Mary mixes 5.00 lb of water at 200°F with 7.00 lb of water at 65.0°F. Find the final temperature of the mixture.
-Specific Heat of water is 1.00 BTU/lbs°F
To find the final temperature of the mixture, we can use the principle of conservation of energy.
The formula for the principle of conservation of energy is:
(heat gained by water 1) + (heat gained by water 2) = (heat lost by water 1 + water 2)
The formula for heat gained/lost is:
heat = mass * specific heat * change in temperature
Given:
- Water 1: mass = 5.00 lb, initial temperature = 200°F
- Water 2: mass = 7.00 lb, initial temperature = 65.0°F
- Specific heat of water = 1.00 BTU/lb°F
First, let's calculate the heat gained by water 1 and water 2:
Heat gained by water 1 = 5.00 lb * 1.00 BTU/lb°F * (final temperature - 200°F)
Heat gained by water 2 = 7.00 lb * 1.00 BTU/lb°F * (final temperature - 65.0°F)
Now, let's calculate the heat lost by the mixture:
Heat lost by the mixture = (5.00 lb + 7.00 lb) * 1.00 BTU/lb°F * (final temperature - x)
where x is the final temperature of the mixture.
Since energy is conserved, we can set up the equation:
(5.00 lb * 1.00 BTU/lb°F * (final temperature - 200°F)) + (7.00 lb * 1.00 BTU/lb°F * (final temperature - 65.0°F)) = (12.00 lb * 1.00 BTU/lb°F * (final temperature - x))
Now, we can solve for the final temperature:
5.00 * (final temperature - 200) + 7.00 * (final temperature - 65) = 12.00 * (final temperature - x)
Expanding the equation:
5.00 * final temperature - 1000 + 7.00 * final temperature - 455 = 12.00 * final temperature - 12.00 * x
Combining like terms:
12.00 * final temperature - 1455 = 12.00 * final temperature - 12.00 * x
Subtracting 12.00 * final temperature from both sides:
-1455 = -12.00 * x
Dividing both sides by -12.00:
x = -1455 / -12.00
Simplifying:
x = 121.25
Therefore, the final temperature of the mixture is approximately 121.25°F.