To Study the Relationship Between the Temperature of a Hot Body and Time by Plotting a Cooling Curve
1. Aim
To study the relationship between the temperature of a hot body and time by plotting a cooling curve and to verify Newton's Law of Cooling.
2. Apparatus Used
- Calorimeter with lid and stirrer
- Thermometer (0-100°C)
- Laboratory stand with clamp
- Stopwatch or timer
- Hot water (around 80-90°C)
- Graph paper
- Beaker (for initial heating)
- Heat source (bunsen burner or electric heater)
- Heat-resistant gloves
3. Diagram
4. Theory
Newton's Law of Cooling states that the rate of heat loss of a body is directly proportional to the difference in the temperatures between the body and its surroundings. Mathematically, this can be expressed as:
Where:
- $\frac{d\theta}{dt}$ is the rate of cooling
- $\theta$ is the temperature of the body at time $t$
- $\theta_0$ is the temperature of the surroundings (ambient temperature)
- $k$ is the cooling constant which depends on the nature of the body
By solving this differential equation, we get:
Where $C$ is a constant of integration.
If $\theta_1$ is the temperature at time $t_1$ and $\theta_2$ is the temperature at time $t_2$, then:
This shows that if we plot $\ln(\theta - \theta_0)$ vs $t$, we should get a straight line with slope $-k$, thus verifying Newton's Law of Cooling.
5. Formula
Newton's Law of Cooling:
$$\frac{d\theta}{dt} = -k(\theta - \theta_0)$$Integrated form:
$$\theta - \theta_0 = (\theta_i - \theta_0)e^{-kt}$$Taking natural logarithm:
$$\ln(\theta - \theta_0) = \ln(\theta_i - \theta_0) - kt$$For verification:
$$\text{Plot of } \ln(\theta - \theta_0) \text{ vs } t \text{ should give a straight line with slope } -k$$6. Procedure
- Fill the calorimeter about two-thirds with water and heat it to about 80-90°C.
- Place the calorimeter on the laboratory stand and insert the thermometer through the hole in the lid.
- Ensure that the thermometer bulb is completely immersed in water but not touching the sides or bottom of the calorimeter.
- Note the ambient temperature ($\theta_0$) of the laboratory.
- Start the stopwatch when the temperature of water is around 80°C and record this as the initial temperature at time $t = 0$.
- Record the temperature every minute for about 30-40 minutes or until the temperature approaches room temperature.
- Gently stir the water before each reading to ensure uniform temperature throughout.
- Record all readings in the observation table.
- Plot a graph of temperature ($\theta$) vs time ($t$) to obtain the cooling curve.
- Calculate $\ln(\theta - \theta_0)$ for each temperature reading and plot a graph of $\ln(\theta - \theta_0)$ vs time ($t$).
- Determine the slope of the straight line to find the cooling constant $k$.
7. Observation Table
| S.No. | Time (min) | Temperature, $\theta$ (°C) | $\theta - \theta_0$ (°C) | $\ln(\theta - \theta_0)$ |
|---|---|---|---|---|
| 1 | 0 | |||
| 2 | 1 | |||
| 3 | 2 | |||
| 4 | 3 | |||
| 5 | 4 | |||
| 6 | 5 | |||
| . . . Continue for at least 30 readings . . . | ||||
Room temperature ($\theta_0$): _______ °C
8. Calculations
1. Calculate $\theta - \theta_0$ for each reading by subtracting the room temperature from the observed temperature.
2. Calculate $\ln(\theta - \theta_0)$ for each reading.
3. Plot the following graphs:
- Graph 1: Temperature ($\theta$) vs Time ($t$) - This gives the cooling curve
- Graph 2: $\ln(\theta - \theta_0)$ vs Time ($t$) - This should give a straight line
4. For Graph 2, find the slope of the straight line using the method of best fit or:
5. The negative of this slope gives the cooling constant $k$.
6. Using this $k$ value, verify Newton's Law of Cooling by comparing the theoretical values with the experimental values:
Where $\theta_i$ is the initial temperature at $t = 0$.
9. Result
1. The cooling curve showing the relationship between temperature and time has been plotted.
2. The plot of $\ln(\theta - \theta_0)$ vs time ($t$) gives a straight line with slope = _______.
3. The cooling constant $k$ = _______ min-1.
4. Newton's Law of Cooling has been verified as the experimental values closely match the theoretical values predicted by the law.
5. The rate of cooling is directly proportional to the temperature difference between the body and its surroundings, as stated by Newton's Law of Cooling.
10. Precautions
- Ensure the thermometer bulb is completely immersed in water but does not touch the sides or bottom of the calorimeter.
- Keep the calorimeter away from direct air currents or drafts.
- Stir the water gently before taking each reading to ensure uniform temperature distribution.
- Read the thermometer at eye level to avoid parallax error.
- Ensure that the stopwatch is started exactly when the first temperature reading is noted.
- The calorimeter should be well-insulated to minimize heat loss through conduction.
- Maintain a constant room temperature throughout the experiment.
- Handle hot water with care to prevent burns.
- Wait for the mercury/alcohol column in the thermometer to stabilize before recording the reading.
- Take sufficient number of readings to get a smooth curve.
11. Sources of Error
- Heat loss through radiation might not be the only mode of heat transfer; conduction and convection might also contribute.
- The temperature of the surroundings might not remain constant throughout the experiment.
- The thermometer might have inherent calibration errors.
- The calorimeter might not be perfectly insulated, leading to unaccounted heat losses.
- The stirring process might add a small amount of thermal energy to the system.
- The time lag between reading the temperature and noting it down can introduce errors.
- The thermal capacity of the thermometer and calorimeter is neglected in the calculations.
- Air currents in the laboratory might cause irregular cooling.
- Newton's Law of Cooling is an approximation and might not be perfectly applicable for all temperature ranges.