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PHYS 163 LAB #13 Newton's Law of Cooling DATA & ANALYSIS t (min) Tw (oC) ± Tr (oC) ± X=Tw-Tr ± Xcalc ± % Diff 0 84.0 0.5 23.0 0.5 61.0 1.0 61.0 17.5 0.00 62.0 60.0 2 82.0 0.5 23.0 0.5 59.0 1.0 59.4 17.2 0.73 60.0 58.0 4 80.0 0.5 23.0 0.5 57.0 1.0 57.9 16.8 1.57 58.0 56.0 6 79.0 0.5 23.0 0.5 56.0 1.0 56.4 16.7 0.75 57.0 55.0 8 78.0 0.5 23.0 0.5 55.0 1.0 55.0 16.5 0.05 56.0 54.0 10 76.0 0.5 23.0 0.5 53.0 1.0 53.6 16.2 1.05 54.0 52.0 12 75.0 0.5 23.0 0.5 52.0 1.0 52.2 16.0 0.36 53.0 51.0 14 73.0 0.5 23.0 0.5 50.0 1.0 50.8 15.6 1.67 51.0 49.0 16 72.0 0.5 23.0 0.5 49.0 1.0 49.5 15.4 1.10 50.0 48.0 18 71.0 0.5 23.0 0.5 48.0 1.0 48.3 15.2 0.57 49.0 47.0 20 70.0 0.5 23.0 0.5 47.0 1.0 47.0 15.0 0.07 48.0 46.0 0.5 23.0 0.5 45.0 1.0 45.8 14.6 1.80 46.0 44.0 0.5 23.0 0.5 44.0 1.0 44.7 14.3 1.46 45.0 43.0 0.5 23.0 0.5 43.0 1.0 43.5 14.1 1.16 44.0 42.0 22 24 26 68.0 67.0 66.0 steep shallow 28 65.0 0.5 23.0 0.5 42.0 1.0 42.4 13.9 0.92 43.0 41.0 30 64.0 0.5 23.0 0.5 41.0 1.0 41.3 13.7 0.73 42.0 40.0 Ave. = Chart Title 70.0 0.9% Chart Title 70.0 y = 60.626e-0.013x 60.0 50.0 40.0 30.0 20.0 10.0 0.0 0 5 10 15 20 25 30 35 High Low 57.47925 22.51445 57.68655 23.33973 57.8946 24.19527 57.9989 24.63473 58.10339 25.08217 58.31294 26.00158 58.418 26.47384 58.62869 27.44427 58.73431 27.94273 58.84013 28.45026 58.94614 28.967 59.15873 30.02881 59.26531 30.57422 59.37208 31.12954 59.47905 31.69495 59.58621 32.27062 80 #13 Newton's Law of Cooling Objective The major objective of this experiment is to investigate the law of de- cay by measuring the temperature of a container of water which is cooling after having been heated to a high temperature. In so doing, the student will make use of the graphical analysis techniques that were introduced in Experiment #3 Introduction and Theory Processes of Growth and Decay There exist in nature many processes for which the rate at which a certain quantity changes with time is directly proportional to the value of the quantity at that instant of time. If the process is one in which the change per unit time interval is positive, it is called a growth process, if negative, it is a decay process. In this experiment we shall be dealing spe- cifically with decay processes and becoming familiar with the general mathematical expression which describes them. As was stated above, the law of decay holds only in situations where there is a direct relation between the time rate of change and the size of the quantity being measured. Processes which obey this law are described by an equation of the form Ax - kx (13.1) At where x is the size of the quantity and Aris the change in x which occurs during a measured time interval At. Here k is a proportionality constant, the negative sign in front of it indicating a decay process. (A growth pro- cess would have a positive sign.) In the limit where the time interval shrinks to zero, we can express the quotient on the left side of Eq. 13.1 as a derivative, so that dx =- kx (13.2) dt In other words, the rate at which the process occurs is directly propor- tional to the quantity x; the larger the value of x, the more rapidly the de- cay occurs. Separating variables in Eq. 13.2 and integrating, we obtain (as the student should verify) an exponential function x = xoekt (13.3) 81 where xo is the size of x at t = 0. Eqs. 13.2 and 13.3 are equivalent expres- sions for the law of decay. In this experiment you will investigate another physical situation which can be described by Eq. 13.3, namely the exponential decrease in the temperature of an object that has been heated to several degrees above room temperature and is then allowed to cool. Using graphical analysis, you will then find the equation that relates x to t. This specific application of this law is known as Newton's Law of Cooling. Before discussing the specifics of this particular experiment, it will be helpful to consider one of the classic examples of exponential decrease, the decay of radioactive nuclei. Consider a sample of N uranium-238 nu- clei. Each such nucleus is unstable, and will eventually decay into a stable lead-206 nucleus. The nuclei decay randomly (refer to Experiment #4), and the number of uranium nuclei diminishes exponentially to zero. There- fore the decay rate must also steadily decrease, because as the uranium becomes depleted there will be fewer decays per unit time interval. Ex- pressed mathematically, AN =-yN, or At N = Noen where No is the initial number of nuclei and y is the proportionality con- stant, which, as was noted earlier, can be determined experimentally. Newton's Law of Cooling for a Warm Object It was Isaac Newton who, three centuries ago, suggested