Please help!
suppose you have some 20 degrees celsius water and some boiling water. How much of each would you use to make 4 kilograms of water at 45 degrees celsuis?
let M be the cool water, m be the boiling water.
Mc(45-20)+mc((45-100)=0
where M+m=4
M(25)+(4-M)(-55)=0 solve
To solve this problem, we need to use the concept of heat transfer, specifically the principle of conservation of energy. The heat gained by the cold water should equal the heat lost by the hot water.
Let's break down the problem into steps:
Step 1: Determine the initial temperature and mass of each water sample:
- Let's assume the initial temperature of the 20 degrees Celsius water is T1 = 20°C.
- Since we don't know the initial temperature of the boiling water, we will use T2 for now.
- Let's assume the mass of the 20 degrees Celsius water is m1 = x kilograms.
- Since we don't know the mass of the boiling water, we will use m2 for now.
Step 2: Calculate the heat gained by the cold water:
- We can use the specific heat capacity (c) of water, which is approximately 4.18 J/g°C, to calculate the heat gained by the cold water.
- The formula for heat transfer is Q = mcΔT, where Q is the heat gained (in joules), m is the mass of the water (in grams), c is the specific heat capacity of water (in J/g°C), and ΔT is the change in temperature (final temperature - initial temperature).
- Plugging in the values, we have Q1 = m1c(Tf - T1).
Step 3: Calculate the heat lost by the hot water:
- Similar to step 2, we can calculate the heat lost by the hot water using the same formula.
- However, the initial temperature is unknown, so we'll use T2 instead.
- Q2 = m2c(T2 - Tf).
Step 4: Apply the principle of conservation of energy:
- According to the principle of conservation of energy, the heat gained by the cold water should equal the heat lost by the hot water.
- Therefore, Q1 = Q2.
Step 5: Calculate the final temperature:
- In this step, we'll solve for T2, the initial temperature of the boiling water, using the equation Q1 = Q2.
- Set the equation Q1 = Q2 and substitute the expressions from steps 2 and 3: m1c(Tf - T1) = m2c(T2 - Tf).
- Rearrange the equation to solve for T2: T2 = Tf + (m1c(Tf - T1))/(m2c).
Step 6: Calculate the masses of the cold and hot water needed to achieve the desired final temperature:
- Since we know the mass of the final water sample is 4 kilograms, we can calculate the mass of the cold water using the equation m1 + m2 = 4 kilograms.
- Substitute the mass of the cold water (m1) and the calculated value of T2 into the equation from step 5.
- Rearrange the equation to solve for m2: m2 = 4 - m1.
Now that we have the necessary formulas and steps, you can plug in the given values and calculate the mass of the cold and hot water needed.