Class 11 Physics Notes Chapter 6 (Chapter 6) – Lab Manual (English) Book

Detailed Notes with MCQs of Experiment 6 from your Lab Manual, which deals with the cooling of a hot body. This is an important concept, often tested in various government exams, as it relates directly to Newton's Law of Cooling and heat transfer principles.

Experiment 6: To study the relationship between the temperature of a hot body and time by plotting a cooling curve.

1. Aim:
To observe how the temperature of a hot body (like water in a calorimeter) decreases over time when exposed to cooler surroundings and to represent this relationship graphically (plotting a cooling curve).

2. Underlying Principle: Newton's Law of Cooling

• Heat Transfer: When a body is hotter than its surroundings, it loses heat to the surroundings through conduction, convection, and radiation. In this experiment, convection and radiation are usually the dominant modes.
• Statement: Newton's Law of Cooling states that, for small temperature differences between a body and its surroundings, the rate at which the body loses heat is directly proportional to the temperature difference between the body and the surroundings.
• Mathematical Formulation:
  • Let 'T' be the temperature of the body at any time 't'.
  • Let 'T₀' be the constant temperature of the surroundings.
  • The rate of loss of heat (-dQ/dt) is given by:
    -dQ/dt ∝ (T - T₀)
-dQ/dt = k (T - T₀)
    where 'k' is a constant of proportionality that depends on the nature of the surface (emissivity) and the surface area of the body.
  • We also know that the heat lost by the body results in a decrease in its temperature. If 'm' is the mass and 's' is the specific heat capacity of the body, then dQ = ms dT. (Here, dT is the change in temperature, which is negative as the body cools, so dQ is negative, representing heat loss).
    Therefore, dQ/dt = ms (dT/dt).
  • Combining these, we get:
    -ms (dT/dt) = k (T - T₀)
dT/dt = - (k/ms) (T - T₀)
dT/dt = -K (T - T₀)
    where K = k/ms is another constant called the 'cooling constant'. Its unit is s⁻¹ or min⁻¹.
  • Interpretation: The negative sign indicates that the temperature 'T' decreases with time 't'. The equation shows that the rate of fall of temperature (-dT/dt) is directly proportional to the excess temperature (T - T₀) over the surroundings.
• Conditions for Validity:
  • The temperature difference (T - T₀) should be relatively small (ideally < 30-40°C).
  • Heat loss should occur primarily through convection and radiation (forced convection can alter results).
  • The temperature of the surroundings (T₀) must remain constant during the experiment.
  • The temperature should be uniform throughout the body (achieved by stirring in case of liquids).

3. Apparatus Required:

• Newton's Law of Cooling apparatus (or a double-walled vessel acting as a constant temperature enclosure)
• Calorimeter with a stirrer and a lid (with holes for thermometer and stirrer)
• Two thermometers (0-100°C range, with 0.5°C or 0.1°C least count)
• Stopwatch
• Heating arrangement (burner or heater)
• Water
• Beaker, Clamp stand

4. Experimental Procedure (Simplified Steps):

1. Fill the space between the double walls of the enclosure with water at room temperature (this acts as the constant temperature surroundings, T₀). Note this temperature using one thermometer.
2. Heat water in a beaker to about 80-90°C.
3. Fill the calorimeter about two-thirds full with this hot water.
4. Place the calorimeter inside the enclosure. Suspend the second thermometer in the hot water (ensure the bulb is fully immersed but doesn't touch the bottom or sides). Place the lid.
5. Start the stopwatch simultaneously when the temperature reaches a convenient value (e.g., 80°C).
6. Keep stirring the water gently and continuously.
7. Record the temperature (T) of the water in the calorimeter at regular intervals (e.g., every 1 minute).
8. Continue recording until the temperature falls by about 25-30°C (or for a sufficient duration, e.g., 30 minutes).
9. Record the final temperature of the water in the enclosure (T₀) to ensure it remained constant.

5. Observations:

• Least count of the thermometer = ... °C

• Least count of the stopwatch = ... s

• Temperature of surroundings (T₀) = ... °C

• Observation Table:

  Serial No.	Time (t) (min)	Temperature of hot water (T) (°C)	Temperature Difference (T - T₀) (°C)	logₑ(T - T₀)
  1	0	T₁	T₁ - T₀	...
  2	1	T₂	T₂ - T₀	...
  3	2	T₃	T₃ - T₀	...
  ...	...	...	...	...

6. Graph Plotting:

• Cooling Curve (T vs t): Plot time (t) on the x-axis and temperature (T) on the y-axis.
  • Shape: The curve will be exponential decay. It starts steep (rapid cooling) and becomes less steep (slower cooling) as time passes. It approaches the surrounding temperature (T₀) asymptotically.
  • Interpretation: The slope of the tangent to the curve at any point (dT/dt) represents the instantaneous rate of cooling. The slope is maximum initially and decreases with time.
• (Optional) Verification Graph (logₑ(T - T₀) vs t): Plot time (t) on the x-axis and logₑ(T - T₀) on the y-axis.
  • Shape: If Newton's Law is strictly obeyed, this graph should be a straight line with a negative slope.
  • Interpretation: The slope of this line is equal to -K (the negative of the cooling constant). This plot helps verify the law more directly.