that the rate at which an object cools is proportional to the temperature difference be- tween it and its surroundings. If we let T be the temperature of the object (in this case water in a container) at some instant and T, be the tempera- ture of the environment (room temperature), then the temperature excess X of the water above room temperature is defined as X = T - T (13.4) Since the rate of decrease of temperature is proportional to X, we have -- (13.5) AX - kX At or, expressed in exponential form, x = xoekt (13.6) where Xo = (T-T), is the initial temperature difference between the wa- ter and the room. 82 One can readily calculate the amount of energy (in calories) given up by the water as it cools. Recall that the energy Q gained or lost by a sub- stance with mass m and specific heat c when undergoing a temperature change AT is Q = mcat. If the room temperature remains approximately constant, then the change in the temperature of the water AT equals the change in the temperature excess AX, and we can write the heat lost (in calories) as Q = MCAT = mc(T - T.) = mc(X - Xo) (13.7) In summary, Newton's law of cooling states that an object cools at a rate proportional to the temperature difference between it and its surround- ings. The proportionality constant k of the decay-rate formula is deter- mined from the slope of a graph of ln X versus t. Apparatus Check that you have the following equipment on your lab table: electric hot plate - glass flask (boiler) two thermometers one small metal container with aluminum foil cover - stand with holder for container and clamp for thermometer - stopwatch (or wall clock or wristwatch with a second hand). - heat protective gloves Experimental Procedure 1. Heat the water to boiling, pour it into the cup, insert the thermometer, and cover the cup opening with foil. 2. Start the stopwatch when the temperature is exactly 80°C and in your spreadsheet record both the water temperature and room temperature (along with their uncertainties) at two-minute intervals until you have 16 readings (30 minutes). The times should all be recorded to two deci- mal places, and the temperatures to one decimal place. 83 Analysis of Data 1. Plot a graph of X versus t, including vertical error bars. Insert the ap- propriate trendline and determine the equation of the resulting curve. This equation should have the form of Eq. 13.6. 2. Use the equation found in the preceding step to calculate the value of X for each value of t. Compare the results with each corresponding exper- imental value of X by % difference. Calculate and record the average % difference. 3. Using Eq. 13.7, calculate the amount of heat lost by one gram of water to the surrounding air in cooling from its initial temperature to the final temperature after 30 minutes. Recall that the specific heat of water is 1.00 cal/g-°C. Conclusion Comment on the following: 1. How well did your data fit Newton's Law of Cooling, Equation 13.6? Your evidence should include a discussion of your graphical results, in- cluding error bars. 2. Do any of your experimental and calculated values of X differ by more than 10%? If so, explain why this might have occurred. 3. How does the amount of heat lost computed in step 3 of Analysis of Da- ta compare with, for instance, the amount of energy given up per gram when water freezes at 0°C (i.e., the heat of fusion for water), approxi- mately 80 cal/g? (Note: There is no connection between the two values; you need not compute a percent difference.)
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Running head: Newton’s Law of Cooling

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Instructor……………………

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Laboratory course section……………….

Date……………...

Newton’s Law of Cooling for a warm object

Lab partners………………………

Newton’s Law of Cooling

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LAB REPORT
Newton’s Law of Cooling for a Warm Object
Introduction
The cooling rate of an object is relative to the temperature distinction amongst it and its
environment.
The heat lost by an object to the environment can be computed as;

Q = mc△T = mc(T-T0) = mc(X-X0)

Where, Q is heat lost, m is mass of water, c is the specific heat capacity of water and △T
is the temperature change.

Objective
To determine the decay law by measuring the temperature of a container of cooling water
after being heat to a high temperature.

Apparatus


Electric hot plate



Glass flask (boiler)



Two thermometers



One small metal container with aluminum foil cover



Stand with holder for container and clamp for thermometer



St...


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