Mixing Hot and Cold Water

Suppose that a hot drink and a cold drink are mixed together and you would like to predict the temperature of the mixture. To do this, we need to know the temperatures of the drinks before they are mixed, T1 and T2, and the amounts of each used in the mixture, V1 and V2.

Translated into math language:

T1V1 + T2 V2 = Tfinal(V1 + V2)

YOU NEED:

1 CBL Unit
TI-82/83 Calculator
Hot and Cold Water
2 Large Styrofoam Cups or Coffee Mugs (about 16 oz. capacity)

INSTRUCTIONS:

1. In this activity, you will record the temperature of water in two cups, then find the temperature when the contents of the cups are mixed together. To start, label one cup or mug as "Cup 1" and the other as "Cup 2."

2. Fill Cup 1 with 100 ml of cold water (about 10 °C) and Cup 2 with 150 ml of warm water (about 50 °C). IMPORTANT: Do not put any ice in the cold water cup.

3. Turn ON the CBL unit and plug the TI temperature probe into channel one. Turn the CBL into a thermometer by selecting MODE. The display should be reading a temperature in Celsius. Check to make sure the CBL is functioning properly by placing the temperature probe into your hand and verifying a temperature increase. If the CBL is not functioning correctly, shut it off by selecting 2nd, ON (may have to hold the ON key down for a second to get the CBL to shut down). Begin step 3 again.

4. Measure the temperature of cup 1 and cup 2 (be sure to allow the temperature reading to stabilize for 5 seconds before recording the reading). Mix the two cups and measure the new temperature, Tfinal.

5. Repeat this experiment with 4 other volumes and temperatures of water.


QUESTIONS:

1. Consider the equation given in the introductory section relating volumes and temperatures for mixed solutions:

T1V1 + T2 V2 = Tfinal(V1 + V2)

Use algebra and solve this equation for the mixture temperature Tfinal.

3. How does this value compare to the measured value of Tfinal listed in the our data table? What might have caused the difference between the calculated and measured mixture temperature values?

5. Exactly what volume of warm water, V2, should be added to Cup 1 so that the mixture temperature equals the average of T1 and T2? To do this, you will need to solve the mixture equation for V2

T1V1 + T2 V2 = Tfinal(V1 + V2)

EXTENSION:

The solve( feature of the TI-83 calculator makes it easy to solve the equation above for X. This feature finds a solution of an expression set equal to zero, for a specified variable, when given an initial guess. The syntax for this command is:

solve(expression, variable, guess)

Use the manual to learn how to use this feature and see if you can verify your answers.

APPLICATION PROBLEMS:

1. The directions on a box of instant cocoa tell you to prepare the drink by adding 6 oz. of hot water to the package contents. What amounts of cold water (8 °C) and boiling water (100 °C) should be combined to obtain 6 oz. of cocoa at a temperature of 68 °C?

2. Suppose the thermostat of your school's swimming pool malfunctions causing the water temperature to climb to 93 °F. The recommended temperature for competition is 78 °F. If the pool holds 200,000 gallons of water, how many gallons must be drained from the pool and replaced with tap water (42 °F) to make it ready for competition?

3. Solution A is 5% acid. Solution B is 17% acid. A chemist wants to mix the two to get 500 ml of a solution that is 12% acid. How much of each should be used?


Modified from an experiment from "Real World Math with the CBL", Texas Instruments, 1995. Prepared for SMART Center Workshop, July, 1996.
Revised 7/9/96.
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