7. Results:

1. The cooling curve (T vs t) shows that the temperature of the hot water decreases exponentially with time.
2. The rate of cooling is faster initially when the temperature difference between the water and surroundings is large, and it slows down as the temperature difference decreases.
3. (If plotted) The graph between logₑ(T - T₀) and time 't' is found to be a straight line (approximately), which verifies Newton's Law of Cooling within the experimental limits.

8. Precautions:

1. The temperature of the surroundings (T₀) should remain constant. Use a double-walled enclosure.
2. Stir the water in the calorimeter gently and continuously for uniform temperature distribution.
3. The thermometer bulb should be fully immersed in the hot water but should not touch the base or sides of the calorimeter.
4. Start the stopwatch and note the initial temperature accurately.
5. Record temperatures and time intervals precisely.
6. Avoid draughts of air which might affect the cooling rate unnaturally.
7. The temperature difference (T - T₀) should ideally not exceed 30-40°C for the law to hold accurately.

9. Sources of Error:

1. The surrounding temperature (T₀) may not remain perfectly constant.
2. Heat loss through conduction (e.g., via the lid, thermometer) is ignored but present.
3. Non-uniform temperature distribution within the water if stirring is inadequate.
4. Errors in reading the thermometer and stopwatch (parallax error, least count error, reaction time).
5. Heat loss is assumed only due to the temperature difference, but evaporation can also contribute, especially at higher temperatures if the lid is not proper.

Key Takeaways for Exams:

• Understand the statement and mathematical form of Newton's Law of Cooling.
• Know the relationship: Rate of cooling (-dT/dt) ∝ (T - T₀).
• Recognize the exponential shape of the T vs t cooling curve.
• Understand that the slope of the T vs t curve represents the rate of cooling.
• Know the shape of the logₑ(T - T₀) vs t graph (straight line with negative slope).
• Be aware of the conditions under which the law is valid (small temperature difference).
• Remember the key precautions for the experiment.

Multiple Choice Questions (MCQs):

1. According to Newton's Law of Cooling, the rate of loss of heat (-dQ/dt) by a body is directly proportional to:
   a) Its temperature (T)
   b) The temperature of the surroundings (T₀)
   c) The difference in temperature between the body and surroundings (T - T₀)
   d) The fourth power of its absolute temperature (T⁴)

2. The cooling curve plotted between temperature (T) and time (t) for a hot body cooling according to Newton's Law is:
   a) A straight line with a negative slope
   b) A parabola
   c) An exponential decay curve
   d) A hyperbolic curve

3. In the experiment to verify Newton's Law of Cooling, why is the water in the calorimeter stirred continuously?
   a) To increase the rate of cooling
   b) To ensure uniform temperature throughout the water
   c) To prevent heat loss by conduction
   d) To measure the surrounding temperature accurately

4. If a graph is plotted between logₑ(T - T₀) and time (t), where T is the temperature of the cooling body and T₀ is the surrounding temperature, the graph is approximately a:
   a) Straight line with positive slope
   b) Straight line with negative slope
   c) Curve increasing exponentially
   d) Curve decreasing exponentially

5. Two identical objects A and B are at the same initial temperature T, higher than the surroundings T₀. Object A has a dull black surface, while object B has a shiny polished surface. Which object will cool down faster initially?
   a) Object A
   b) Object B
   c) Both will cool at the same rate
   d) Depends on the specific heat capacity

6. Newton's Law of Cooling is generally valid when:
   a) The temperature difference (T - T₀) is very large
   b) The temperature difference (T - T₀) is small
   c) Heat loss is primarily by conduction
   d) The body is a perfect blackbody

7. The unit of the cooling constant 'K' in the equation dT/dt = -K(T - T₀) is:
   a) °C/s
   b) J/s
   c) s⁻¹
   d) °C⁻¹

8. What does the slope of the tangent to the cooling curve (T vs t) at any point represent?
   a) The total heat lost
   b) The instantaneous temperature difference (T - T₀)
   c) The instantaneous rate of cooling (-dT/dt)
   d) The cooling constant (K)

9. A key precaution in the Newton's Law of Cooling experiment is to:
   a) Use a calorimeter made of glass
   b) Heat the water to boiling point initially
   c) Ensure the surrounding temperature (T₀) remains constant
   d) Measure temperature every 10 minutes

10. If the temperature difference between a body and its surroundings is doubled, according to Newton's Law of Cooling, the rate of cooling becomes approximately:
    a) Half
    b) Double
    c) Four times
    d) Remains the same

Answer Key for MCQs:

1. c) The difference in temperature between the body and surroundings (T - T₀)
2. c) An exponential decay curve
3. b) To ensure uniform temperature throughout the water
4. b) Straight line with negative slope
5. a) Object A (black surfaces are better emitters/radiators)
6. b) The temperature difference (T - T₀) is small
7. c) s⁻¹ (or min⁻¹)
8. c) The instantaneous rate of cooling (-dT/dt) (Note: Slope is dT/dt, which is negative. Rate of cooling is -dT/dt, which is positive)
9. c) Ensure the surrounding temperature (T₀) remains constant
10. b) Double

Study these notes carefully, focusing on the concepts, the graphical representations, and the reasoning behind the experimental procedure and precautions. Good luck with your preparation!