Thermal Physics: Temperature, Expansion, Ideal Gases, and Kinetic Theory

Temperature Scales and Conversions

Temperature Measurement and Units

Temperature is a fundamental physical quantity that measures the average kinetic energy of particles in a substance. Common temperature scales include Fahrenheit (°F), Celsius (°C), and Kelvin (K).

• Fahrenheit (°F): Water freezes at 32°F and boils at 212°F.

• Celsius (°C): Water freezes at 0°C and boils at 100°C.

• Kelvin (K): The absolute temperature scale, where 0 K is absolute zero (the lowest possible temperature).

Temperature Conversions:

• $T_C = \frac{5}{9}(T_F - 32)$

• $T_F = \frac{9}{5}T_C + 32$

• $T_K = T_C + 273.15$

Example: Convert 25°C to Kelvin: $T_K = 25 + 273.15 = 298.15$ K

Thermal Expansion

Linear Thermal Expansion

When the temperature of a material increases, its size (length, area, or volume) increases due to the increased kinetic energy of its particles.

• Linear Expansion: Change in length is proportional to the change in temperature.

• $\Delta L = \alpha L_0 \Delta T$

• $\alpha$ = linear expansion coefficient (unit: $K^{-1}$)

• $L_0$ = initial length

Example: A metal rod of length 2.0 m expands by 0.002 m when heated by 50°C. If $\alpha = 2 \times 10^{-5} K^{-1}$, $\Delta L = 2 \times 10^{-5} \times 2.0 \times 50 = 0.002$ m.

Volume Thermal Expansion

For solids and liquids, the volume also expands with temperature.

• $\Delta V = \beta V_0 \Delta T$

• $\beta$ = volume expansion coefficient (for solids, $\beta \approx 3\alpha$)

• $V_0$ = initial volume

Example: A glass container of $V_0 = 500$ cm³ is heated from 20°C to 80°C. If $\beta = 9 \times 10^{-5} K^{-1}$, $\Delta V = 9 \times 10^{-5} \times 500 \times 60 = 2.7$ cm³.

Ideal Gases and the Ideal Gas Law

Properties of Ideal Gases

An ideal gas is a simplified model of a real gas that assumes:

• Gas molecules have negligible volume.

• Particles move randomly and do not interact except during elastic collisions.

• Particles are in constant, straight-line motion.

Ideal Gas Law

The ideal gas law relates pressure, volume, temperature, and the number of particles:

• $PV = nRT$

• $PV = Nk_BT$

• $n$ = number of moles, $N$ = number of particles, $R$ = universal gas constant, $k_B$ = Boltzmann constant

Example: Calculate the pressure of 1 mole of gas in a 22.4 L container at 273 K: $P = \frac{nRT}{V} = \frac{1 \times 0.0821 \times 273}{22.4} = 1$ atm

Kinetic Molecular Theory

Average Kinetic Energy of Ideal Gas Particles

The kinetic molecular theory connects macroscopic properties (P, V, T) with the microscopic motion of gas molecules.

• Average kinetic energy per particle: $\langle KE \rangle = \frac{3}{2}k_BT$

• Temperature is directly proportional to the average kinetic energy of particles.

Example: At $T = 300$ K, $\langle KE \rangle = \frac{3}{2} \times 1.38 \times 10^{-23} \times 300 = 6.21 \times 10^{-21}$ J

Root-Mean-Square (RMS) Speed of Ideal Gases

The RMS speed is a measure of the average speed of gas particles.

• $v_{rms} = \sqrt{\frac{3k_BT}{m}}$

• Depends on temperature and mass of gas particles.

Example: For $O_2$ molecules at $T = 300$ K, $m = 5.31 \times 10^{-26}$ kg, $v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 300}{5.31 \times 10^{-26}}} \approx 484$ m/s

Most Probable and Average Speeds

• Most probable speed: $v_{mp} = \sqrt{\frac{2k_BT}{m}}$

• Average speed: $v_{avg} = \sqrt{\frac{8k_BT}{\pi m}}$

• Relationship: $v_{avg} > v_{rms} > v_{mp}$

Phase Diagram

Phases of Matter and Phase Diagrams

A phase diagram shows the states of matter (solid, liquid, gas) as a function of pressure and temperature.

• Triple point: All three phases coexist.

• Critical point: Above this, liquid and gas are indistinguishable.

Example: Water's triple point is at 0.01°C and 611 Pa.

Mean Free Path

Definition and Calculation

The mean free path is the average distance a gas particle travels before colliding with another particle.

• $\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$

• $d$ = diameter of particle, $n$ = number density

Example: For $d = 3 \times 10^{-10}$ m, $n = 2.5 \times 10^{25}$ m$^{-3}$, $\lambda = \frac{1}{\sqrt{2} \pi (3 \times 10^{-10})^2 \times 2.5 \times 10^{25}} \approx 2.5 \times 10^{-7}$ m

Internal Energy of Ideal Monatomic Gases

Internal Energy Calculation

The internal energy of an ideal monatomic gas is the sum of the kinetic energies of all particles.

• Per particle: $U = \frac{3}{2}k_BT$

• For $N$ particles: $U_{total} = N \frac{3}{2}k_BT = n \frac{3}{2}RT$

Example: For 2 moles of gas at 300 K, $U_{total} = 2 \times \frac{3}{2} \times 8.314 \times 300 = 7,482$ J

Specific Heat and Temperature Change

Specific Heat Capacity

Specific heat ($c$) is the amount of heat required to raise the temperature of 1 kg of a substance by 1 K.

• $Q = mc\Delta T$

• $m$ = mass, $c$ = specific heat, $\Delta T$ = temperature change

Example: Heating 0.5 kg of water ($c = 4186$ J/kg·K) by 10 K: $Q = 0.5 \times 4186 \times 10 = 20,930$ J

Solving Calorimetry Problems

Calorimetry

Calorimetry involves mixing substances at different temperatures and calculating the final equilibrium temperature.

• Heat lost by hot object = Heat gained by cold object

• $m_1c_1(T_{final} - T_1) = m_2c_2(T_2 - T_{final})$

Example: Mixing 0.2 kg of water at 80°C with 0.1 kg at 20°C, find $T_{final}$.

Additional info: These notes cover foundational concepts in thermal physics, including temperature scales, thermal expansion, ideal gases, kinetic theory, phase diagrams, mean free path, internal energy, specific heat, and calorimetry. All equations are provided in LaTeX format for clarity.