming Tf_pasta = Tf_water:
Tf_water = [1800 * 0.25 * (20 - Tf_pasta)] / (4186 * 2) + 100
Tf_pasta = 20 - [(Tf_water - 100) * 4186 * 2] / (1800 * 0.25)
Now, plug in the value of Tf_water into the equation for Tf_pasta to get the final answer.
For part b) of the question, we need to calculate the time taken for the water to return to a boil. To do this, we can use the equation:
Energy input = Energy required to heat water + Energy required to boil water
The energy input is given as 1 kW, which is equal to 1000 J/s.
The energy required to heat the water can be calculated using the equation: Energy = c_water * m_water * ΔT, where ΔT is the change in temperature needed to reach boiling point (100 °C - Ti_water).
The energy required to boil the water can be calculated using the equation: Energy = heat of vaporization * m_water, where heat of vaporization is the amount of heat energy required to convert a unit mass of a substance from the liquid phase to the gas phase.
Once we have the total energy required, we can divide it by the energy input rate (1 kW) to find the time taken.
I hope this explanation helps you in solving the problem correctly. Let me know if you have any further questions!