Welcome to Thermodynamics!
Welcome to one of the most exciting parts of Physics! Thermodynamics is the study of heat, temperature, and energy. It explains everything from why your morning coffee cools down to how stars burn in the middle of space.
In this chapter, we will look at how energy moves through materials, how gases behave on a tiny molecular level, and how we can tell the temperature of a star just by looking at its light. Don't worry if it seems like a lot at first—we'll break it down into small, easy-to-manage chunks!
1. Heating and Changing State
When we give energy to an object, two things can happen: it gets hotter, or it changes its state (like ice melting into water).
Specific Heat Capacity
If you heat a block of lead and a block of aluminum of the same mass, the lead will get hot much faster. This is because every material has a different Specific Heat Capacity (\(c\)).
Definition: The energy required to raise the temperature of 1 kg of a substance by 1 Kelvin (or 1°C).
The formula is: \( \Delta E = mc\Delta\theta \)
Where:
• \( \Delta E \) = Energy transferred (Joules, \(\text{J}\))
• \( m \) = Mass (\(\text{kg}\))
• \( c \) = Specific heat capacity (\(\text{J}\cdot\text{kg}^{-1}\cdot\text{K}^{-1}\))
• \( \Delta\theta \) = Change in temperature (\(\text{K}\) or \(^\circ\text{C}\))
Specific Latent Heat
Have you ever noticed that when a pot of water is boiling, the temperature stays at 100°C even though the stove is still on? That extra energy is being used to break the bonds between molecules to turn the liquid into gas. This is called Latent Heat.
Definition: The energy required to change the state of 1 kg of a substance without changing its temperature.
The formula is: \( \Delta E = L\Delta m \)
Where \( L \) is the Specific Latent Heat (\(\text{J}\cdot\text{kg}^{-1}\)).
Quick Review: Use \( mc\Delta\theta \) when the temperature is changing. Use \( L\Delta m \) when the state is changing (temperature stays flat!).
Takeaway: Energy goes into either moving molecules faster (higher temperature) or pulling them apart (change of state).
2. Internal Energy and Absolute Zero
What is Internal Energy?
Everything around you is made of tiny particles that are constantly moving and interacting. Internal Energy is the sum of the random distribution of kinetic and potential energies associated with the molecules of a system:
1. Kinetic Energy: Due to the random motion of the molecules.
2. Potential Energy: Due to the electrostatic interactions (bonds) between the molecules.
Absolute Zero
If you keep cooling an object down, its molecules move slower and slower. Eventually, you reach a point where they have the minimum possible internal energy. This is Absolute Zero (0 Kelvin or -273.15°C).
Did you know? You can't actually go colder than Absolute Zero because you can't have less than minimum kinetic energy!
The Kelvin Scale: To convert from Celsius to Kelvin:
\( T(\text{K}) = \theta(^\circ\text{C}) + 273 \)
Takeaway: Absolute temperature is directly proportional to the average kinetic energy of the particles. At 0 K, thermal kinetic energy is at its minimum.
3. The Kinetic Theory of Gases
Gases can be complicated, so Physicists use an Ideal Gas model.
Assumptions of the Ideal Gas Model
In the kinetic model of an ideal gas, we make key assumptions:
• Molecules are point masses with negligible volume compared to the volume of the container.
• Collisions between molecules and with the container walls are perfectly elastic (no kinetic energy is lost).
• There are no intermolecular forces between molecules except during collisions.
• Molecules are in continuous, rapid, random motion.
• The duration of a collision is negligible compared to the time between collisions.
The Ideal Gas Equation
For an ideal gas, the pressure, volume, and absolute temperature are linked by the ideal gas equation:
\( pV = NkT \)
Where:
• \( p \) = Pressure (\(\text{Pa}\))
• \( V \) = Volume (\(\text{m}^3\))
• \( N \) = Number of molecules
• \( k \) = Boltzmann Constant (\( 1.38 \times 10^{-23}\text{ J}\cdot\text{K}^{-1} \))
• \( T \) = Absolute temperature (Must be in Kelvin!)
The Kinetic Theory Equation and Molecular Speed
Why does a gas exert pressure? When gas molecules collide with the container walls, they undergo a change in momentum, exerting a force on the walls (\( F = \frac{\Delta p}{\Delta t} \)).
The kinetic theory derivation yields:
\( pV = \frac{1}{3}Nm\langle c^2\rangle \)
Here, \( \langle c^2\rangle \) is the mean square speed of the molecules. The root-mean-square (r.m.s.) speed is given by:
\( c_{\text{rms}} = \sqrt{\langle c^2\rangle} \)
Energy and Temperature
By combining \( pV = NkT \) and \( pV = \frac{1}{3}Nm\langle c^2\rangle \), we find the link between molecular kinetic energy and temperature:
\( \frac{1}{2}m\langle c^2\rangle = \frac{3}{2}kT \)
This shows that the average translational kinetic energy of a gas molecule is directly proportional to the absolute temperature (\( T \)).
Common Mistake: Always use Kelvin for gas law calculations. If you use Celsius, your ratios and calculations will be wrong!
Takeaway: Pressure comes from molecular collisions with container walls. Absolute temperature is a direct measure of the average kinetic energy of the molecules.
4. Black Body Radiation
All objects with a temperature above absolute zero emit Electromagnetic Radiation.
What is a Black Body?
A Black Body Radiator is a theoretical object that is a perfect absorber and emitter of electromagnetic radiation. It absorbs all radiation incident upon it and emits radiation whose spectrum depends solely on its temperature. Stars like our Sun are very close approximations to black body radiators.
Wien’s Law
Hotter objects emit radiation with a shorter peak wavelength:
\( \lambda_{\text{max}}T = 2.898 \times 10^{-3}\text{ m}\cdot\text{K} \)
• \( \lambda_{\text{max}} \) = Peak emission wavelength (\(\text{m}\))
• \( T \) = Surface temperature in Kelvin (\(\text{K}\))
The Stefan-Boltzmann Law
The total power (luminosity) emitted by a black body depends on its surface area and temperature:
\( L = \sigma AT^4 \)
Where:
• \( L \) = Luminosity / total power (\(\text{W}\))
• \( \sigma \) = Stefan-Boltzmann constant (\( 5.67 \times 10^{-8}\text{ W}\cdot\text{m}^{-2}\cdot\text{K}^{-4} \))
• \( A \) = Surface area (for a spherical star, \( A = 4\pi r^2 \))
• \( T \) = Absolute temperature in Kelvin (\(\text{K}\))
Watch out! Luminosity is proportional to \( T^4 \). Doubling the absolute temperature increases the radiated power by a factor of \( 2^4 = 16 \)!
Takeaway: A star's temperature can be determined by its peak emission wavelength (Wien's Law), and its total power output depends strongly on temperature and size (Stefan-Boltzmann Law).
Core Practicals in this Chapter
CP 12: Calibrate a Thermistor
You'll use a potential divider circuit to determine how a thermistor's resistance changes with temperature, calibrating it for use as a thermostat.
CP 13: Specific Latent Heat
Determine the specific latent heat of a phase change (e.g., melting ice or boiling water) by measuring electrical energy input and change in mass.
CP 14: Boyle's Law (Pressure and Volume)
Investigate the inverse relationship between the pressure and volume of a fixed mass of gas at constant temperature (\( pV = \text{constant} \)).
Practice substituting values carefully into these equations and ensuring temperatures are always converted to Kelvin!