Lecture 15: Temperature and Heat

Funda‌mental Concepts of Temperature and Motion

• Temperature: A physical quantity that defines the average kinetic energy of the molecules within an object. It provides an objective quantification of the "hotness" or "coldness" of a substance.

• Symbol: The uppercase letter $T$ is used to represent temperature in physics.

• Human Context: From a practical perspective, temperature dictates behavioral decisions such as selecting appropriate clothing for seasons or determining feasible indoor and outdoor activities.

• Scientific Importance: Objective temperature scales allow scientists to bypass subjective sensations and conduct precise experiments in fields like thermodynamics and climate change.

• Relationship to Kinetic Energy: In accordance with principles established in earlier physics lessons, kinetic energy is the energy of motion. Thus, substances at higher temperatures exhibit greater internal molecular motion compared to those at lower temperatures.

• Diffusion Demo: A comparison of two beakers—one with cold water (appearing translucent due to water vapor condensation) and one with hot water—demonstrates this. Blue food dye added to the hot water distributes and swirls much faster, proving the greater kinetic energy of the hot water molecules.

Origins and Physics of Temperature Scales

• Temperature scales are classified into two categories: empirical scales and absolute scales.

• Empirical Scales: These scales use experimentally derived upper and lower limits based on physical phenomena.

  • Fahrenheit Scale: Proposed in the 1700s. The lower limit ($0$) was set using a frozen salt water solution. The upper limit ($90$) was initially established using human body temperature. The scale contains 89 gradations between these points.

  • Fahrenheit Limitations: It is considered non-scientific because it is not easily repeatable; variations in salt concentration or the health of the human test subject (e.g., a fever) can result in different definitions of $0$ and $90$.

  • Celsius Scale: Developed in the 1700s. The cold end is defined by the freezing point of pure water ($0$) and the hot end by the boiling point of pure water ($100$) at one atmosphere of pressure. It contains 99 gradations.

  • Celsius Advantages: It is a scientific, repeatable scale. Any researcher using pure water at standard pressure will arrive at an identical scale.

• Conversion Formulas:

  • Celsius to Fahrenheit: Multiply the Celsius temperature by $1.8$ and add $32$. $F = (C \times 1.8) + 32$. Example: $10\,^{\circ}\text{C} = 50\,^{\circ}\text{F}$.

  • Fahrenheit to Celsius: Subtract $32$ from the Fahrenheit temperature, then multiply the result by five-ninths ($\frac{5}{9}$). $C = (F - 32) \times \frac{5}{9}$. Example: $86\,^{\circ}\text{F} = 30\,^{\circ}\text{C}$.

• Absolute Scales: These are defined by a lower limit based on a physical limitation rather than an arbitrary freezing point.

  • Kelvin Scale: Based on absolute zero, the lowest possible temperature where molecular motion ceases. As far as science knows, nothing can be colder than absolute zero.

  • Kelvin Relationship to Celsius: A change in one Kelvin ($1\,\text{K}$) is equal to a change in one degree Celsius ($1\,^{\circ}\text{C}$). Absolute zero is equivalent to $-273\,^{\circ}\text{C}$.

  • Kelvin Conversions: To find Kelvin, add $273$ to the Celsius temperature ($K = C + 273$). To find Celsius, subtract $273$ from the Kelvin value ($C = K - 273$).

  • SI Unit: Kelvin is the official SI unit for temperature, though Celsius is frequently used in general physics contexts.

Thermometry and Measurement Devices

• Thermometers: Devices designed to measure temperature by utilizing the predictable expansion or electrical changes of substances in response to heat.

• Analog Thermometers: Use a liquid that remains in liquid form through a wide range of expected temperatures.

  • Modern Consumer Versions: Use dyed alcohol because it remains liquid throughout standard ambient temperature ranges human experience.

  • Historical Versions: Previously used mercury, which has a vast liquid range but is highly toxic and has been phased out of consumer products.

• Digital Thermometers: Contain electronic components whose electrical properties change as a function of temperature. A microcontroller relates these properties to specific temperatures and displays the numerical value on an LCD screen.

• Calibration/Building Demo: A thermometer made with dyed water in a bulb.

  • Shortcomings: Water freezes at $0\,^{\circ}\text{C}$ and boils at $100\,^{\circ}\text{C}$, making it useless for measuring extremes.

  • Process: Marking a line for room temperature, placing the bulb in freezing water to establish the $0$ mark at equilibrium, and placing it in boiling water. (In the demo, the water expanded so vigorously it exited the top of the tube, indicating a need for a longer tube or less expansive liquid).

The Nature and Flow of Heat

• Heat: Energy transferred from one object to another due to a difference in temperature.

• Symbol and Units: The uppercase letter $Q$ denotes heat. Units include calories ($\text{cal}$) and Joules ($\text{J}$). In many thermodynamic contexts, calories are preferred.

• Direction of Flow: Heat spontaneously flows only from hot objects to cold objects. It never flows spontaneously in the opposite direction.

• Gravity Analogy: Similar to a ball rolling down a ramp due to gravity. The ball will stay at the bottom unless external work (by a person or machine) is done to move it back up.

• Work and Refrigeration: To move heat against its natural flow (from cold to hot), mechanical work must be performed. This is how refrigerators and air conditioners operate, and it requires electrical energy, making it costly to run in summer.

Specific Heat Capacity and Thermal Inertia

• Definition: Specific heat capacity is a measure of how difficult it is to change the temperature of a substance, analogous to "thermal inertia."

