A 200 gram sample of hot water at 89°C is mixed with 200 grams of cool water at 16°C. Assume there is no loss of heat energy to the surroundings, what is the final temperature of the mixture when they reach thermal equilibrium?
the sum of heats gained is zero.
200(cw)(tf-89)+200(cw)(Tf-16)=0
2tf=89+15
tf= 52C
To find the final temperature of the mixture when they reach thermal equilibrium, we can use the principle of conservation of energy.
The amount of heat lost by the hot water will be equal to the amount of heat gained by the cool water. We can calculate the amount of heat gained or lost using the formula:
Q = mcΔT,
where Q is the heat gained or lost, m is the mass of the water, c is the specific heat capacity of water, and ΔT is the change in temperature.
For the hot water:
Q_hot = m_hot * c * ΔT_hot,
where m_hot is the mass of the hot water, c is the specific heat capacity of water (approximately 4.18 J/g°C), and ΔT_hot is the change in temperature of the hot water.
Similarly, for the cool water:
Q_cool = m_cool * c * ΔT_cool,
where m_cool is the mass of the cool water, c is the specific heat capacity of water, and ΔT_cool is the change in temperature of the cool water.
Since the hot water is losing heat and the cool water is gaining heat, the equation becomes:
Q_hot = -Q_cool.
Substituting the values:
m_hot * c * ΔT_hot = -m_cool * c * ΔT_cool.
After rearranging the equation, we get:
ΔT_hot / ΔT_cool = -m_cool / m_hot.
We can now plug in the given values:
ΔT_hot / (16 - ΔT_hot) = -200 g / 200 g.
Simplifying the equation:
ΔT_hot / (16 - ΔT_hot) = -1.
Now, we can solve for ΔT_hot:
ΔT_hot = -16 + ΔT_hot.
Rearranging the equation:
2ΔT_hot = -16.
Simplifying the equation:
ΔT_hot = -8°C.
Since the change in temperature is negative, it means the hot water lost heat. Thus, the hot water's final temperature at thermal equilibrium will be:
Final temperature = initial temperature + ΔT_hot.
Final temperature = 89°C - 8°C = 81°C.
Hence, the final temperature of the mixture when they reach thermal equilibrium will be 81°C.