🔋College Physics I – Introduction Unit 13 – Temperature and Gas Laws

Key Concepts and Definitions

• Temperature measures the average kinetic energy of particles in a substance
• Heat is the transfer of thermal energy from a hotter object to a cooler one
• Thermal equilibrium occurs when two objects in contact have the same temperature and no net heat transfer
• Absolute zero is the lowest possible temperature, where particles have minimal kinetic energy (-273.15°C or 0 K)
• Pressure is the force per unit area exerted by a gas on its container's walls
  • Measured in pascals (Pa), atmospheres (atm), or millimeters of mercury (mmHg)
• Volume is the amount of space occupied by a substance
• Molar mass is the mass of one mole of a substance
  • Measured in grams per mole (g/mol)

Temperature Scales and Conversions

• Celsius scale defines 0°C as the freezing point and 100°C as the boiling point of water at sea level
• Fahrenheit scale sets 32°F as the freezing point and 212°F as the boiling point of water at sea level
• Kelvin scale is an absolute temperature scale with 0 K as absolute zero and no negative values
  • To convert from Celsius to Kelvin, add 273.15 (K = °C + 273.15)
• Converting between Celsius and Fahrenheit:
  • °F = (9/5)°C + 32
  • °C = (5/9)(°F - 32)
• Converting between Kelvin and Fahrenheit:
  • K = (5/9)(°F - 32) + 273.15
  • °F = (9/5)(K - 273.15) + 32

Thermal Expansion

• Most substances expand when heated and contract when cooled due to increased particle motion
• Linear thermal expansion describes length changes in solids: $\Delta L = \alpha L_0 \Delta T$
  • $\Delta L$ is the change in length, $\alpha$ is the linear expansion coefficient, $L_0$ is the initial length, and $\Delta T$ is the temperature change
• Volumetric thermal expansion describes volume changes in solids, liquids, and gases: $\Delta V = \beta V_0 \Delta T$
  • $\Delta V$ is the change in volume, $\beta$ is the volumetric expansion coefficient, $V_0$ is the initial volume
• Thermal expansion can cause structural damage (cracked sidewalks) or be used in applications (bimetallic strips in thermostats)

Ideal Gas Law

• The ideal gas law relates pressure, volume, temperature, and amount of an ideal gas: $PV = nRT$
  • $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the universal gas constant (8.314 J/mol·K), and $T$ is the absolute temperature in Kelvin
• An ideal gas consists of point particles with no intermolecular forces and elastic collisions
• Real gases behave like ideal gases at high temperatures and low pressures
• The ideal gas law can be used to calculate changes in gas properties under different conditions (e.g., pressure change in a sealed container when heated)

Kinetic Theory of Gases

• Kinetic theory explains gas behavior based on the motion of its particles
• Gas particles are in constant, random motion and collide elastically with each other and the container walls
• The average kinetic energy of gas particles is directly proportional to the absolute temperature
• Pressure results from gas particles colliding with the container walls
  • Increasing temperature or decreasing volume increases collision frequency and pressure
• The root-mean-square (rms) speed of gas particles depends on temperature and molar mass: $v_\text{rms} = \sqrt{\frac{3RT}{M}}$
  • $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass

Gas Processes and Thermodynamic Laws

• Isothermal process occurs at constant temperature (e.g., slow compression or expansion)
  • For an ideal gas, $PV = \text{constant}$ (Boyle's law)
• Isobaric process takes place at constant pressure (e.g., heating a gas in an open container)
  • For an ideal gas, $V/T = \text{constant}$ (Charles's law)
• Isochoric (isovolumetric) process happens at constant volume (e.g., heating a gas in a sealed container)
  • For an ideal gas, $P/T = \text{constant}$ (Gay-Lussac's law)
• Adiabatic process occurs without heat transfer to or from the surroundings (e.g., rapid compression or expansion)
  • For an ideal gas, $PV^\gamma = \text{constant}$, where $\gamma$ is the ratio of specific heats
• The first law of thermodynamics states that energy is conserved in a closed system
  • Change in internal energy $\Delta U = Q + W$, where $Q$ is heat added and $W$ is work done
• The second law of thermodynamics indicates that entropy (disorder) tends to increase in closed systems

Real-World Applications

• Pressure cookers use high pressure to raise the boiling point of water, reducing cooking time
• Hot air balloons rise due to the lower density of heated air compared to the surrounding cooler air
• Automotive engines rely on the expansion of heated gases to drive pistons and generate power
• Refrigerators and air conditioners use the principles of gas compression and expansion to transfer heat
• Weather patterns, such as wind and precipitation, are influenced by temperature and pressure differences in the atmosphere
• Scuba divers must be aware of gas compression and expansion effects to avoid decompression sickness (the bends)

Problem-Solving Strategies

• Identify the given information and the desired quantity to solve for
• Determine the appropriate formula or principle to apply based on the problem scenario
• Convert any given values to the proper units for consistency (e.g., Celsius to Kelvin, atmospheres to pascals)
• Substitute the known values into the formula and solve for the unknown variable
• Check the reasonableness of the answer by considering the problem context and the expected magnitude of the result
• For multi-step problems, break the solution down into smaller, manageable parts
• When dealing with gas processes, identify the type of process (isothermal, isobaric, isochoric, or adiabatic) to select the suitable equation
• Remember to use absolute temperature (Kelvin) when applying the ideal gas law or kinetic theory equations