• Symbol: Represented by the lowercase letter $c$.

• Units: Standard units are calories per gram degree Celsius ($\text{cal/g}\,^{\circ}\text{C}$). This tells us the heat required to change the temperature of one gram of a substance by $1\,^{\circ}\text{C}$.

• Primary Equation: $Q = mcΔT$.

  • $Q$: Heat (calories).

  • $m$: Mass (grams).

  • $c$: Specific heat capacity.

  • $ΔT$: Change in temperature (Final temperature minus Initial temperature).

  • Constraint: This equation applies only for temperature changes and does not account for phase changes (like boiling or melting).

• Water Characteristics: Water has a high specific heat capacity of $1\,\text{calorie/g}\,^{\circ}\text{C}$.

  • Applications: Water is an excellent coolant because it can absorb significant heat with minimal temperature change.

  • Climate Impact: Large bodies of water (oceans, Lake Michigan) moderate regional climates, leading to milder seasonal shifts. Conversely, deserts have low moisture and low specific heat capacity, leading to extreme daily temperature fluctuations.

Mathematical Examples and Calorimetry

• Example 1: Heating Water on a Hotplate

  • Data: $m = 347.6\,\text{g}$, $T_{initial} = 21.2\,^{\circ}\text{C}$, $T_{final} = 44.9\,^{\circ}\text{C}$.

  • $ΔT = 44.9 - 21.2 = 23.7\,^{\circ}\text{C}$.

  • $Q = 347.6\,\text{g} \times 1\,\text{cal/g}\,^{\circ}\text{C} \times 23.7\,^{\circ}\text{C} = 8238.12\,\text{calories}$.

  • Cooling: If the same beaker cools back down, $ΔT$ is $-23.7\,^{\circ}\text{C}$. The resulting negative $Q$ value indicates heat leaving the substance.

• Example 2: Mixture in Thermal Equilibrium

  • Data: $173.9\,\text{g}$ of copper ($c = 0.094\,\text{cal/g}\,^{\circ}\text{C}$) and $364.3\,\text{g}$ of water ($c = 1\,\text{cal/g}\,^{\circ}\text{C}$) heated from $22.3\,^{\circ}\text{C}$ to $45.2\,^{\circ}\text{C}$.

  • $ΔT = 22.9\,^{\circ}\text{C}$.

  • $Q_{water} = 364.3 \times 1 \times 22.9 = 8342.47\,\text{cal}$.

  • $Q_{copper} = 173.9 \times 0.094 \times 22.9 = 374.34\,\text{cal}$.

  • $Q_{total} = 8342.47 + 374.34 = 8716.81\,\text{cal}$.

• Example 3: Conservation of Energy (Mixing Substances)

  • Principle: Assuming no heat loss to the environment, $Q_{hot} + Q_{cold} = 0$.

  • Data: $60\,\text{g}$ of aluminum ($c = 0.215$) at $80\,^{\circ}\text{C}$ mixed with $200\,\text{g}$ of water at $20\,^{\circ}\text{C}$.

  • $(60 \times 0.215 \times (T_f - 80)) + (200 \times 1 \times (T_f - 20)) = 0$.

  • $12.9(T_f - 80) + 200(T_f - 20) = 0$.

  • $12.9T_f - 1032 + 200T_f - 4000 = 0$.

  • $212.9T_f = 5032$.

  • $T_f = 23.64\,^{\circ}\text{C}$.

  • Logic: The result makes sense as it is closer to the starting temperature of the water, which both had a higher mass and a higher specific heat capacity.

Thermal Expansion and Practical Repercussions

• Molecular Basis: As internal molecules gain kinetic energy and motion, the object expands. Conversely, cooling leads to contraction.

• Ball and Hoop Demo: A room-temperature ball passes through a hoop. Once heated with a blowtorch, the ball expands and no longer fits. Heating the hoop increases the opening size, allowing the ball to pass again.

• Engineering Implications:

  • Bridges: Must use expansion joints to allow for stretching in heat without buckling.

  • Power Lines: Expand in summer heat. Heavy electrical loads (AC use) add more heat, causing lines to sag further. If they touch vegetation, they short circuit. This was a contributing cause of the Northeast Blackout of 2003.

  • Dentistry: Fillings must be engineered to have a thermal expansion rate identical to tooth enamel to prevent discomfort or damage when eating hot or cold foods. Composite fillings are engineered for this matching rate.

• Bimetallic Strips: Composed of two different metals bonded together. When heated, the metals expand at different rates, causing the strip to curl.

  • Physics: The faster-expanding metal undergoes tension; the slower metal undergoes compression.

  • Applications: Used in analog thermostats. A coil unfurls as a room cools, eventually hitting a sensor to trigger the furnace. It recoils when the room warms, turning the heat off. Also used in fridges and AC units.

• Anomalous Expansion of Water:

  • Water is densest at $4\,^{\circ}\text{C}$. Below this temperature, it expands as it approaches freezing.

  • Ecological Significance: Ice floats due to this expansion. This keeps the cold water at the bottom of deep lakes at $4\,^{\circ}\text{C}$ during winter, allowing aquatic life to survive under an insulating layer of ice.

• Thermal Stress: Objects need time for molecules to adjust to size changes. Rapid temperature shifts create huge internal stress.

  • Glass Demo: A plain glass survives heating with a blowtorch but Shatters instantly when plunged into ice water.

  • Safety: This is why tempered glass like Pyrex is required for materials subjected to wide temperature swings in ovens or stoves.